Abstract
We derive integral tests for the existence and absence of arbitrage in a financial market with one risky asset which is either modeled as stochastic exponential of an Itô process or a positive diffusion with Markov switching. In particular, we derive conditions for the existence of the minimal martingale measure. We also show that for Markov switching models the minimal martingale measure preserves the independence of the noise and we study how the minimal martingale measure can be modified to change the structure of the switching mechanism. Our main mathematical tools are new criteria for the martingale and strict local martingale property of certain stochastic exponentials.
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1 Introduction
The absence of arbitrage is of fundamental interest in many areas of financial mathematics. Our goal is to provide a systematic discussion for a financial market with one risky asset modeled via its discounted price process \(P = (P_t)_{t \in [0, T]}\), which we assume to be either the stochastic exponential of an Itô process, i.e. to have dynamics
or to be a positive diffusion with Markov switching, i.e. to have dynamics
where \(\xi = (\xi _t)_{t \in [0, T]}\) is a continuous-time Markov chain and \(W = (W_t)_{t \in [0, T]}\) is a Brownian motion.
For semimartingale markets the classical concepts of no arbitrage are the notions of no free lunch with vanishing risk (NFLVR) as defined by Delbaen and Schachermayer [15, 16] and no feasible free lunch with vanishing risk (NFFLVR) as defined by Sin [52]. The difference between (NFLVR) and (NFFLVR) is captured by the concept of a financial bubble in the sense of Cox and Hobson [9]. For our market it is well-known that (NFLVR) is equivalent to the existence of an equivalent local martingale measure (ELMM), see [16], and that (NFFLVR) is equivalent to the existence of an equivalent martingale measure (EMM), see [6, 52, 55]. The no arbitrage condition used in the stochastic portfolio theory of Fernholz [22] is no relative arbitrage (NRA). In complete markets Fernholz and Karatzas [21] showed that (NRA) is equivalent to the existence of a strict martingale density (SMD). A weaker concept is no unbounded profit with bounded risk (NUPBR), which is known to be equivalent to the existence of a strict local martingale density (SLMD), see [7]. (NUPBR) is considered to be the minimal notion needed for portfolio optimization, see [35].
The first findings of this article are integral tests for the existence and non-existence of SMDs, ELMMs and EMMs. For (1.1) the tests are formulated in terms of Markovian upper and lower bounds on the volatility coefficient \(\sigma \) and for (1.2) the tests depend on \(x \mapsto \sigma (x, j)\) with \(j\) in the state space of the Markov chain \(\xi \). The main novelty of our results is that they apply in the presence of multiple sources of risk. Beside the Markov switching framework, this is for instance the case in diffusion models with a change point, which represents a change of the economical situation caused for instance by a sudden adjustment in the interest rates or a default of a major financial institution. In general, the question whether (NFLVR) and/or (NFFLVR) hold for a model with a change point is difficult, see [24] for some results in this direction. Our integral tests provide explicit criteria, which are easy to verify. For many applications of the Markov switching model (1.2) it is important to know how the change to an ELMM affects the dynamics of the Markov chain \(\xi \). As a second contribution, we study this question form a general perspective for independent sources of risk modeled via martingale problems. In particular, we show that the minimal local martingale measure (MLMM), see [25], preserves the independence and the laws of the sources of risk. To our knowledge, this property has not been reported in the literature. A third contribution of this article are integral tests for the martingale property of certain stochastic exponentials driven by Itô processes or switching diffusions. These characterizations are our key tools to study the absence of arbitrage.
We comment on related literature. For continuous semimartingale models the absence of arbitrage has been studied by Criens [11], Delbaen and Shirakawa [17], Lyasoff [40] and Mijatović and Urusov [43]. Criens, Delbaen and Shirakawa and Mijatović and Urusov proved integral tests for the existence of SMDs, ELMMs and EMMs in diffusion frameworks. Our results can be viewed as generalizations to an Itô process or Markov switching framework. For a model comparable to (1.1), Lyasoff proved that the existence of an ELMM is determined by the equivalence of a probability measure to the Wiener measure. The structure of this characterization is very different from our results. In Sect. 3.3 below we comment in more detail on the results in [11, 17, 40, 43]. The martingale property of stochastic exponentials is under frequent investigation. At this point we mention the articles of Blanchet and Ruf [3], Cheridito et al. [5], Criens [11] and Kallsen and Muhle-Karbe [34]. Criens used arguments based on Lyapunov functions and contradictions to verify the martingale property of certain stochastic exponential in a multi-dimensional diffusion setting. We transfer these techniques to a general Itô process setting. Cheridito et al. and Kallsen and Muhle-Karbe related the martingale property of a stochastic exponential to an explosion probability via a method based on the concept of local uniqueness as defined in [30]. This technique traces back to work of Jacod and Mémin [29] and Kabanov et al. [31, 32]. We use a similar argument for the Markov switching setting. The main difficulties are the proofs of explosion criteria and local uniqueness. Both approaches have a close relation to the work of Blanchet and Ruf, where a tightness criterion for the martingale property of non-negative local martingales has been proven. The connection between Lyapunov functions, explosion and tightness is for instance explained in [12].
Let us also comment on consecutive problems and extensions of our results: In case the discounted price process \(P\) is a positive Itô process of the type
our results on the martingale property of stochastic exponentials can be used to obtain characterizations for no arbitrage with a similar structure as for the model (1.2). Moreover, in case \(P\) is the stochastic exponential of a diffusion with Markovian switching, i.e.
our martingale criteria yield conditions for no arbitrage with a similar structure as for (1.1). It is also interesting to ask about multi-dimensional models. In this case, results in the spirit of [11] can be proven by similar arguments as used in this article. However, the conditions are rather complicated to formulate and space consuming. Therefore, we restrict ourselves to the one-dimensional case.
The article is structured as follows. In Sect. 2 we give conditions for the martingale and strict local martingale property of certain stochastic exponentials. In Sect. 3.1 we study the model (1.1) and in Sect. 3.2 we study the model (1.2). In Sect. 4 we show that the MLMM preserves independence and laws for sources of risk and we explain how the MLMM can be modified to affect the law of an additional source of risk. The proofs are collected in the remaining sections.
2 Martingale property of stochastic exponentials
Fix a finite time horizon \(0< T < \infty \) and let \((\Omega , \mathcal {F}, \mathbf {F}, \mathbb {P})\) be a complete filtered probability space with right-continuous and complete filtration \(\mathbf {F}= (\mathcal {F}_t)_{t \in [0, T]}\). Moreover, fix a state space \(I \triangleq (l, r)\) with \(- \infty \le l < r \le + \infty \).
In the following two sections we provide conditions for the martingale and strict local martingale property of certain stochastic exponentials.
2.1 The general case
Assume that \(S = (S_t)_{t \in [0, T]}\) is an \(I\)-valued Itô process with deterministic initial value \(S_0 \in I\) and dynamics
where \(W = (W_t)_{t \in [0, T]}\) is a one-dimensional Brownian motion and \(b = (b_t)_{t \in [0, T]}\) and \(\sigma = (\sigma _t)_{t \in [0, T]}\) are real-valued progressively measurable processes. It is implicit that \(b\) and \(\sigma \) are such that the integrals are well-defined, i.e. a.s.
We assume that -a.e. \( \sigma \not = 0,\) which latter will correspond to the assumption that we consider an asset price process with a non-vanishing volatility.
Let \(c = (c_t)_{t \in [0, T]}\) be a real-valued progressively measurable process such that a.s.
and let \(N = (N_t)_{t \in [0, T]}\) be a local martingale such that a.s. \(\Delta N \ge - 1\) and \([N, W] = 0\). We ask for conditions under which the non-negative local martingale
is a true or a strict local martingale. Here, \(\mathcal {E}\) denotes the stochastic exponential. The structure of \(Z\) is very important in mathematical finance, because \(Z\) is the prototype of a strict local martingale density, see Lemma 3.2 below.
Let \(\underline{a}, \overline{a} :I \rightarrow (0, \infty ), \underline{u}, \overline{u} :I \rightarrow \mathbb {R}\) and \(\zeta :[0, T] \rightarrow \mathbb {R}_+\) be Borel functions such that
In case \((f, g)\) is one of the pairs \((\underline{u}, \underline{a}), (\underline{u}, \overline{a}), \dots \) we set
where \(x_0 \in I\) is fixed. Let \(l_n \searrow l, r_n \nearrow r\) be sequences such that \(l< l_{n+1}< l_n< r_n< r_{n +1} < r\).
The first main result of this section is the following:
Theorem 2.1
Assume the following:
- (M1)
The sequence
$$\begin{aligned} \tau _n \triangleq \inf (t \in [0, T] :S_t \not \in (l_n, r_n)), \quad n \in \mathbb {N}, \end{aligned}$$is a localizing sequence for \(Z\), i.e. \(Z_{\cdot \wedge \tau _n}\) is a martingale for every \(n \in \mathbb {N}\). We use the convention that \(\inf (\emptyset ) \triangleq \infty \).
- (M2)
For
-a.a. \((t, \omega ) \in [0, T] \times \Omega \)
$$\begin{aligned} \sigma ^2_t(\omega )&\le \zeta (t) \overline{a}(S_t(\omega )),\\ \underline{u}(S_t(\omega )) \sigma _t^2 (\omega )&\le b_t(\omega ) + c_t(\omega ) \sigma _t(\omega ),\\ \overline{u}(S_t(\omega )) \sigma _t^2 (\omega )&\ge b_t(\omega ) + c_t(\omega ) \sigma _t(\omega ). \end{aligned}$$ - (M3)
\(\lim _{x \nearrow r} v(\overline{u}, \overline{a})(x) = \lim _{x \searrow l} v(\underline{u}, \overline{a}) (x) = \infty .\)
Then, \(Z\) is a martingale.
The proof of this theorem is given in Sect. 5.
Remark 2.2
(M3) is independent of the choice of \(x_0\), see [36, Problem 5.5.28].
Next, we provide a counterpart to Theorem 2.1. Let \(\mathscr {H}\) be the set of all Borel functions \(h :\mathbb {R}_+ \rightarrow \mathbb {R}_+\) which are starting at zero, are strictly increasing and satisfy
and let \(\mathscr {K}\) be the set of all Borel functions \(\kappa :\mathbb {R}_+ \rightarrow \mathbb {R}_+\), which are starting at zero, are strictly increasing and concave and satisfy
In case \((f, g)\) is one of the pairs \((\underline{u}, \underline{a}), (\underline{u}, \overline{a}), \dots \) we say that \((f, g)\) satisfies the Yamada–Watanabe (YW) conditions, if for every \(n \in \mathbb {N}\) there exist \(h_n \in \mathscr {H}\) and \(\kappa _n \in \mathscr {K}\) such that and for all \(x, y \in [l_n, r_n]\)
The second main result of this section is the following:
Theorem 2.3
Assume one of the following conditions:
- (SL1)
The pair \((\underline{u}, \underline{a})\) satisfies the YW conditions, for
-a.a. \((t, \omega ) \in [0, T] \times \Omega \)
$$\begin{aligned} \begin{aligned} \underline{a}(S_t(\omega ))&\le \sigma ^2_t(\omega ),\\ \underline{u}(S_t(\omega )) \sigma _t^2 (\omega )&\le b_t(\omega ) + c_t(\omega ) \sigma _t(\omega ), \end{aligned} \end{aligned}$$(2.3)and \( \lim _{x \nearrow r} v(\underline{u}, \underline{a})(x) < \infty . \)
- (SL2)
The pair \((\overline{u}, \underline{a})\) satisfies the YW conditions, for
-a.a. \((t, \omega ) \in [0, T] \times \Omega \)
$$\begin{aligned} \underline{a}(S_t(\omega ))&\le \sigma ^2_t(\omega ),\\ \overline{u}(S_t(\omega )) \sigma _t^2 (\omega )&\ge b_t(\omega ) + c_t(\omega ) \sigma _t(\omega ), \end{aligned}$$and \(\lim _{x \searrow l} v(\overline{u}, \underline{a})(x) < \infty .\)
Then, \(Z\) is a strict local martingale.
The proof of this theorem is given in Sect. 5. In Sect. 2.3 below we comment on the assumptions of Theorems 2.1 and 2.3 and related literature.
2.2 Markov switching case
In this section we consider a special case of the setting from Sect. 2.1 and assume that \(S\) is a switching diffusion. Before we introduce the setting in detail, we clarify terminology: A process is called a Feller–Markov chain if it is a Markov chain which is a Feller process in the sense that the corresponding transition semigroup is a self-map on the space of continuous functions vanishing at infinity. For conditions implying that a Markov chain is Feller–Markov we refer to [2]. It is also important to stress that whenever we have fixed a filtration and a Markov chain, we presume that the Markov chain is Markovian for the given filtration. All non-explained terminology for Markov chains, such as irreducible, recurrent, etc., can be found in [44].
We assume that \(S = (S_t)_{t \in [0, T]}\) is an \(I\)-valued Itô process with deterministic initial value \(S_0 \in I\) and dynamics
where \(W = (W_t)_{t \in [0, T]}\) is a one-dimensional Brownian motion, \(\xi = (\xi _t)_{t \in [0, T]}\) is a continuous-time irreducible Feller–Markov chain with state space \(J \triangleq \{1, \dots , N\}, 1 \le N \le \infty ,\) and deterministic initial value \(j_0 \in J\), and \(b :I \times J \rightarrow \mathbb {R}\) and \(\sigma :I \times J \rightarrow \mathbb {R}\backslash \{0\}\) are Borel functions such that
It is implicit that the integrals in (2.4) are well-defined. We allow \(N = \infty \) in which case \(J= \mathbb {N}\). A process of the type (2.4) is called a switching diffusion and the elements of \(J\) are called regimes.
Let \(c :I \times J \rightarrow \mathbb {R}\) be a Borel function such that
Lemma 2.4
Almost surely \(\int _0^T c^2(S_s, \xi _s) ds < \infty \).
Proof
Set \(F \triangleq \{\xi _s :s \in [0, T]\}\), \(m \triangleq \min _{s \in [0, T]}\,S_s\) and \(M \triangleq \max _{s \in [0, T]}\,S_s\). Using that \(\xi \) only makes finitely many jumps in the finite time interval \([0, T]\), the occupation times formula for continuous semimartingales and (2.6), we obtain a.s.
where \(L^S\) denotes the local time of \(S\). The lemma is proven. \(\square \)
We are interested in the martingale property of the non-negative local martingale
This definition coincides with (3.3) for the choices \(c = c(S, \xi )\) and \(N = 0\).
Before we state the main result of this section, we fix some notation. Because \(L^2_\text{ loc } (I) \subset L^1_\text{ loc } (I)\), (2.5) and (2.6) imply that
Thus, we can set
for \((x, j) \in I \times J\) and a fixed \(x_0 \in I\).
We say that \(\sigma \)satisfies the Engelbert–Schmidt (ES) conditions for\(j \in J\) if one of the following two conditions holds:
- (ES1)
For every compact set \(K \subset I\) there are Borel functions \(f :K \rightarrow [0, \infty ]\) and \(h :\mathbb {R} \rightarrow [0, \infty ]\) and a constant \(c > 0\) such that the following properties are satisfied:
- (i)
\(\frac{f}{\sigma ^2 (\cdot , j)} \in L^1 (K)\).
- (ii)
For every neighborhood \(U\) of the origin
$$\begin{aligned} \int _U \frac{dy}{h(y)} = \infty . \end{aligned}$$ - (iii)
For all \(x, x + y \in K, y \in (- c, c)\)
$$\begin{aligned} |\sigma (x + y, j) - \sigma (x, j)|^2 \le f(x) h(y). \end{aligned}$$
- (i)
- (ES2)
For every compact set \(K \subset I\) there are Borel functions \(g :K \rightarrow \mathbb {R}\) and \(h :\mathbb {R} \rightarrow [0, \infty ]\) and a constant \(c > 0\) such that the following properties are satisfied:
- (i)
\(g\) is increasing.
- (ii)
For every neighborhood \(U\) of the origin
$$\begin{aligned}\int _U \frac{dy}{h(y)} = \infty .\end{aligned}$$ - (iii)
For all \(x, x + y \in K, y \in (- c, c)\backslash \{0\}\)
$$\begin{aligned} |\sigma (x + y, j) - \sigma (x, j)|^2 \le h(y) \frac{|g(x + y) - g(x)|}{|y|}. \end{aligned}$$ - (iv)
\(\inf _{x \in K} \sigma (x, j) > 0\).
- (i)
We say that the Markov chain \(\xi \) is recurrent if it is a recurrent Markov chain when extended to the infinite time interval \(\mathbb {R}_+\).
The following theorem gives an almost complete answer to the question when \(Z\) is a true or strict local martingale. A proof is given in Sect. 6.
Theorem 2.5
-
(i)
Suppose that \(c\) is bounded on compact subsets of \(I \times J\), that \(\sigma \) satisfies the ES conditions for all \(j \in J\) and that
$$\begin{aligned} \lim _{x \nearrow r} v(x, j) = \lim _{x \searrow l} v(x, j) = \infty \text { for all } j \in J. \end{aligned}$$(2.7)Then, \(Z\) is a martingale.
-
(ii)
Assume that \(\xi \) is recurrent and that there exists a \(j \in J\) such that \(\sigma \) satisfies the ES conditions for \(j\) and
$$\begin{aligned} \lim _{x \nearrow r} v(x, j)< \infty \text { or } \lim _{x \searrow l} v(x, j) < \infty . \end{aligned}$$(2.8)Then, \(Z\) is a strict local martingale.
Remark 2.6
The proof of Theorem 2.5 (ii) is based on a contradiction argument. In case (2.8) holds and \(Z\) is a martingale there exists an \(I\)-valued switching diffusion with an explosive regime \(j\). The recurrence of \(\xi \) simplifies the proof that this switching diffusion reaches the regime \(j\), which leads to a contradiction. In case the initial regime \(j_0\) is already explosive, more precisely if \(\sigma \) satisfies the ES conditions for \(j_0\) and \(\lim _{x \nearrow r} v(x, j_0) < \infty \) or \(\lim _{x \searrow l} v(x, j_0) < \infty \), the proof needs no change even without the assumption that \(\xi \) is recurrent.
Noting that \(\xi \) is recurrent in case \(N< \infty \), we obtain the following:
Corollary 2.7
Suppose that \(c\) is bounded on compact subsets of \(I \times J\), that \(\sigma \) satisfies the ES conditions for all \(j \in J\) and that \(N < \infty \). Then, \(Z\) is a martingale if and only if (2.7) holds.
Proof
If \(N < \infty \), the recurrence of \(\xi \) follows from [44, Theorems 1.5.6, 3.4.1]. Now, the claim is due to Theorem 2.5. \(\square \)
In financial applications, \(N\) can be interpreted as the number of states of the business cycle and therefore \(N < \infty \) is a reasonable assumption.
2.3 Comments on related literature
The martingale property of non-negative local martingales is under frequent investigation. We mention a few related works: A general semimartingale setting has been considered in [13, 27, 30] and a diffusion and/or jump-diffusion setting has been studied in [5, 34, 39, 43, 50, 53].
To the best of our knowledge, for a general Itô process or Markov switching setting Theorems 2.1, 2.3 and 2.5 are the first results which provide integral tests for the martingale property of certain stochastic exponentials.
For the diffusion case
a complete characterization of the martingale property of the non-negative local martingale
has been proven in [43]. We stress that in [43] the diffusion \(S\) is allowed to explode, which is a feature not included in our framework. Provided \(S\) is non-explosive, the main theorem of [43] shows that \(Z\) is a martingale if and only if
where \( u \triangleq \frac{b + c \sigma }{\sigma ^2} \) and \(v\) is defined as in (2.2). The same condition is implied by either Theorems 2.1 and 2.3, or Corollary 2.7. For the strict local martingale property we require that \(\sigma \) satisfies the ES conditions, which are not imposed in [43].
The key idea underlying Theorems 2.1, 2.3 and 2.5 is a local change of measure combined with either a Lyapunov-type argument (in case of Theorem 2.1), a comparison with one-dimensional diffusions (in case of Theorem 2.3) or a local uniqueness property (in case of Theorem 2.5).
The idea of using a local change of measure is not new. It has for instance been used in [5, 11, 13, 50, 53]. The Lyapunov and comparison arguments were inspired by [11], where a multi-dimensional diffusions has been studied. To use the ideas in our general setting, we prove a new Lyapunov condition for Itô processes and we transport the comparison arguments from a multi-dimensional diffusion setting to a one-dimensional Itô process framework, see Sect. 5 below. The idea of relating local uniqueness to the martingale property of a stochastic exponential traces back to [29, 31, 32]. More recently, the method was used in [5, 11, 34, 53]. Although the terminology suggests the converse, local uniqueness is a strong version of uniqueness in law. In the proof of Theorem 2.5 we deduce local uniqueness from pathwise uniqueness by a Yamada–Watanabe-type argument.
3 On the absence and existence of arbitrage
Let \(0< T < \infty \) be a finite time horizon and let \((\Omega , \mathcal {F}, \mathbf {F}, \mathbb {P})\) be a complete filtered probability space with right-continuous and complete filtration \(\mathbf {F}= (\mathcal {F}_t)_{t \in [0, T]}\). We consider a financial market consisting of one risky asset with discounted price process \(P = (P_t)_{t \in [0, T]}\), which is assumed to be a positive continuous semimartingale with deterministic initial value.
Recall the following classical terminology: A probability measure \(\mathbb {Q}\) is called an equivalent (local) martingale measure (E(L)MM) if \(\mathbb {Q}\sim \mathbb {P}\) and \(P\) is a (local) \(\mathbb {Q}\)-martingale. A strictly positive local \(\mathbb {P}\)-martingale \(Z = (Z_t)_{t \in [0, T]}\) with \(Z_0 = 1\) is called a strict (local) martingale density (S(L)MD) if \(ZP\) is a (local) \(\mathbb {P}\)-martingale.
In the following we study existence and non-existence of SMDs, ELMMs and EMMs in case \(P\) is either the stochastic exponential of an Itô process or a positive switching diffusion. In case \(P\) is a positive Itô process or the stochastic exponential of a real-valued switching diffusion similar results can be deduced from the martingale criteria in Sect. 2.
3.1 Stochastic exponential model
Suppose that \(P\) is the stochastic exponential of the real-valued Itô process \(S = (S_t)_{t \in [0, T]}\) with deterministic initial value \(S_0 \in \mathbb {R}\) and dynamics
where \(W = (W_t)_{t \in [0, T]}\) is a one-dimensional Brownian motion and \(b = (b_t)_{t \in [0, T]}\) and \(\sigma = (\sigma _t)_{t \in [0, T]}\) are real-valued progressively measurable processes such that the stochastic integrals in (3.1) are well-defined. We assume that -a.e. \( \sigma \not = 0, \) which corresponds to the assumption that \(P\) has a non-vanishing volatility.
3.1.1 Absence of arbitrage
In the following we study when a SMD, ELMM or EMM exists. As a minimal condition we assume that (NUPBR) holds. This is equivalent to the existence of a market price of risk\(\theta = (\theta _t)_{t \in [0, T]}\), i.e. a real-valued progressively measurable process such that a.s.
and
![](http://media.springernature.com/lw150/springer-static/image/art%3A10.1007%2Fs11579-020-00262-1/MediaObjects/11579_2020_262_Equ12_HTML.png)
We define the continuous local martingale
Integration by parts and (3.2) yield that
which shows that \(ZP\) is a local martingale or, equivalently, that \(Z\) is a SLMD. We observe the following:
- (O1)
If \(ZP\) is a martingale, then \(Z\) is a SMD by definition.
- (O2)
If \(Z\) is a martingale, we can define a probability measure \(\mathbb {Q}\) by the Radon–Nikodym derivative \(\frac{d\mathbb {Q}}{d\mathbb {P}} \triangleq Z_T\) and \(\mathbb {Q}\) is an ELMM by (3.4) and [30, Proposition III.3.8].
- (O3)
If \(ZP\) and \(Z\) are martingales, then \(\mathbb {Q}\) as defined in (O2) is an EMM by [30, Proposition III.3.8].
In summary, to prove the existence of a SMD, ELMM and EMM we have to identify conditions for the martingale property of \(ZP\) and \(Z\). The following is the main result of this section:
Theorem 3.1
Suppose the following:
- (L1)
The sequence
$$\begin{aligned} \tau _n \triangleq \inf (t \in [0, T] :|S_t| \ge n ), \quad n \in \mathbb {N}, \end{aligned}$$(3.5)is a localizing sequence for \(Z\).
- (L2)
There are Borel functions \(\overline{a} :\mathbb {R} \rightarrow (0, \infty )\) and \(\zeta :[0, T] \rightarrow \mathbb {R}_+\) such that
$$\begin{aligned} \frac{1}{\overline{a}} \in L^1_\text{ loc }(\mathbb {R}), \quad \zeta \in L^1([0, T]), \end{aligned}$$and \(\sigma ^2_t(\omega ) \le \zeta (t) \overline{a} (S_t (\omega ))\) for
-a.a. \((t, \omega ) \in [0, T] \times \Omega \).
Then, \(Z\) is a martingale, \(\mathbb {Q}\) defined by \(\frac{d\mathbb {Q}}{d\mathbb {P}} \triangleq Z_T\) is an ELMM and
is a \(\mathbb {Q}\)-Brownian motion such that
If in addition
then \(\mathbb {Q}\) is an EMM and \(Z\) is a SMD.
Proof
We apply Theorem 2.1 with \(I \triangleq \mathbb {R}, l_n \triangleq - n, r_n \triangleq n\) and \(c \triangleq - \theta \). Note that (L1) equals (M1). Furthermore, set \(\underline{u} (x) \equiv \overline{u} (x) \triangleq 0\). Then, (L2) implies (M2), because (3.2) implies -a.e. \(b + c \sigma = 0\). Finally, note that
which shows that (M3) holds due to [36, Problem 5.5.27]. We conclude that \(Z\) is a martingale and that \(\mathbb {Q}\) is an ELMM by (O2).
Next, we assume that (3.7) holds. We apply Theorem 2.1 with \(I \triangleq \mathbb {R}, l_n \triangleq -n, r_n \triangleq n\) and \(c \triangleq \sigma - \theta \) to show that the local martingale
is a martingale. In this case, \(\mathbb {Q}\) is an EMM and \(Z\) is a SMD by (O1) and (O3). By (L1), the set \(\{Z_{T \wedge \gamma \wedge \tau _n} :\gamma \ \text{ stopping } \text{ time }\)} is uniformly integrable (see [30, Proposition I.1.47]). Thus,
where the \(\sup _{\gamma }\) is meant to be the supremum over all stopping times \(\gamma \le T\). Due to [30, Proposition I.1.47], we conclude that (M1) holds for \(Z'\). Note that (3.2) implies that -a.e. \(b + c \sigma = \sigma ^2\). Thus, we set \(\underline{u} (x)\equiv \overline{u}(x) \triangleq 1\) and note that (L2) implies (M2) for \(Z'\). Using Fubini’s theorem and (3.7), we obtain that
Because
[36, Problem 5.5.27] yields that \(\lim _{x \searrow - \infty } v(1, \overline{a}) (x) = \infty \). Hence, (M3) holds for \(Z'\). We conclude that \(Z'\) is a martingale and the proof is complete. \(\square \)
In our setting there might exist several ELMMs and it is an important question which ELMM should be chosen for applications. The ELMM from Theorem 3.1 is the minimal local martingale measure (MLMM) as defined in [25].Footnote 1 For financial interpretations of the MLMM we refer to [25] and for a general overview on possible applications we refer to [23]. In Theorem 4.3 below we discover a new property of the MLMM: The MLMM preserves independence and laws of sources of risk.
In the following paragraph we relate the assumptions (L1) and (L2) to so-called weakly equivalent local martingale measures (WELMM) as introduced in [37]. We explain the connection from a general point of view under the assumptions that \(\mathcal {F} = \mathcal {F}_T\) and that (NUPBR) holds. With slight abuse of notation, let \(Z = (Z_t)_{t \in [0, T]}\) be a SLMD with localizing sequence \((\tau _n)_{n \in \mathbb {N}}\). For every \(n \in \mathbb {N}\) we can define a probability measure \(\mathbb {Q}^n\) by the Radon–Nikodym derivative \(\frac{d\mathbb {Q}^n}{d\mathbb {P}} \triangleq Z_{T \wedge \tau _n}\). It is easy to see that \(\mathbb {Q}^n\) is an ELMM for the stopped process \(P_{\cdot \wedge \tau _n}\). In other words, for every \(n \in \mathbb {N}\) the notion (NFLVR) holds for all admissible strategies which invest riskfree after \(\tau _n\). Roughly speaking, this observation suggests that (NFLVR) holds in case we can take the limit \(n \rightarrow \infty \). As explained in Section 2.4.2 of [37], Alaoglu’s theorem yields that \((\mathbb {Q}^n)_{n \in \mathbb {N}}\) has an accumulation point \(\mathsf {Q}\) for the weak\(^*\) topology on the dual of \(L^\infty (\Omega , \mathcal {F}, \mathbb {P})\), which is a finitely additive probability such that \(\mathsf {Q}(A) = 0\) for all \(A \in \mathcal {F}\) with \(\mathbb {P}(A) = 0\), see the “Appendix” of [14]. We use the sans-serif typeface to highlight that \(\mathsf {Q}\) is not necessarily a probability measure, because it may fail to be countably additive. Note that \(\mathsf {Q} = \mathbb {Q}^n\) on \(\mathcal {F}_{\tau _n}\) for every \(n \in \mathbb {N}\). Using this fact, it follows that for all \(A \in \mathcal {F}\) with \(\mathsf {Q}(A) = 0\) we also have \(\mathbb {P}(A) = 0\), which shows that \(\mathsf {Q}\) and \(\mathbb {P}\) have the same null-sets. Indeed, if \(A \in \mathcal {F} = \mathcal {F}_T\) is such that \(\mathsf {Q}(A) = 0\), we have \(A \cap \{\tau _n> T\} \in \mathcal {F}_{\tau _n}\) and consequently
for all \(n \in \mathbb {N}\). This implies \(\mathbb {P}(A \cap \{\tau _n > T\}) = 0\) and, because \(\mathbb {P}\)-a.s. \(\tau _n \nearrow \infty \) as \(n \rightarrow \infty \), we conclude that \(\mathbb {P}(A) = 0\). Following [37], we call \(\mathsf {Q}\) a WELMM. The main difference between WELMMs and ELMMs, and therefore between (NUPBR) and (NFLVR), is that a WELMM is not necessarily a measure.
The idea of condition (L1) is to identify a WELMM, which, as explained above, is a natural candidate for an ELMM. Assuming that \((\tau _n)_{n \in \mathbb {N}}\) is given by (3.5) means controlling the MPR via the size of the asset. This assumption is reasonable from a modeling perspective, because, as explained by Lyasoff [40, p. 488], “excessively large expected instantaneous net returns from risky securities entail excessively large demands for money (to invest in such securities), which, in turn, means higher and higher interest rates, which, in turn, means lower and lower market price of risk”. In the diffusion settings of Mijatović and Urusov [42], (L1) is implied by the local integrability condition [42, Eq. 3.2] on the MPR, see [43, Lemma 6.3].
Condition (L2) takes care on the countable additivity of the candidate WELMM, which corresponds to problems arising when \(n \rightarrow \infty \). Indeed, \(\mathsf {Q}\) is countably additive if and only if
which is also the condition we check in the proof of Theorem 2.1. If \(\mathsf {Q}\) is countably additive, then (3.8) follows from the monotone convergence theorem and the fact that \(\mathbb {P}\)-a.s. \(\tau _n \nearrow \infty \) as \(n \rightarrow \infty \). Conversely, assume that (3.8) holds. Let \((E_k)_{k \in \mathbb {N}} \subset \mathcal {F}\) be a decreasing sequence with \(\bigcap _{k \in \mathbb {N}} E_k = \emptyset \). Then, because \(E_k \in \mathcal {F} = \mathcal {F}_T\), we have \(E_k \cap \{\tau _n > T\} \in \mathcal {F}_{\tau _n}\), which yields that
Thus, \(\mathsf {Q}\) is continuous at zero, which implies that it is countably additive.
3.1.2 Existence of a financial bubble
In Theorem 3.1 we gave conditions for the existence of an ELMM. In this section, we derive a counterpart to (3.7), which implies the existence of a financial bubble in the sense of [9].
As we explain next, the question when no SMD exists is strongly connected to the question when a non-negative local martingale is a strict local martingale. We recall the following:
Lemma 3.2
If \(Z\) is a SLMD, then there exists a market price of risk \(\theta = (\theta _t)_{t \in [0, T]}\) and a local martingale \(N = (N_t)_{t \in [0, T]}\) such that a.s. \(\Delta N > -1, [N, W] = 0\) and
Proof
See [51, Theorem 1]. \(\square \)
In case \(Z\) is a SMD, (3.9) holds and
is a martingale by definition. If this is not the case, we have a contradiction and no SMD exists.
The following is the main result of this section:
Theorem 3.3
Suppose there exists a Borel function \(\underline{a} :\mathbb {R}\rightarrow (0, \infty )\) such that \((1, \underline{a})\) satisfies the YW conditions (see Sect. 2.1 for this terminology), \(\underline{a} (S_t (\omega )) \le \sigma ^2_t (\omega )\) for -a.a. \((t, \omega ) \in [0, T] \times \Omega \) and
Then, no SMD exists.
Proof
We use Theorem 2.3 with \(I \triangleq \mathbb {R}\) and \(\underline{u} \triangleq 1\) to show that \(ZP\) as defined in (3.10) is a strict local martingale. Because \(\theta \) is a MPR, -a.e. \(b + (\sigma - \theta ) \sigma = \sigma ^2 = \underline{u}(S) \sigma ^2\). Furthermore, Fubini’s theorem and (3.11) yield that
Thus, the conditions from part (ii) of Theorem 2.3 hold and we conclude that \(ZP\) is a strict local martingale. Consequently, as explained above, no SMD exists. \(\square \)
The conditions (3.7) and (3.11) provide a test for the MLMM to be an EMM or not. In a diffusion setting the conditions boil down to a single sufficient and necessary condition, which is also given in [11, Proposition 5.2].
3.1.3 Example: diffusion models with a change point
Fontana et al. [24] study (NUPBR) and (NFLVR) for a model with a change point. The authors are interested in the influence of filtrations, which represent different levels of information. Under a weak form of the \(\mathcal {H}'\)-hypothesis the model can be included into our framework. More precisely, in this case \(S\) is of the form
where \(\tau \) is a stopping time. The coefficient \(\sigma ^{(i)}\) is assumed to be positive, continuous and Lipschitz continuous in the second variable uniformly in the first, see [24, Condition I]. Theorem 3.1 provides local conditions for (NFLVR). In particular, for the special cases described in [24, Section 3.3], Theorem 3.1 yields that (NFLVR) always holds, because
where \(\mu ^{(i)}\) is locally bounded. This extends the observation from [24] that (NUPBR) holds in these cases. Furthermore, if in addition to (3.12) for \(i = 1, 2\)
then even (NFFLVR) holds. The notion (NFFLVR) has not been studied in [24].
3.2 Diffusion model with Markov switching
In this section, we assume that \(P\) is a positive continuous semimartingale with deterministic initial value \(P_0 \in (0, \infty )\) and dynamics
where \(W = (W_t)_{t \in [0, T]}\) is a one-dimensional Brownian motion, \(\xi = (\xi _t)_{t \in [0, T]}\) is a continuous-time irreducible Feller–Markov chain with state space \(J \triangleq \{1, \dots , N\}\), \(1 \le N \le \infty \), and deterministic initial value \(j_0 \in J\), and \(b :(0, \infty ) \times J \rightarrow \mathbb {R}\) and \(\sigma :(0, \infty ) \times J \rightarrow \mathbb {R}\backslash \{0\}\) are Borel functions such that
We can interpret \(N\) as the number of all possible states of the business cycle. The assumption of irreducibilily means that we exclude all states of the business cycle which are not attainable from the initial state. We assume \(\xi \) to be a Feller process for technical reasons. In case \(N < \infty \) any Markov chain with values in \(J\) is a Feller process, because all real-valued functions on \(J\) are continuous and vanishing at infinity. Due to Lemma A.1 in the “Appendix”, the sources of risk \(\xi \) and \(W\) are independent. The lemma even shows that it is not possible to model \(\xi \) and \(W\) as Markov processes for a superordinate filtration without their independence. This observation gives a novel interpretation for the independence assumption, which is typically interpreted as the price process being influenced by the business cycle and an additional independent source of risk represented by the driving Brownian motion.
3.2.1 Absence and existence of arbitrage
We impose the following two assumptions: The coefficient \(b\) is bounded on compact subsets of \((0, \infty ) \times J\), \(\sigma ^2\) is bounded away from zero on compact subsets of \((0, \infty ) \times J\) and \(\sigma \) satisfies the ES conditions for all \(j \in J\), see Sect. 2.2 for this terminology.
We define
which is a Borel map bounded on compact subsets of \((0, \infty ) \times J\). The process \(\theta _t \triangleq \theta (P_t, \xi _t)\) is a MPR. We define the continuous local martingale \(Z\) as in (3.3). Note that the observations (O1) – (O3) in Sect. 3.1 also hold in this setting. We call the E(L)MM \(\mathbb {Q}\) with Radon–Nikodym derivative \(\frac{d\mathbb {Q}}{d\mathbb {P}} = Z_T\) the minimal (local) martingale measure (M(L)MM). The following theorem provides conditions for the existence of the M(L)MM and for \(Z\) to be a SMD.
Theorem 3.4
-
(i)
Assume that
$$\begin{aligned} \int _0^1 \frac{z}{\sigma ^2(z, j)} dz= \infty \quad \text {for all } \; j \in J. \end{aligned}$$(3.13)Then, \(Z\) is a martingale and the probability measure \(\mathbb {Q}\) defined by the Radon–Nikodym derivative \(\frac{d\mathbb {Q}}{d\mathbb {P}} \triangleq Z_T\) is an ELMM. Moreover, \(B\) as defined in (3.6) is a \(\mathbb {Q}\)-Brownian motion such that
$$\begin{aligned}P = P_0 + \int _0^\cdot \sigma (P_t, \xi _t)dB_t. \end{aligned}$$If in addition
$$\begin{aligned} \int _1^\infty \frac{z}{\sigma ^2(z, j)} dz = \infty \quad \text {for all }\; j \in J, \end{aligned}$$(3.14)then \(\mathbb {Q}\) is an EMM.
-
(ii)
If (3.14) holds, then \(Z\) is a SMD.
Proof
The claim follows similar to the proof of Theorem 3.1 when Theorem 2.5 is used instead of Theorem 2.1. \(\square \)
Theorem 2.5 suggests that in case \(\xi \) is recurrent, the conditions in Theorem 3.4 are sufficient and necessary. The following theorem makes this precise.
Theorem 3.5
Suppose that \(\xi \) is recurrent.
- (i)
If there exists a \(j \in J\) such that
$$\begin{aligned} \int _0^1 \frac{z}{\sigma ^2(z, j)} dz < \infty , \end{aligned}$$(3.15)then \(Z\) is a strict local martingale and the MLMM does not exist.
- (ii)
If there exists a \(j \in J\) such that
$$\begin{aligned} \int _1^\infty \frac{z}{\sigma ^2 (z, j)} dz < \infty , \end{aligned}$$(3.16)then \(Z\) is no SMD. In particular, the MMM does not exist.
Proof
The claim follows similar to the proof of Theorem 3.3 when Theorem 2.5 is used instead of Theorem 2.3. \(\square \)
Recalling that in case \(N < \infty \) the Markov chain \(\xi \) is recurrent, we obtain the following:
Corollary 3.6
Suppose that \(N < \infty \).
With \(N = 1\) we recover [42, Corollary 3.4, Theorems 3.6 and 3.11]. Corollary 3.6 means that the M(L)MM exists if and only if the M(L)MM exists for all markets with fixed regimes. We will see in the next section that in case one of the frozen markets allows arbitrage, it is not possible to find a risk-neutral market in which the business cycle has Markovian dynamics.
3.2.2 Non-existence of structure preserving ELMMs and EMMs
Let \(\mathcal {L}_{\text{ sp }}\) the set of all ELMMs \(\mathbb {Q}\) such that \(\xi \) is an irreducible recurrent Feller–Markov chain on \((\Omega , \mathcal {F}, \mathbf {F}, \mathbb {Q})\) and let \(\mathcal {M}_\text{ sp }\) be the set of all EMMs in \(\mathcal {L}_\text{ sp }\). The main result of this section is the following:
Theorem 3.7
-
(i)
Suppose there exists a \(j \in J\) such that (3.15) holds and \(\sigma \) satisfies the ES conditions for \(j\). Then, \(\mathcal {L}_\text{ sp } = \emptyset \).
-
(ii)
Suppose there exists a \(j \in J\) such that (3.16) holds and \(\sigma \) satisfies the ES conditions for \(j\). Then, \(\mathcal {M}_\text{ sp } = \emptyset \).
Proof
The result follows from the contradiction argument used in the proof of Theorem 2.3, where Theorem 6.1 has to be used instead of Theorem 5.3. \(\square \)
In Sect. 4 we show that an equivalent change to the MLMM does not affect the Markov chain \(\xi \). Thus, Theorem 3.7 generalizes Theorem 3.5.
3.2.3 Example: Markov switching CEV model
We consider a version of the CEV model (see [10]) with Markov switching. Take \(\beta :J \rightarrow (0, \infty )\) and assume that
Furthermore, assume that \(b :(0, \infty ) \times J \rightarrow \mathbb {R}\) is locally bounded such that
for all \(j \in J\). Then, the discounted asset price process \(P\) exists due to Theorem 6.1 below. Let \(Z\) be defined as in (3.3) with \(\theta _t = \frac{b(S_t, \xi _t)}{\sigma (S_t, \xi _t)}\). In case \(N< \infty \), Corollary 3.6 shows the following:
- (a)
The MLMM exists if and only if \(\beta (j) \ge 1\) for all \(j \in J\).
- (b)
The MMM exists if and only if \(\beta (j) = 1\) for all \(j \in J\).
- (c)
\(Z\) is a SMD if and only if \(\beta (j) \le 1\) for all \(j \in J\).
3.3 Comments on related literature
For continuous semimartingale markets the existence and non-existence of SMDs, ELMMs and EMMs has been studied in [11, 17, 40, 43]. We comment on these works in more detail.
In [17, 43] a one-dimensional diffusion framework has been considered. We discuss the results from [43] and refer to [43, Remark 3.2] for comments on the relation between [17] and [43]. In [43] it is assumed that the price process \(P = (P_t)_{t \in [0, T]}\) is a \([0, \infty )\)-valued diffusion such that
where \(b :(0, \infty ) \rightarrow \mathbb {R}\) and \(\sigma :(0, \infty ) \rightarrow \mathbb {R} \backslash \{0\}\) are Borel functions satisfying
see also [36, Definition 5.5.20]. In the following we assume that \(P\) cannot explode to zero. In [43] the notions (NFLVR) and (NFFLVR) are also studied in case \(P\) can explode to zero and (NFLVR), (NFFLVR) and (NRA) are further studied for the infinite time horizon. For the non-explosive case the results from [43] are as follows:
- (a)
(NFLVR) \(\Leftrightarrow \)\(\frac{b}{\sigma } \in L^2_\text{ loc } ((0, \infty ))\) and \(\int _0^1 \frac{x}{\sigma (x)} dx = \infty \), see [43, Corollary 3.4].
- (b)
(NFFLVR) \(\Leftrightarrow \)\(\frac{b}{\sigma } \in L^2_\text{ loc } ((0, \infty ))\) and \(\int _0^1 \frac{x}{\sigma (x)} dx = \int _1^\infty \frac{x}{\sigma (x)} dx = \infty \), see [43, Theorem 3.6].
- (c)
If \(\frac{b}{\sigma } \in L^2_\text{ loc }((0, \infty ))\), then (NRA) \(\Leftrightarrow \)\(\int _1^\infty \frac{x}{\sigma (x)} dx = \infty \), see [43, Theorem 3.11].
Applying Corollary 3.14 with \(N = 1\) shows versions of (a) – (c) under slightly more restrictive regularity assumptions on \(b\) and \(\sigma \). The novelty of Corollary 3.14 or more generally Theorems 3.4 and 3.5 is their scope of application.
A multi-dimensional diffusion setting has been studied in [11]. We explain the one-dimensional version: Assume that the price process \(P = (P_t)_{t \in [0, T]}\) is the stochastic exponential of
where \(b, \sigma :\mathbb {R} \rightarrow \mathbb {R}\) are locally bounded Borel functions such that \(\sigma ^2\) is locally bounded away from zero. In this setting, [11, Propositions 5.1] shows that (NFLVR) always holds and [11, Proposition 5.2] implies that (NFFLVR) \(\Leftrightarrow \) (NRA) \(\Leftrightarrow \)\(\int _1^\infty \frac{dx}{\sigma ^2(x)} = \infty \). Under slightly different regularity assumptions on \(b\) and \(\sigma \), the same observation follows from Theorems 3.1 and 3.3. The novelty of Theorems 3.1 and 3.3 is that no diffusion structure is needed. In particular, the coefficients \(b\) and \(\sigma \) are allowed to depend on the path of \(S\) or several sources of risk. In [11] the main interest lies in the multi-dimensional setting. We stress that it is possible to extend our results to a multi-dimensional framework. The structure of the conditions will be similar as in [11].
In [40] the price process \(P = (P_t)_{t \in [0, T]}\) is assumed to be the stochastic exponential of
where \(X = (X_t)_{t \in [0, T]}\) is a continuous process, \(\alpha \) and \(\theta \) are suitable processes such that the integrals are well-defined and -a.e. \(\alpha \not = 0\). The process \(X\) is called information process. This setting is closely related to those from Sect. 3.1. Let \(\mathscr {W}\) be the Wiener measure and let \(\nu \) be the law of \( - \int _0^\cdot \theta _s ds + W. \) The main result from [40] is the following: If a.s. \(\int _0^T \theta _s^2 ds < \infty \), then (NFLVR) \(\Leftrightarrow \mathscr {W} \sim \nu \), see [40, Proposition 2.3]. This result is very different from ours, which are intended to give easy to verify conditions for a large class of models.
4 Modifying minimal local martingale measures
In Sect. 3.2.1 we proved conditions for the existence of the minimal (local) martingale measure in a Markov switching framework. We ask the following consecutive questions:
- 1.
Does the MLMM change the dynamics of the Markov chain?
- 2.
Is it possible to modify the MLMM such that the dynamics of the Markov chain are changed in a tractable manner?
In this section we answer these questions from a general perspective under an independence assumption, which holds in our Markov switching framework.
4.1 Martingale problems
To characterize additional sources of risk in our financial market, we introduce a martingale problem.
Let \(J\) be a Polish space, define \(D(\mathbb {R}_+, J)\) to be the space of all càdlàg functions \(\mathbb {R}_+ \rightarrow J\) and \(\mathcal {D}\) to be the \(\sigma \)-field generated by the coordinate process \(X = (X_t)_{t \ge 0}\), i.e. \(X_t(\omega ) = \omega (t)\) for \(\omega \in D(\mathbb {R}_+, J)\) and \(t \in \mathbb {R}_+\). We equip \(D(\mathbb {R}_+, J)\) with the Skorokhod topology, which renders it into a Polish space. It is well-known that \(\mathcal {D}\) is the Borel \(\sigma \)-field on \(D(\mathbb {R}_+, J)\). We refer to [20, 30] for more details. Let \({\mathbf {D}}^o \triangleq (\mathcal {D}^o_t)_{t \ge 0}\) be the filtration induced by \(X\), i.e. \(\mathcal {D}^o_t \triangleq \sigma (X_s, s \in [0, t])\), and let \({\mathbf {D}}\triangleq (\mathcal {D}_t)_{t \ge 0}\) be its right-continuous version, i.e. \(\mathcal {D}_t \triangleq \bigcap _{s > t} \mathcal {D}^o_s\) for all \(t \in \mathbb {R}_+\).
Let \((B_n)_{n \in \mathbb {N}}\) be an increasing sequence of nonempty open sets in \(J\) such that \(\bigcup _{n \in \mathbb {N}}B_n = J\) and define
Due to [20, Proposition 2.1.5], \(\rho _n\) is a \({\mathbf {D}}^o\)-stopping time and, due to [20, Problem 4.27], \(\rho _n \nearrow \infty \) as \(n \rightarrow \infty \). We will use the sequence \((\rho _n)_{n \in \mathbb {N}}\) as a localizing sequence for test martingales of our martingale problem. We fix this sequence, because for some arguments we need a common localizing sequence consisting of \({\mathbf {D}}^o\)-stopping times.
The input data for our martingale problem is the following:
- (i)
A set \(A \subseteq C(J, \mathbb {R})\), where \(C(J, \mathbb {R})\) denotes the space of continuous functions \(J\rightarrow \mathbb {R}\).
- (ii)
A map \(L :A \rightarrow \mathcal {PM}\) such that for all \(f \in A, t \in \mathbb {R}_+\) and \(\omega \in D (\mathbb {R}_+, J)\)
$$\begin{aligned} \int _0^{t} \big |Lf(\omega , s)\big | ds < \infty , \end{aligned}$$where \(\mathcal {PM}\) denotes the space of all \({\mathbf {D}}\)-progressively measurable processes.
- (iii)
An initial value \(j_0 \in J\).
- (iv)
A time horizon \(0 < T \le \infty \).
We use the convention that in case \(T = \infty \) the interval \([0, T]\) is identified with \(\mathbb {R}_+\).
Definition 4.1
- (i):
-
Let \((\Omega ^o, \mathcal {F}^o, \mathbf {F}^o, \mathbb {P}^o)\) be a filtered probability space with right-continuous filtration \(\mathbf {F}^o = (\mathcal {F}^o_t)_{t \in [0, T]}\), supporting a càdlàg, adapted, \(J\)-valued process \(\xi = (\xi _t)_{t \in [0, T]}\). We say that \(\xi \) is a solution process to the martingale problem\((A, L, j_0, T)\), if for all \(f \in A\) and \(n \in \mathbb {N}\) the process
$$\begin{aligned} M^{f, n} \triangleq f (\xi _{\cdot \wedge \rho _n (\xi )}) - f(\xi _0) - \int _0^{\cdot \wedge \rho _n (\xi )} L f (\xi , s) ds \end{aligned}$$(4.2)is a martingale, \(\mathbb {P}^o (\xi _0 = j_0) = 1\) and for all \(t \in [0, T]\) there exists a constant \(C = C(f, n, t) > 0\) such that a.s. \(\sup _{s \in [0, t]} |M^{f, n}_s| \le C\).
- (ii):
-
We say that the martingale problem has a solution if there exists a filtered probability space which supports a solution process.
- (iii):
-
We say that the martingale problem satisfies uniqueness if the laws (seen as Borel probability measures on \(D(\mathbb {R}_+, J)\)) of any two solution processes, possibly defined on different filtered probability spaces, coincide.
- (iv):
-
If for all \(j_0 \in J\) the martingale problem \((A, L, j_0, T)\) has a solution and satisfied uniqueness, we call the martingale problem \((A, L, T)\)well-posed.
Martingale problems were introduced by Stroock and Varadhan [54] in a diffusion setting. Martingale problems for semimartingales were studied in [27] and Markovian martingale problems with a Polish state space were studied in [20]. Our definition is unifying in the sense that it deals with non-Markovian processes and a Polish state space. Most of the conditions for existence and uniqueness given in [20, 27, 54] also apply to our setting.
Example 4.2
(Martingale problem for Markov chains) Suppose that \(J = \{1, \dots , N\}\) with \(1 \le N \le \infty \). We equip \(J\) with the discrete topology. Let \(\xi = (\xi _t)_{t \ge 0}\) be a Feller–Markov chain with initial value \(j_0 \in J\) and \(Q\)-matrix \(Q\). Due to [46, Theorem 5], the generator \((\mathcal {L}, D(\mathcal {L}))\) of \(\xi \) is given by \( \mathcal {L} = Q\) and \( D(\mathcal {L}) = \{f \in C_0(J) :Q f \in C_0(J)\}, \) where \(C_0(J)\) denotes the space of all continuous functions \(J \rightarrow \mathbb {R}\) which are vanishing at infinity. Due to Dynkin’s formula (see [47, Proposition VII.1.6]) the process \(\xi \) solves the martingale problem \((D(\mathcal {L}), \mathcal {L}, j_0, \infty )\) and, due to [38, Theorem 3.33], the martingale problem satisfies uniqueness.
Conversely, in case \(\xi \) is a solution process to the martingale problem \((\mathcal {L}, D(\mathcal {L}), j_0, \infty )\), where \((\mathcal {L}, D(\mathcal {L}))\) given as above is the generator of a Feller process, \(\xi \) is a Feller–Markov chain with \(Q\)-matrix \(Q\), see [20, Theorem 3.4.2] and [38, Theorem 3.33].
4.2 How to modify the MLMM
Fix a finite time horizon \(0< T < \infty \) and let \((\Omega , \mathcal {F}, \mathbf {F}, \mathbb {P})\) be a complete filtered probability space with right-continuous and complete filtration \(\mathbf {F}= (\mathcal {F}_t)_{t \in [0, T]}\), which supports a solution process \(\xi = (\xi _t)_{t \in [0, T]}\) to the martingale problem \((A, L, j_0, T)\). Let \(W = (W_t)_{t \in [0, T]}\) be a one-dimensional Brownian motion such that \(\sigma (W_t, t \in [0, T])\) and \(\sigma (\xi _t, t \in [0, T])\) are independent. We think of \(W\) and \(\xi \) as two independent sources of risk influencing the market. The independence assumption is satisfied when \(\xi \) is a Feller–Markov chain, see Lemma A.1 in the “Appendix”.
In the following theorem we find a new property of the MLMM. To wit, we show that the MLMM preserves the independence of the sources of risk and their laws. Because the M(L)MM is often used for pricing, this observation is important for analytical and numerical computations. We prove the following theorem in Sect. 7.
Theorem 4.3
Let \(c = (c_t)_{t \in [0, T]}\) be a real-valued progressively measurable process such that a.s.
and define
Suppose further that \(Z\) is a martingale and that the martingale problem \((A, L, j_0, T)\) satisfies uniqueness. Define \(\mathbb {Q}\) by the Radon–Nikodym derivative \(\frac{d\mathbb {Q}}{d\mathbb {P}} \triangleq Z_T\). Then, \(\sigma ( B_t, t \in [0, T])\) and \(\sigma (\xi _t, t \in [0,T])\) are \(\mathbb {Q}\)-independent, \(B\) is a \(\mathbb {Q}\)-Brownian motion and \(\xi \) is a solution process to the martingale problem \((A, L, j_0, T)\) on \((\Omega , \mathcal {F}, \mathbf {F}, \mathbb {Q})\).
Let us outline an important consequence of Theorem 4.3: If the MLMM exists, then its density is of the same type as \(Z\) in Theorem 4.3 and it follows that the joint law of the sources of risk remains unchanged by an equivalent change to the MLMM. In particular, in the setting of Sect. 2.2 this means that \(\xi \) stays a Markov chain after a change to the MLMM.
We ask further whether it is possible to modify the MLMM such that the law of \(\xi \) can be affected in a tractable manner. An answer to this question is provided by the next theorem. A proof can be found in Sect. 8.
Theorem 4.4
Let \(f \in A\) be strictly positive and suppose that the process
is a martingale. Set
and
Suppose that for every \(g \in A^*\) and \(n \in \mathbb {N}\) there exists a constant \(C = C(g, n) > 0\) such that a.s.
Define the probability measure \(\mathbb {Q}\) by the Radon–Nikodym derivative \( \frac{d\mathbb {Q}}{d\mathbb {P}} \triangleq Z_T. \) Then, \(\sigma (\xi _t, t \in [0, T])\) and \(\sigma (W_t, t \in [0, T])\) are \(\mathbb {Q}\)-independent, \(W\) is a \(\mathbb {Q}\)-Brownian motion and \(\xi \) is a solution process for the martingale problem \((A^*, L^*, j_0, T)\) on \((\Omega , \mathcal {F}, \mathbf {F}, \mathbb {Q})\).
Remark 4.5
-
(i)
For all \(\omega \in D(\mathbb {R}_+, J)\) and \(g \in A^*\)
$$\begin{aligned} \int _0^T\Big (\Big |\frac{L f(\omega , s)}{f(\omega (s))}\Big | + \big | L^* g(\omega , s)\big | \Big )ds < \infty , \end{aligned}$$because \(f\) and \(g\) are continuous and the set \(\{\omega (t) :t \in [0, T]\} \subseteq J\) is relatively compact, see [20, Problem 16, p. 152]. Consequently, \(Z\) and the martingale problem \((A^*, L^*, j_0, T)\) are well-defined.
-
(ii)
In view of [20, Corollary 2.3.3], the process (4.3) is always a local martingale by the definition of the martingale problem.
We explain an application of Theorem 4.4: Suppose that the MLMM exists. Then, using the change of measure described in Theorem 4.4, the MLMM can be changed further such that the law of \(\xi \) gets affected as described in the theorem, while the local martingale property of the price process is preserved. We stress that in this manner the MLMM induces a family of ELMMs, which is often infinite. In a Markov switching framework with \(N < \infty \) the following proposition explains how the \(Q\)-matrix of the driving Feller–Markov chain can be changed.
Proposition 4.6
Suppose that \(J = \{1, \dots , N\}\) with \(N < \infty \) and
for a \(Q\)-matrix \(Q = (q_{ij})_{i, j \in J}\) and \(f \in A \triangleq \mathbb {R}^N\). Let \(f \in (0, \infty )^N\) and \(A^*,L^*\) as in Theorem 4.4. Then, \(A^* = \mathbb {R}^N\) and
for \(Q^* = (q^*_{ij})_{i, j \in J}\) with
Proof
See [45, Proposition 5.1]. \(\square \)
A useful criterion for the martingale property of (4.3) is given by Theorem 4.9 below. We consider it as an extension of results from [5, 27, 54]. In the following \(X = (X_t)_{t \ge 0}\) denotes the coordinate process on \(D(\mathbb {R}_+,J)\).
Definition 4.7
A set \(\widetilde{A} \subseteq A\) is called a determining set for the martingale problem \((A, L, \infty )\) if for all \(j_0 \in J\) a Borel probability measure \(\mu \) on \(D(\mathbb {R}_+, J)\) is the law of a solution process to the martingale problem \((A, L, j_0, \infty )\) if and only if for all \(f \in \widetilde{A}\) and \(n \in \mathbb {N}\) the process
is a \(\mu \)-martingale and \(\mu (X_0 = j_0) = 1\).
Example 4.8
(Determining set for Feller–Markov chains) Let \(J, A\) and \(L\) be as in Example 4.2. Note that
Because \(C_0(J)\) equipped with the uniform metric is a separable metric space, \(G\) is a separable metric space when equipped with the taxicap uniform metric. Hence, we find a countable set \(\widetilde{A} \subseteq A\) such that for each \((f, g) \in G\) there exists a sequence \((f_n)_{n \in \mathbb {N}} \subset \widetilde{A}\) with
Due to [20, Proposition 4.3.1], \(\widetilde{A}\) is a determining set for the martingale problem \((A, L, \infty )\).
A proof for the following theorem can be found in Sect. 8.
Theorem 4.9
Let \(f, A^*\) and \(L^*\) be as in Theorem 4.4. Moreover, assume there exists a countable determining set for the martingale problem \((A^*, L^*, \infty )\) and that
where \(K\) maps \(A^*\) into the space of Borel functions \(J \rightarrow \mathbb {R}\). Finally, assume that the martingale problem \((A^*, L^*, \infty )\) is well-posed and that \((\rho _n (\xi ))_{n \in \mathbb {N}}\) is a localizing sequence for the local martingale (4.3), see Remark 4.5. Then, the process (4.3) is a martingale.
Roughly speaking, this theorem shows that in Markovian settings we can modify the law of \(\xi \) whenever the martingale problem \((A^*, L^*, \infty )\) is well-posed.
Remark 4.10
The existence of a solution to the martingale problem \((A^*, L^*, j_0, T)\) is often necessary for the martingale property of \(Z\), see Theorem 4.4.
5 Proof of Theorems 2.1 and 2.3
The following section is divided into three parts. In the first part we prove Lyapunov-type conditions for non-explosion of Itô processes, in the second part we prove non-existence conditions for Itô processes and in the third part we deduce Theorems 2.1 and 2.3.
5.1 Criteria for non-explosion
In this section we pose ourselves into a version of the setting from Sect. 2.1. Let \(I = (l, r)\) be as in Sect. 2.1 and \((\Omega ,\mathcal {F})\) be a measurable space which supports three real-valued processes \(S = (S_t)_{t \in [0, T]}, b = (b_t)_{t \in [0,T]}\) and \(\sigma = (\sigma _t)_{t \in [0, T]}\). For every \(n \in \mathbb {N}\) we fix a probability measure \(\mathbb {Q}^n\) and a right-continuous \(\mathbb {Q}^n\)-complete filtration \(\mathbf {F}^n= (\mathcal {F}^n_t)_{t \in [0, T]}\) on \((\Omega , \mathcal {F})\) such that \(S, b\) and \(\sigma \) are \(\mathbf {F}^n\)-progressively measurable. We set \(\tau _n\) as in Theorem 2.1, i.e.
where \(l_n \searrow l, r_n \nearrow r\) are sequences such that \(l< l_{n+1}< l_n< r_n< r_{n +1} < r\). Moreover, suppose that \(\mathbb {Q}^n\)-a.s.
where \(W^n = (W^n_t)_{t \in [0, T]}\) is a Brownian motion on \((\Omega , \mathcal {F}, \mathbf {F}^n, \mathbb {Q}^n)\). It is implicit that the integrals are well-defined. We also assume that
![](http://media.springernature.com/lw218/springer-static/image/art%3A10.1007%2Fs11579-020-00262-1/MediaObjects/11579_2020_262_Equ30_HTML.png)
and we fix a Borel function \(\zeta :[0, T] \rightarrow \mathbb {R}_+\) such that \(\zeta \in L^1([0, T])\).
5.1.1 A Lyapunov criterion
In this section we give a Lyapunov-type condition for
For \(f \in C^1(I, \mathbb {R})\) with locally absolutely continuous derivative, it is well-known that there exists a -null set \(N^f \subset I\) such that \(f\) has a second derivative \(f''\) on \(I \backslash N^f\). In this case, we set
Theorem 5.1
Let \(V :I \rightarrow (0, \infty )\) be differentiable with locally absolutely continuous derivative such that
Suppose there exists a -null set \(N \subset I\) such that
![](http://media.springernature.com/lw485/springer-static/image/art%3A10.1007%2Fs11579-020-00262-1/MediaObjects/11579_2020_262_Equ33_HTML.png)
Then, (5.2) holds.
Proof
Let \(L^S\) be the local time of the continuous \(\mathbb {Q}^n\)-semimartingale \(S_{\cdot \wedge \tau _n}\). The occupation times formula yields that \(\mathbb {Q}^n\)-a.s.
which implies that \(\mathbb {Q}^n\)-a.s . We will use this fact in the following without further reference.
Set
Using a generalized version of Itô’s formula (see [49, Lemma IV.45.9]), we obtain that the process
is a local \(\mathbb {Q}^n\)-martingale. We deduce from (5.4) and the fact that non-negative local martingales are supermartingales, that \(\mathbb {Q}^n\)-a.s.
W.l.o.g. we assume that \(S_0 \in (l_{1}, r_{1})\). Note that for all \(n \in \mathbb {N}\) we have \(\mathbb {Q}^n\)-a.s. \(S_{\tau _n} \in \{l_n, r_n\}\) on \(\{\tau _n \le T\}\). We conclude that for all \(n \in \mathbb {N}\)
By (5.3) there exists a sequence \((n_k)_{k \in \mathbb {N}} \subset \mathbb {N}\) with \(n_k \rightarrow \infty \) as \(k \rightarrow \infty \) such that \( V(l_{n_k}) \wedge V(r_{n_k}) > 0\) for all \(k \in \mathbb {N} \) and \( \lim _{k \rightarrow \infty } V(l_{n_k}) \wedge V(r_{n_k}) = \infty . \) We deduce from
that
Because \(\{\tau _n \le T\}^c = \{\tau _n = \infty \}\), we obtain
which implies (5.2). The proof is complete. \(\square \)
5.1.2 An integral test
Let \(\overline{a} :I \rightarrow (0, \infty )\) and \(\underline{u}, \overline{u} :I \rightarrow \mathbb {R}\) be Borel functions such that
Recall from Sect. 2 that in case \((f, g)\) is one of the pairs \((\underline{u}, \overline{a}), (\overline{u}, \overline{a})\) we set
for a fixed \(x_0 \in I\). The main result of this section is the following:
Theorem 5.2
Suppose that
Moreover, for all \(n \in \mathbb {N}\) assume that for -a.a. \((t, \omega ) \in [0, T] \times \Omega \)
Then, (5.2) holds.
Proof
Due to [36, Lemma 5.5.26], there are differentiable functions \(U_1 :[x_0, r) \rightarrow [1, \infty )\) and \(U_2 :(l, x_0] \rightarrow [1, \infty )\) with locally absolutely continuous derivatives and a -null set \(N' \subset I\) such that \(U_1\) is increasing, \(U_2\) is decreasing, \(U_1 (x_0) = U_2(x_0) = 1, U'_1 (x_0) = U'_2(x_0) = 0\), for all \(x \in [x_0, r) \backslash N'\) and for all \(y \in (l, x_0] \backslash N'\)
\(1 + v(\overline{u}, \overline{a}) \le U_1\) and \(1 + v(\underline{u}, \overline{a}) \le U_2\). We define
which is a differentiable function with locally absolutely continuous derivative. In particular, \(V' \ge 0\) on \([x_0, r)\), \(V' \le 0\) on \((l, x_0]\), \( \frac{1}{2} V'' + \underline{u} V' \ge 0 \) on \((l, x_0] \backslash N'\) and \( \frac{1}{2} V'' + \overline{u} V' \ge 0 \) on \([x_0, r)\backslash N'\). Furthermore,
due to the assumption (5.6). Let \(\widetilde{N}\) be the set of all \((t, \omega ) \in [0, T] \times \Omega \) such that (5.7) holds. For all \((t, \omega ) \in \widetilde{N} :S_t (\omega ) \in [x_0, r) \backslash N'\)
In the same manner we see that for all \((t, \omega ) \in \widetilde{N}:S_t (\omega ) \in (l, x_0] \backslash N'\)
We conclude that (5.4) holds for \(N = N'\). The claim follows from Theorem 5.1. \(\square \)
5.2 Criteria for non-existence
In this section we give a converse to Theorem 5.2. As in Sect. 2, let \(I = (l, r)\) with \(- \infty \le l < r \le + \infty \) and let \(\underline{a} :I \rightarrow (0, \infty )\) and \(\underline{u}, \overline{u} :I \rightarrow \mathbb {R}\) be Borel functions such that
If \((f, g)\) is one of the pairs \((\underline{u}, \underline{a}), (\overline{u}, \underline{a})\), we set \(v(f, g)\) as in (5.5).
Let \(0< T < \infty \), \((\Omega , \mathcal {F})\) be a measurable space with right-continuous filtration \(\mathbf {F}= (\mathcal {F}_t)_{t \in [0, T]}\) and \(s_0 \in I\). Suppose that \((\Omega , \mathcal {F}, \mathbf {F})\) supports three progressively measurable processes \(S = (S_t)_{t \in [0, T]}, b = (b_t)_{t \in [0, T]}\) and \(\sigma = (\sigma _t)_{t \in [0, T]}\). We define \(\mathcal {I}\) be the set of all pairs \((\mathbb {Q}, B)\) consisting of a probability measure \(\mathbb {Q}\) on \((\Omega , \mathcal {F})\) and an \((\mathbf {F}, \mathbb {Q})\)-Brownian motion \(B = (B_t)_{t \in [0, T]}\) with the properties that \(S\) is \(\mathbb {Q}\)-a.s. \(I\)-valued and
where it is implicit that the integrals are well-defined.
Theorem 5.3
-
(i)
Suppose that the pair \((\underline{u}, \underline{a})\) satisfies the YW conditions (see Sect. 2.1 for this terminology) and
$$\begin{aligned} \lim _{x \nearrow r} v ( \underline{u}, \underline{a})(x) < \infty . \end{aligned}$$Then, there exists no pair \((\mathbb {Q}, B) \in \mathcal {I}\) such that for
-a.a. \((t, \omega ) \in [0, T] \times \Omega \)
$$\begin{aligned} \begin{aligned} \underline{a} (S_t(\omega ))&\le \sigma ^2_t(\omega ),\\ \underline{u} (S_t(\omega )) \sigma ^2_t(\omega )&\le b_t(\omega ). \end{aligned} \end{aligned}$$(5.8) -
(ii)
Suppose that the pair \((\overline{u}, \underline{a})\) satisfies the YW conditions and
$$\begin{aligned} \lim _{x \searrow l} v(\overline{u}, \underline{a})(x) < \infty . \end{aligned}$$Then, there exists no pair \((\mathbb {Q}, B)\in \mathcal {I}\) such that for
-a.a. \((t, \omega ) \in [0, T] \times \Omega \)
$$\begin{aligned} \begin{aligned} \underline{a} (S_t(\omega ))&\le \sigma ^2_t(\omega ),\\ \overline{u} (S_t(\omega )) \sigma ^2_t(\omega )&\ge b_t(\omega ). \end{aligned} \end{aligned}$$(5.9)
Proof
(i). We use a comparison and contradiction argument as in the proof of [11, Theorem 4.1]. For contradiction, assume that \((\mathbb {Q}, B)\in \mathcal {I}\) is such that (5.8) holds. W.l.o.g. we assume that \(\mathbf {F}\) is \(\mathbb {Q}\)-complete. In the following we work on \((\Omega , \mathcal {F}, \mathbf {F}, \mathbb {Q})\). Because \(\underline{a}\) is positive and continuous and a.s
![](http://media.springernature.com/lw343/springer-static/image/art%3A10.1007%2Fs11579-020-00262-1/MediaObjects/11579_2020_262_Equ259_HTML.png)
the function
is a.s. finite, continuous and strictly increasing, which implies that the same holds for the function
see [47, pp. 179 – 180]. Furthermore, we have a.s. \(\phi _t \le t \text { for all } t \in [0, T]\). We redefine \(\phi _t\) to be zero on the null sets where the previously mentioned properties fail. Because \(\mathbf {F}\) is complete, this modification of \((\phi _t)_{t \in [0, T]}\) is an increasing and continuous sequence of finite stopping times.
Next, we set \(\mathbf {F}_\phi \triangleq (\mathcal {F}_{\phi _t})_{t \in [0, T]}\). The following lemma follows from [47, Propositions V.1.4, V.1.5].
Lemma 5.4
Suppose that \((H_t)_{t \in [0, T]}\) is progressively measurable. Then, the time-changed process \((H_{\phi _t})_{t \in [0, T]}\) is \(\mathbf {F}_\phi \)-progressively measurable and a.s.
provided the integrals are well-defined. Moreover, the process \(B_\phi = (B_{\phi _t})_{t \in [0, T]}\) is a continuous local \(\mathbf {F}_\phi \)-martingale with a.s. \( [B_\phi , B_\phi ]= \phi \), and if a.s. \(\int _0^T H^2_s ds < \infty \) then also a.s. \(\int _0^T H_{\phi _s}^2 d\phi _s < \infty \) and a.s.
We deduce from Lemma 5.4 that a.s.
![](http://media.springernature.com/lw411/springer-static/image/art%3A10.1007%2Fs11579-020-00262-1/MediaObjects/11579_2020_262_Equ260_HTML.png)
We will use this observation in the following without further reference.
Applying Lemma 5.4 with
yields that a.s.
Using again Lemma 5.4, we obtain that a.s. for all \(t \in [0, T]\)
where
Due to Lemma 5.4 and (5.10), we obtain that a.s. for all \(t \in [0, T]\)
Consequently, \(B'\) is a continuous local \(\mathbf {F}_\phi \)-martingale with a.s. \([B', B']_t = t\) for \(t \in [0, T]\), i.e. an \(\mathbf {F}_\phi \)-Brownian motion due to Lévy’s characterization. We summarize that
Using a standard extension of \((\Omega , \mathcal {F}, \mathbf {F}_{\phi }, \mathbb {Q})\) we can extend \((B'_t)_{t \in [0, T]}\) to a Brownian motion \((B'_t)_{t \ge 0}\), see, e.g., the proof of [47, Theorem V.1.7].
We will use the following terminology: When we say that \((V_t)_{t \ge 0}\) is a continuous \([l, r]\)-valued process we mean that all its paths are continuous in the \([l, r]\)-topology and absorbed in \(\{l, r\}\), i.e. that \(V_t = V_{\tau (V)}\) for all \(t \ge \tau (V) \triangleq \inf (t \in \mathbb {R}_+ :V_t \not \in I)\). This convention is in line with [36, Definition 5.5.20].
Definition 5.5
Let \(\mu :I \rightarrow \mathbb {R}\) and \(v :I \rightarrow \mathbb {R}\) be Borel functions. We say that an SDE
where \((B^*_t)_{t \ge 0}\) is a one-dimensional Brownian motion, satisfies strong existence and uniqueness up to explosion, if on any complete probability space \((\Omega ^o, \mathcal {F}^o, \mathbb {P}^o)\) with complete right-continuous filtration \(\mathbf {F}^o = (\mathcal {F}^o_t)_{t \ge 0}\), which supports a Brownian motion \((B^*_t)_{t \ge 0}\) and an \(I\)-valued \(\mathcal {F}^o_0\)-measurable random variable \(\psi \), there exists a up to indistinguishability unique adapted continuous \([l, r]\)-valued process \((V_t)_{t \ge 0}\) such that a.s.
where
It is implicit that the integrals are well-defined. The process \((V_t)_{t \ge 0}\) is called the solution process to (5.11) with driver\((B^*_t)_{t \ge 0}\).
Due to [19, Remark 4.50 (2), Theorem 4.53], the SDE
satisfies strong existence and uniqueness up to explosion.
Consequently, there exists a solution process \((Y_t)_{t \ge 0}\) to (5.12) with driver \((B'_t)_{t \ge 0}\). The following lemma is proven after the proof of Theorem 5.3 is complete.
Lemma 5.6
Almost surely \(Y_t \le S_{\phi _t}\) for all \(t \le T \wedge \tau (Y)\).
Because \((Y_t)_{t \ge 0}\) is regular due to [41, Proposition 2.2] and \(\lim _{x \nearrow r} v ( \underline{u}, \underline{a})(x) < \infty \), we deduce from [41, Proposition 2.12] and [4, Theorem 1.1] that \((Y_t)_{t \in [0, T]}\) reaches \(r\) with positive probability. Consequently, due to Lemma 5.6, \((S_t)_{t \in [0, T]}\) reaches \(r\) with positive probability. This is a contradiction.
(ii). For contradiction, assume that \((\mathbb {Q}, B)\in \mathcal {I}\) is such that (5.9) holds. By the same arguments as in part (i), there exists a process \((Y_t)_{t \ge 0}\) such that
and a.s. \(S_{\phi _t} \le Y_t\) for all \(t \le T \wedge \tau (Y).\) Because \(\lim _{x \searrow l} v(\overline{u}, \underline{a})(x) <\infty ,\) the process \((Y_t)_{t \in [0, T]}\) reaches \(l\) with positive probability and again the pathwise ordering gives a contradiction. \(\square \)
Proof of Lemma 5.6
There are functions \(h_n \in \mathscr {H}\) and \(\kappa _n \in \mathscr {K}\) such that for all \(x, y \in [l_n, r_n]\)
We set
Note that for all \(t \in (0, T]\) we have
Thus, [47, Lemma IX.3.3] implies that the local time of \(Y_{\cdot \wedge \rho _n}- S_{\phi _{\cdot \wedge \rho _n}}\) in the origin is a.s. zero. We deduce from Tanaka’s formula that a.s.
Taking expectation, using the martingale property of the Brownian part of \(Y_{\cdot \wedge \rho _n} - S_{\phi _{\cdot \wedge \rho _n}}\) and Jensen’s inequality yields that for all \(t \in [0, T]\)
Finally, Bihari’s lemma (see [11, Lemma E.2]) yields that for all \(t \in [0, T]\)
Consequently, due to the continuous paths of \(Y\) and \(S_\phi \), the claim follows. \(\square \)
5.3 Proof of Theorem 2.1
Because non-negative local martingales are supermartingales, \(Z\) is a martingale if and only if \({\mathbb {E}}^\mathbb {P}[Z_T] = 1\). By (M1), we can define \(\mathbb {Q}^n\) by the Radon–Nikodym derivative \(\frac{d\mathbb {Q}^n}{d\mathbb {P}} = Z_{T \wedge \tau _n}\). We note that the assumption -a.e. \(\sigma \not = 0\) implies (5.1). Due to Girsanov’s theorem, there exists a \(\mathbb {Q}^n\)-Brownian motion \(B^n = (B^n_t)_{t \in [0, T]}\) such that
The monotone convergence theorem yields that
In view of (M2) and (M3), Theorem 5.2 yields that
Thus, \({\mathbb {E}}^\mathbb {P}[Z_T] = 1\) and the proof is complete. \(\square \)
5.4 Proof of Theorem 2.3
For contradiction, assume that \((Z_t)_{t \in [0, T]}\) is a martingale. Define a probability measure \(\mathbb {Q}\) by the Radon–Nikodym derivative \(\frac{d\mathbb {Q}}{d\mathbb {P}} \triangleq Z_T\). By Girsanov’s theorem, there exists a \(\mathbb {Q}\)-Brownian motion \(B = (B_t)_{t \in [0, T]}\) such that
Consequently, in case (SL1) holds we obtain a contradiction to part (i) of Theorem 5.3 and in case (SL2) holds we obtain a contradiction to part (ii) of Theorem 5.3. The proof is complete. \(\square \)
6 Proof of Theorem 2.5
The section is split into two parts: First, we prove existence, non-existence and local uniqueness for switching diffusions and second, we deduce Theorem 2.5.
6.1 Existence and non-existence criteria
As in Sect. 2.2, let \(I = (l, r)\) with \(- \infty \le l < r \le + \infty \) and \(J = \{1, \dots , N\}\) with \(1 \le N \le \infty \). Moreover, let \(u :I \times J \rightarrow \mathbb {R}\) and \(\sigma :I \times J \rightarrow \mathbb {R}\backslash \{0\}\) be Borel functions such that
We fix \(x_0 \in I\) and set
for \((x, j) \in I \times J\). Let \((\Omega , \mathcal {F}, \mathbf {F}, \mathbb {P})\) be a filtered complete probability space with a right-continuous and complete filtration \(\mathbf {F}= (\mathcal {F}_t)_{t \ge 0}\), which supports a Brownian motion \(W = (W_t)_{t \ge 0}\), a \(J\)-valued irreducible continuous-time Feller–Markov chain \(\xi = (\xi _t)_{t \ge 0}\) and an \(I\)-valued \(\mathcal {F}_0\)-measurable random variable \(\phi \). The main result of this section is the following:
Theorem 6.1
-
(i)
Suppose that \(\sigma \) satisfies the ES conditions for all \(j \in J\) (see Sect. 2.2 for this terminology) and that
$$\begin{aligned} \lim _{x \searrow l} v(x, j) = \lim _{x \nearrow r} v(x, j) = \infty \quad \text { for all }\; j \in J. \end{aligned}$$(6.2)Then, there exists an adapted \(I\)-valued continuous process \((Y_t)_{t \ge 0}\) such that
$$\begin{aligned} Y = \phi + \int _0^\cdot u(Y_s, \xi _s)ds + \int _0^\cdot \sigma (Y_s, \xi _s) dW_s, \end{aligned}$$(6.3)where it is implicit that the integrals are well-defined.
-
(ii)
Assume there exists a \(j \in J\) such that \(\sigma \) satisfies the ES conditions for \(j\) and
$$\begin{aligned} \lim _{x \searrow l} v(x, j)< \infty \text { or } \lim _{x \nearrow r} v(x, j) < \infty . \end{aligned}$$Let \(0 < T \le \infty \) be a time horizon. If \(\xi \) is recurrent, then there exists no adapted \(I\)-valued continuous process \(Y = (Y_t)_{t \in [0, T]}\) such that (6.3) holds.
Proof
The case \(N = 1\) concerns classical diffusions for which all claims are known, see [4, 19, 36] for details. We prove the claim under the assumption \(N \ge 2\).
(i). We define the jump times of \(\xi \) inductively by
Because \(\xi \) is irreducible, we have a.s. \(\gamma _n < \infty \) (see [33, Theorem 10.19]) and a.s. \(\gamma _n - \gamma _{n-1} > 0\) for all \(n \in \mathbb {N}\).
We follow the idea from the proof of [26, Theorem IV.9.1] and construct the process \(Y\) explicitly from solutions to the SDEs
where \(W' = (W'_t)_{t \ge 0}\) is a Brownian motion. For the construction we require a strong existence and uniqueness property, which we explain next.
Fix \(j \in J\). It follows from [19, Remark 4.50 (2), Theorem 4.53] and Feller’s test for explosion (see [36, Theorem 5.5.29]) that the SDE (6.4) has a weak solution and that it satisfies pathwise uniqueness for all deterministic initial values. We conclude from [33, Theorem 18.14] that there exists a Borel function \(F^{j} :I \times C(\mathbb {R}_+, \mathbb {R}) \rightarrow C(\mathbb {R}_+, I)\) such that for any one-dimensional Brownian motion \(W' = (W'_t)_{t \ge 0}\) and any \(I\)-valued random variable \(\psi \), which is independent of \(\sigma (W'_t, t \in \mathbb {R}_+)\), the process \(X^j = F^{j}(\psi , W')\) is a solution process to (6.4) with \(X^j_0 = \psi \), which is adapted to the completion of the natural filtration of \(W'\) and \(\psi \), see [36, Definition 5.2.1]. The function \(F^j\) is independent of the law of \(\psi \) and universally adapted (see [33, p. 346] for a definition).
Set \(W^n \triangleq W_{\cdot + \gamma _n} - W_{\gamma _n}\). Due to [47, Proposition V.1.5] and Lévy’s characterization, \(W^n\) is a Brownian motion for the filtration \(\mathbf {F}^n \triangleq (\mathcal {F}_{t + \gamma _n})_{t \ge 0}\). In particular, \(W^n\) is independent of \(\mathcal {F}_{\gamma _n}\). By induction, define
Moreover, set
The process \(Y\) is \(I\)-valued and continuous and, as we explain next, it is also \(\mathbf {F}\)-adapted. Define \(H_t \triangleq Y^n_{t - \gamma _n} \mathbb {1}_{\{\gamma _n < t\}}\). We claim that \((H_t)_{t \ge 0}\) is \(\mathbf {F}\)-progressively measurable. Because \(t \mapsto Y^n_{t - \gamma _n} \mathbb {1}_{\{\gamma _n < t\}}\) is left-continuous and \(s \mapsto Y^n_{t - s} \mathbb {1}_{\{s < t\}}\) is right-continuous, an approximation argument shows that is suffices to explain that \((h_t)_{t \ge 0} \triangleq (Y^n_{t - \zeta } \mathbb {1}_{\{\zeta < t\}})_{t \ge 0}\) is \(\mathbf {F}\)-adapted for any \(\mathbf {F}\)-stopping time \(\zeta \) which takes values in the countable set \(2^{-m} \overline{\mathbb {N}}\) for some \(m \in \mathbb {N}\) and satisfies \(\zeta \ge \gamma _n\). Let \(G \in \mathcal {B}(\mathbb {R})\) and set \(N_{m, t} \triangleq 2^{-m} \overline{\mathbb {N}}\cap [0, t)\). We have
Here, we use that \(\{Y^n_{t - k} \in G\} \in \mathcal {F}_{t - k + \gamma _n} \subseteq \mathcal {F}_{t - k + \zeta }\) and that \(\mathcal {F}_{t - k + \zeta } \cap \{\zeta = k\} \in \mathcal {F}_t\). Thus, \((H_t)_{t \ge 0}\) is \(\mathbf {F}\)-progressively measurable and consequently \((Y_t)_{t \ge 0}\) is \(\mathbf {F}\)-adapted.
We note that
which is an \(\mathbf {F}^{n - 1}\)-stopping time. Thus, \(Y^{n-1}_{\gamma _n - \gamma _{n - 1}}\) is \(\mathcal {F}_{\gamma _n}\)-measurable and therefore independent of \(\sigma (W^n_t, t \in \mathbb {R}_+)\). This yields that the process \(X^{n, j} \triangleq F^{j} (Y^{n - 1}_{\gamma _n - \gamma _{n-1}}, W^n)\) satisfies
Thus, due to classical rules for time-changed stochastic integrals (see [47, Propositions V.1.4, V.1.5]), a.s. for \(t \in [\gamma _n, \gamma _{n+1}]\) on \(\{\xi _{\gamma _n} = j\}\) we have
By induction, a.s. for \(t \in [\gamma _n, \gamma _{n + 1}]\)
Therefore, the process \(Y\) satisfies the SDE
and the proof of (i) is complete.
(ii). For contradiction, assume that \(Y\) satisfies (6.3). Let \(j \in J\) be such that \(\lim _{x \searrow l} v(x, j) < \infty \) or \(\lim _{x \nearrow r} v(x, j) < \infty \). We define
Because \(\xi \) is recurrent, we have a.s. \(\delta < \infty \), see [44, Theorem 1.5.7]. Due to the strong Markov property of \(\xi \) and [33, Lemma 10.18], for all \(G \in \mathcal {B}(\mathbb {R}_+)\) it holds that
where \(q_{jj} < 0\) is the \(j\)-th diagonal element of the \(Q\)-matrix of \(\xi \).
Recall our convention that we call a process \(V = (V_t)_{t \ge 0}\) to be continuous and \([l, r]\)-valued in case all paths are continuous in the \([l, r]\)-topology and absorbed in \(\{l, r\}\), i.e. that \(V_t = V_{\tau (V)}\) for all \(t \ge \tau (V) \triangleq \inf (t \in \mathbb {R}_+ :V_t \not \in I)\).
It follows from [19, Remark 4.50 (2), Theorem 4.53] that the SDE (6.4) satisfies strong existence and uniqueness up to explosion in the sense of Definition 5.5.
Consequently, there exists a continuous \([l, r]\)-valued process \(X = (X_t)_{t \ge 0}\) such that
where \(W^\delta \triangleq W_{\cdot + \delta \wedge T} - W_{\delta \wedge T}\) is a Brownian motion for the filtration \(\mathbf {F}^\delta \triangleq (\mathcal {F}_{t + \delta \wedge T})_{t \ge 0}\). We prove the following lemma after the proof of (ii) is complete.
Lemma 6.2
Almost surely \(Y_{t + \delta } = X_t\) for all \(0 \le t \le \zeta - \delta \) on \(\{\zeta \le T\}\).
Because on \(\{\tau (X) < \infty \}\) we have \(X_{\tau (X)} \not \in I\), Lemma 6.2 implies that
The proof of the following lemma is given after the proof of (ii) is complete.
Lemma 6.3
Suppose that the SDE (5.11) satisfies strong existence and uniqueness up to explosion. Let \(\psi \) be an \(I\)-valued \(\mathcal {F}_0\)-measurable random variable and let \((V_t)_{t \ge 0}\) be the solution process to (5.11) with driver \(W\) and initial value \(\psi \) and let \(\tau \) be a stopping time. Then, all adapted \(I\)-valued continuous processes \((U_t)_{t \ge 0}\) with
are indistinguishable from \((V_{t \wedge \tau })_{t \ge 0}\).
Let \(l_n \searrow l, r_n \nearrow r\) be sequences such that \(l< l_{n+1}< l_n< r_n< r_{n +1} < r\) and set for a function \(\alpha :\mathbb {R}_+ \rightarrow [l, r]\)
We conclude from Lemma 6.3 and Galmarino’s test (see [30, Lemma III.2.43]) that for all \(n \in \mathbb {N}\) the SDE
satisfies weak existence and pathwise uniqueness in the usual sense, see [36, Definitions 5.3.1, 5.3.2]. Thus, due to [33, Theorem 18.14], there exists a Borel function \(F^n :\mathbb {R} \times C(\mathbb {R}_+, \mathbb {R}) \rightarrow C(\mathbb {R}_+, I)\) such that whenever \(X^j\) solves (6.8) with driver \(W= (W_t)_{t \ge 0}\) and (possibly stochastic) initial value \(X^j_0\), then a.s. \(X^j = F^n(X^j_0, W)\).
Lemma 6.3 and Galmarino’s test yield that a.s.
Because strong existence and uniqueness up to explosion holds for the SDE (6.4), for a.a. \(\omega \in \Omega \) there exists an \(\mathbf {F}^\delta \)-adapted continuous \([l, r]\)-valued process \(Y^\omega = (Y^\omega _t)_{t \ge 0}\) such that
We stress that the initial value \(Y_{\delta (\omega ) \wedge T}(\omega )\) is deterministic. Lemma 6.3 and Galmarino’s test yield that a.s.
We prove the following lemma after the proof of (ii) is complete.
Lemma 6.4
For all \(G \in \mathcal {B}(\mathbb {R}_+)\) we have a.s.
Using (6.7), the monotone convergence theorem and then (6.9), we obtain that
using [33, Theorem 5.4] and Lemma 6.4 we further obtain that
which, due to [33, Theorem 5.4] and the independence of \(W^\delta \) and \(\mathcal {F}_{\delta \wedge T}\), equals
and finally, with (6.10) and the monotone convergence theorem, we obtain
Due to Feller’s test for explosion (see [36, Theorem 5.5.29]), \(Y^\omega \) reaches \(l\) or \(r\) in finite time with positive probability. In fact, because \(Y^\omega \) is regular due to [41, Proposition 2.2], [4, Theorem 1.1] implies that \(Y^\omega \) even reaches \(l\) or \(r\) arbitrarily fast with positive probability, i.e. \(\mathbb {P}(\tau (Y^\omega ) \le \varepsilon ) > 0\) for all \(\varepsilon > 0\). Consequently, the identity
implies that for -a.a. \(s \in (0, T)\) we have \(\mathbb {P}(\delta \le T - s) = 0\). However, because \(\xi \) is irreducible, we have \(\mathbb {P}(\xi _t= j) > 0\) for all \(t > 0\). This is a contradiction and the proof of (ii) is complete. \(\square \)
Proof of Lemma 6.2
Define \( \iota \triangleq \zeta \wedge T - \delta \wedge T \). Note that for all \(t \in \mathbb {R}_+\)
which shows that \(\iota \) is an \(\mathbf {F}^\delta \)-stopping time. Moreover, we have for all \(s, t \in \mathbb {R}_+\)
Thus, the random time \(s \wedge \iota + \delta \wedge T\) is an \(\mathbf {F}\)-stopping time. We deduce from classical rules for time-changed stochastic integrals that a.s. for all \(t \in \mathbb {R}_+\)
Because the SDE (6.4) satisfies strong existence and uniqueness up to explosion, Lemma 6.3 implies that a.s. \(Y_{t \wedge \iota + \delta \wedge T} = X_{t \wedge \iota }\) for all \(t \in \mathbb {R}_+\). On \(\{\zeta \le T\} \subseteq \{\delta \le T\}\) we have \(\iota = \zeta - \delta \) and the claim follows.\(\square \)
Proof of Lemma 6.3
Due to localization, we can assume that \(\tau \) is finite. By [47, Proposition V.1.5] and Lévy’s characterization, the process
is an \((\mathcal {F}_{t + \tau })_{t \ge 0}\)-Brownian motion. Due to the strong existence and uniqueness hypothesis, there exists a solution process \(O = (O_t)_{t \ge 0}\) to the SDE
We set
The process \(Z\) has continuous paths and similar arguments as used in the proof of Theorem 6.1 (i) show that it is \(\mathbf {F}\)-adapted. Let
On \(\{\tau \ge t \wedge \theta ^Z_n\}\) we have a.s.
Next, we discuss what happens on the set \(\{\tau < t \wedge \theta ^Z_n\}\). Set
On \(\{\tau < \theta ^Z_n\}\) we have a.s. \( \theta ^Z_n = \theta ^O_n + \tau . \) Moreover, note that
Thus, \(t \wedge (\theta ^O_n + \tau ) - \tau \le \theta ^O_n\). Classical rules for time-changed stochastic integrals yield that on \(\{\tau < t \wedge \theta ^Z_n\}\) a.s.
We conclude that \(Z\) is a solution process of the SDE (5.11) with driver \(W\) and initial value \(\psi \). By the strong existence and uniqueness hypothesis, we conclude that a.s. \(Z = V\). The definition of \(Z\) implies the claim. \(\square \)
Proof of Lemma 6.4
Denote the Wiener measure with initial value \(x \in \mathbb {R}\) by \(\mathscr {W}_x\) and by \(\mu _j\) the law of a Feller–Markov chain with the same \(Q\)-matrix as \(\xi \) and initial value \(j \in J\). Let \(\mathcal {C}\) be the \(\sigma \)-field on \(C(\mathbb {R}_+, \mathbb {R})\) generated by the coordinate process. We deduce from Lemma A.1 in the “Appendix”, [20, Proposition 4.1.5, Theorems 4.4.2, 4.4.6] that \((j, x) \mapsto (\mu _j \otimes \mathscr {W}_x) (F)\) is Borel for every \(F \in \mathcal {D} \otimes \mathcal {C}\) and that the process \((\xi , W)\) is a strong Markov process in the following sense: For all \(F \in \mathcal {D} \otimes \mathcal {C}\) and every a.s. finite stopping time \(\theta \) we have a.s.
For all \(A \in \mathcal {D}\) and \(F \in \mathcal {C}\) the strong Markov properties of \(\xi , W\) and \((\xi , W)\) imply that a.s.
This implies that \(\sigma (\zeta - \delta )\) and \(\sigma (W^\delta _t, t \in \mathbb {R}_+)\) are independent given \(\mathcal {F}_{\delta \wedge T}\). Now, [33, Proposition 5.6] yields that a.s.
By the strong Markov property of \(\xi \) and (6.5), we have for \(F \in \mathcal {F}_\delta \)
The proof is complete. \(\square \)
6.2 Local uniqueness
For the space of continuous functions from \(\mathbb {R}_+\) into \(I\) or \(\mathbb {R}\), we denote by \(\mathcal {C}\) the \(\sigma \)-field generated by the coordinate process. Moreover, we denote by \(\mathbf {C}^o \triangleq (\mathcal {C}^o_t)_{t \ge 0}\) the filtration generated by the corresponding coordinate process and by \(\mathbf {C} \triangleq (\mathcal {C}_t)_{t \ge 0}\) its right-continuous version. The image space will be clear from the context. Let
be a \(\mathbf {C}^o \otimes \mathbf {D}^o\)-stopping time. An example for \(\rho \) is
where \(U \subseteq I\) and \(V \subseteq J\) are open:
Lemma 6.5
\(\tau \) is a \(\mathbf {C}^o \otimes \mathbf {D}^o\)-stopping time.
Proof
See [47, Proposition I.4.5] and [20, Proposition 2.1.5]. \(\square \)
Let \(u :I \times J \rightarrow \mathbb {R}\) and \(\sigma :I \times J \rightarrow \mathbb {R}\backslash \{0\}\) be Borel functions such that (6.1) holds, \(\sigma \) satisfies (6.2) and the ES conditions for all \(j \in J\) (see Sect. 2.2 for this terminology). In other words, we ask that the conditions from part (i) of Theorem 6.1 hold.
For \(i = 1, 2\), let \((\Omega ^i, \mathcal {F}^i, \mathbf {F}^i, \mathbb {P}^i)\) be a filtered probability space with right-continuous complete filtration \(\mathbf {F}^i = (\mathcal {F}^i_t)_{t \ge 0}\). Let \(W^i = (W^i_t)_{t \ge 0}\) be a one-dimensional Brownian motion, \(\xi ^i = (\xi ^i_t)_{t \ge 0}\) be a \(J\)-valued irreducible Feller–Markov chain with \(Q\)-matrix \(Q\) and \(\xi ^i_0 = j_0 \in J\), and let \(X^i = (X^i_t)_{t \ge 0}\) be an adapted continuous \(I\)-valued process such that
It is implicit that the stochastic integrals are well-defined. We stress that \(\xi ^1\) and \(\xi ^2\) have the same law, because they have the same \(Q\)-matrix, see Example 4.2.
The main observation of this section is the following:
Theorem 6.6
\(\mathbb {P}^1 \circ (X^1_{\cdot \wedge \rho (X^1, \xi ^1)}, \xi ^1)^{-1} = \mathbb {P}^2 \circ (X^2_{\cdot \wedge \rho (X^2, \xi ^2)}, \xi ^2)^{-1}\).
Proof
We follow the Yamada–Watanabe-type idea used in [28]. Define
and for \(i =1, 2\)
Denote the Wiener measure by \(\mathscr {W}\) and denote the unique law of \(\xi ^i\) by \(\mu \). Due to Lemma A.1 in the “Appendix”, we have
When the space of continuous functions is equipped with the local uniform topology it is a Polish space and the corresponding Borel \(\sigma \)-field is generated by the coordinate process. Thus, there exist regular conditional probabilities
such that
We define a probability measure \(\mathbb {Q}\) on \((\Omega ^*, \mathcal {F}^*)\) by
With abuse of notation, denote the \(\mathbb {Q}\)-completion of \(\mathcal {F}^*\) again by \(\mathcal {F}^*\) and denote by \(\mathcal {F}^*_t\) the \(\mathbb {Q}\)-completion of
From now on we consider \((\Omega ^*, \mathcal {F}^*, \mathbf {F}^* = (\mathcal {F}^*_t)_{t \ge 0}, \mathbb {Q})\) as underlying filtered probability space. In view of [28, Propositions 4.6, 5.6], for all \(A \in \mathcal {C}_{t}\) the map \( \omega \mapsto Q^i (\omega , A) \) is measurable w.r.t. the \(\mu \otimes \mathscr {W}\)-completion of \(\bigcap _{s > t} (\mathcal {D}^o_s \otimes \mathcal {C}^o_s)\). In other words, [27, Hypothesis 10.43] is satisfied and we deduce from [28, Lemmata 2.7, 2.9], [27, Proposition 10.46] and Lévy’s characterization that \(Z^1\) is a Markov chain with \(Q\)-matrix \(Q\), \(Z^2\) is a Brownian motion and
The proof of the following lemma is given after the proof of Theorem 6.6 is complete.
Lemma 6.7
Almost surely \(Y^1_{\cdot \wedge \rho (Y^1, Z^1) \wedge \rho (Y^2, Z^1)} = Y^2_{\cdot \wedge \rho (Y^1, Z^1) \wedge \rho (Y^2, Z^1)}\).
Due to Galmarino’s test, this implies a.s. \(\rho (Y^1, Z^1) = \rho (Y^2, Z^1)\). Thus, a.s. \(Y^1_{\cdot \wedge \rho (Y^1, Z^1)} = Y^2_{\cdot \wedge \rho (Y^2, Z^1)}\) and the claim follows from the definition of \(\mathbb {Q}\). \(\square \)
Proof of Lemma 6.7
Step 1 Due to localization, we can assume that \(\rho (Y^1, Z^1) \wedge \rho (Y^2, Z^1)\) is finite. Recall the following fact (see [47, Proposition III.3.5]): If \((Z_t)_{t \ge 0}\) is a Feller–Markov chain for the right-continuous filtration \(\mathbf {G}= (\mathcal {G}_t)_{t \ge 0}\) and \(\gamma \) is a finite \(\mathbf {G}\)-stopping time, then \((Z_{t + \gamma })_{t \ge 0}\) is a Feller–Markov chain for a filtration \((\mathcal {G}_{t + \gamma })_{t \ge 0}\) and both chains have the same \(Q\)-matrix. Due to Theorem 4.4 (i), for \(i = 1, 2\) there exists a process \((O^i_t)_{t \ge 0}\) defined by
where
with initial value \(O^i_0 = Y^i_{\rho (Y^1, Z^1) \wedge \rho (Y^2, Z^1)}\). Now, set
As in the proof of Lemma 6.3, we deduce from classical rules for time-changed stochastic integrals that
i.e. that \(V^1\) and \(V^2\) are global solutions. Thus, it remains to show a version of pathwise uniqueness for the global equation (6.11).
Step 2 We use induction. Let \((\zeta _n)_{n \in \mathbb {N}}\) be the stopping times
We stress that \(\zeta _n \nearrow \infty \) as \(n \rightarrow \infty \). Almost surely on \(\{t \le \zeta _1\}\) we have
Recalling that under the assumptions from Theorem 6.1 (i) the SDE (6.4) satisfies strong existence and uniqueness (up to explosion), we deduce from Lemma 6.3 that a.s. \(V^1_t = V^2_t\) for all \(t \le \zeta _1\). In case \(N = 1\), we have \(\zeta _1 = \infty \) and the proof is complete. In the following, we assume that \(N \ge 2\) in which case a.s. \(\zeta _n < \infty \) for all \(n \in \mathbb {N}\). Suppose that \(n \in \mathbb {N}\) is such that a.s. \(V^1_t = V^2_t\) for all \(t \le \zeta _{n}\). Using classical rules for time-changed stochastic integrals, we obtain that a.s. on \(\{t \le \zeta _{n + 1} - \zeta _n\} \cap \{Z^1_{\zeta _n} = j\}\)
where
We conclude again from Lemma 6.3 that a.s. \(V^1_{t + \zeta _n} = V^2_{t + \zeta _n}\) for all \(t \le \zeta _{n + 1} - \zeta _n\). Consequently, a.s. \(V^1_t = V^2_t\) for all \(t \le \zeta _{n +1}\) and our claim follows. \(\square \)
6.3 Proof of Theorem 2.5
(i). Recall that \(J = \{1, \dots , N\}\) with \(1 \le N \le \infty \). For \(n \in \mathbb {N}\) define
Because \(c\) is assumed to be bounded on compact subsets of \(I \times J\), Novikov’s condition implies that \((\tau _n)_{n \in \mathbb {N}}\) is a localizing sequence for \(Z\). We define \(\mathbb {Q}^n\) by the Radon–Nikodym derivative \(\frac{d\mathbb {Q}^n}{d\mathbb {P}} \triangleq Z_{T \wedge \tau _n}\). By Girsanov’s theorem,
is a \(\mathbb {Q}^n\)-Brownian motion such that
We deduce from Lemma A.1, Example 4.2 and Theorem 4.3 that under \(\mathbb {Q}^n\) the process \(\xi \) remains a Feller–Markov chain with unchanged \(Q\)-matrix. W.l.o.g. we extend \(W, \xi \) and \(\mathbf {F}\) to the infinite time interval \(\mathbb {R}_+\). Applying Theorem 6.1 with \(u \triangleq b + c \sigma \) yields that on \((\Omega , \mathcal {F}, \mathbf {F}, \mathbb {P})\) there exists an adapted continuous \(I\)-valued process \(X = (X_t)_{t \ge 0}\) such that
We set
It follows from Lemma 6.5 and Theorem 6.6 that
Consequently, using Galmarino’s test, we obtain that
Now, it follows as in the proof of Theorem 2.1 that \(Z\) is a martingale.
(ii). This result follows similar as Theorem 2.3, where Theorem 6.1 has to be used instead of Theorem 5.3. We omit the details. \(\square \)
7 Proof of Theorem 4.3
Step 1 Let \(g \in A\) and set
Due to the definition of the martingale problem \((A, L, T)\), the process \(M^g\) is a local martingale with localizing sequence \((\rho _n (\xi ))_{n \in \mathbb {N}}\). Thus, the quadratic variation process \([M^g, W]\) is well-defined. Our first step is to show that a.s. \([M^g, W] = 0\). We explain that \(WM^g\) is a local martingale for the completed right-continuous version of the natural filtration of \(\xi \) and \(W\). Let \(0 \le s < t \le T\), \(G \in \sigma (W_r, r \in [0, s]) \triangleq \mathcal {W}_s\) and \(F \in \sigma (\xi _r, r \in [0, s]) \triangleq \mathcal {E}_s\). The independence assumption yields that
By a monotone class argument, we have
for all \(B \in \mathcal {W}_s \vee \mathcal {E}_s\). Due to the downwards theorem (see [48, Theorem II.51.1]), the process \(W M^g_{\cdot \wedge \rho _m(\xi )}\) is a martingale for the completed right-continuous version \(\mathbf {G}\triangleq (\mathcal {G}_t)_{t \in [0, T]}\) of \((\mathcal {W}_t \vee \mathcal {E}_t)_{t \in [0, T]}\). Consequently, because \(\rho _m (\xi )\nearrow \infty \) as \(m \rightarrow \infty \), \(W M^g\) is a local \(\mathbf {G}\)-martingale. By the tower rule, also \(W\) and \(M^g\) are local \(\mathbf {G}\)-martingales. Integration by parts implies that
where the stochastic integrals are defined as local \(\mathbf {G}\)-martingales. Here, we use that \([W, M^g]\) can be defined independently of the filtration. We deduce that the process \([W, M^g]\) is a continuous local \(\mathbf {G}\)-martingale of finite variation and hence a.s. \([W, M^g] = 0\).
Step 2 In this step we identify the laws of \(B\) and \(\xi \) under \(\mathbb {Q}\). Clearly, \(B\) is a \(\mathbb {Q}\)-Brownian motion due to Girsanov’s theorem. Next, we show that on \((\Omega , \mathcal {F}, \mathbf {F}, \mathbb {Q})\) the process \(\xi \) is a solution process for the martingale problem \((A, L, j_{0}, T)\). By Step 1 and Girsanov’s theorem, the process
is a local \(\mathbb {Q}\)-martingale. The equivalence \(\mathbb {Q}\sim \mathbb {P}\) implies that \(\mathbb {Q}(\xi _0 = j_0) = 1\) and that \(M^g_{\cdot \wedge \rho _n(\xi )}\) is \(\mathbb {Q}\)-a.s. bounded. Thus, the claim follows.
Step 3 We prove \(\mathbb {Q}\)-independence of \(B\) and \(\xi \) borrowing an idea from [20, Theorem 4.10.1]. We define \(C^2_b(\mathbb {R})\) to be the set of all bounded twice continuously differentiable functions \(\mathbb {R} \rightarrow \mathbb {R}\) with bounded first and second derivative. Suppose that \(f \in C^2_b(\mathbb {R})\) with \(\inf _{x \in \mathbb {R}} f(x) > 0\) and define
By Itô’s formula, we have
Thus, \(K^f\) is a \(\mathbb {Q}\)-martingale, as it is a bounded local \(\mathbb {Q}\)-martingale. Recall that the quadratic variation process is not affected by an equivalent change of measure. By Step 1, \(\mathbb {Q}\)-a.s. \([B, M^g] = 0\). Due to integration by parts, we obtain that
which implies that \(K^f M^g_{\cdot \wedge \rho _m(\xi )}\) is a \(\mathbb {Q}\)-martingale, as it is a bounded local \(\mathbb {Q}\)-martingale.
Let \(\zeta \) be a stopping time such that \(\zeta \le T\) and set
Because \(K^f M^g_{\cdot \wedge \rho _m(\xi )}, K^f\) and \(M^f_{\cdot \wedge \rho _m(\xi )}\) are \(\mathbb {Q}\)-martingales (see also Step 2), the optional stopping theorem implies that for all stopping times \(\psi \le T\)
Consequently, by [47, Proposition II.1.4], \(M^g_{\cdot \wedge \rho _m(\xi )}\) is a \(\widetilde{\mathbb {Q}}\)-martingale. Because \(\widetilde{\mathbb {Q}} \sim \mathbb {Q}\), this implies that on \((\Omega , \mathcal {F}, \mathbf {F}, \widetilde{\mathbb {Q}})\) the process \(\xi \) is a solution process for the martingale problem \((A, L, j_0, T)\). The uniqueness assumption for the martingale problem \((A, L, j_0, T)\) implies that
for all
where \(G_1, \dots , G_n \in \mathcal {B}(J)\) and \(0 \le t_1< \dots < t_n \le T\). We fix \(\Gamma \) such that \(\mathbb {Q}(\Gamma ) > 0\) and define
Using the definition of \(\widetilde{\mathbb {Q}}\), (7.2), the fact that \(K^f\) is a \(\mathbb {Q}\)-martingale and the optional stopping theorem, we obtain
Because \(\zeta \) was arbitrary, we conclude that \(K^f\) is a \(\widehat{\mathbb {Q}}\)-martingale. Furthermore, \(\widehat{\mathbb {Q}}(B_0 = 0) = 1\) follows from the fact that \(B\) is a \(\mathbb {Q}\)-Brownian motion. Finally, due to [20, Proposition 4.3.3], the process \(B\) is a \(\widehat{\mathbb {Q}}\)-Brownian motion. We conclude that
for all \(F_1, \dots , F_k \in \mathcal {B}(\mathbb {R})\) and \(0 \le s_1< \dots < s_k \le T\). By the definition of \(\widehat{\mathbb {Q}}\), we have proven that
which implies that the \(\sigma \)-fields \(\sigma (\xi _t, t \in [0, T])\) and \(\sigma (B_t, t \in [0, T])\) are \(\mathbb {Q}\)-independent. The proof is complete. \(\square \)
8 Proof of Theorem 4.4
Because \(\sigma (\xi _t, t \in [0, T])\) and \(\sigma (W_t, t \in [0, T])\) are assumed to be \(\mathbb {P}\)-independent, it follows as in the proof of Theorem 4.3 that a.s. \([Z, W] = 0\). Thus, Girsanov’s theorem implies that \(W\) is a \(\mathbb {Q}\)-Brownian motion.
Take \(0 \le s_1< \dots< s_m \le T, 0 \le t_1< \dots < t_n \le T\),\((G_k)_{k \le m} \subset \mathcal {B}(J)\) and \((F_k)_{k \le n} \subset \mathcal {B}(\mathbb {R})\), and set
The \(\mathbb {P}\)-independence of \(\sigma (\xi _t, t \in [0, T])\) and \(\sigma (W_t, t \in [0, T])\) and the uniqueness of the Wiener measure yield that
We conclude that \(\sigma (\xi _t, t \in [0, T])\) and \(\sigma (W_t, t \in [0, T])\) are \(\mathbb {Q}\)-independent.
For \(g \in A^*\) we set
The processes \(K^f\) and \(K^{fg}\) are local \(\mathbb {P}\)-martingales. We set
Integration by parts implies that
Using again integration by parts and the identity \(L^* g = \frac{1}{f} (L (fg) - g Lf)\) yields
We conclude that \(Z M^g\) is a local \(\mathbb {P}\)-martingale and it follows from [30, Proposition III.3.8] that \(M^g\) is a local \(\mathbb {Q}\)-martingale. Due to the equivalence \(\mathbb {Q}\sim \mathbb {P}\), we conclude that on \((\Omega , \mathcal {F}, \mathbf {F}, \mathbb {Q})\) the process \(\xi \) is a solution process to the martingale problem \((A^*, L^*, j_0, T)\). This completes the proof.\(\square \)
9 Proof of Theorem 4.9
Let \((X_t)_{t \ge 0}\) be the coordinate process on \(D (\mathbb {R}_+, J)\) and denote
Define by \(\mu \triangleq \mathbb {P}\circ \xi ^{-1}\) a Borel probability measure on \(D (\mathbb {R}_+, J)\). We have to show that
It follows from [27, Lemma 2.9] that \(M^f\) is a local \(\mu \)-martingale with localizing sequence \((\rho _n)_{n \in \mathbb {N}}\). For all \(n \in \mathbb {N}\), define a Borel probability measure \(\mu _n\) on \(D (\mathbb {R}_+, J)\) via the Radon–Nikodym derivative
The following lemma is proven after the proof of Theorem 4.9 is complete.
Lemma 9.1
Let \(\mu ^*\) be the unique law of a solution process to the martingale problem \((A^*, L^*, j_0, \infty )\). For all \(n \in \mathbb {N}\) we have \(\mu _n = \mu ^*\) on \(\mathcal {D}^o_{T \wedge \rho _n}\).
Recalling that \(\{\rho _n > T\} \in \mathcal {D}^o_{T \wedge \rho _n}\), Lemma 9.1 implies that
This completes the proof. \(\square \)
Proof of Lemma 9.1
We adapt the proof of [30, Theorem III.2.40]. To simplify our notation, we set \(T \wedge \rho _n\triangleq \rho \). We denote by \(\mu _j\) the unique law of a solution process to the martingale problem \((A^*, L^*, j, \infty )\).
Step 1 We show that \(j \mapsto \mu _j (G)\) is Borel for all \(G \in \mathcal {D}\), following the strategy outlined in [54, Exercise 6.7.4]. Recall that we assume that \(A^*\) contains a countable determining set \(\widetilde{A}\). Let \(\mathcal {P}\) be the space of Borel probability measures on \(D(\mathbb {R}_+, J)\) equipped with the topology of convergence in distribution. Note that a \({\mathbf {D}}^o\)-adapted process is a \({\mathbf {D}}\)-martingale if and only if it is a \({\mathbf {D}}^o\)-martingale. The implication \(\Rightarrow \) follows from the downward theorem (see [48, Theorem II.51.1]) and the implication \(\Leftarrow \) follows from the tower rule. For \(g \in \widetilde{A}\) set
Define \(\mathcal {I}\) to be the set of all \(\nu \in \mathcal {P}\) such that \(\nu \circ X_0^{-1} = \delta _j\) for some \(j \in J\). Moreover, let \(\mathcal {M}\) be the set of all \(\nu \in \mathcal {P}\) such that
for all \(g \in \widetilde{A}\), all rational \(s < t, m \in \mathbb {N}\) and \(G\) in a countable determining class of \(\mathcal {D}^o_s\). By the uniqueness assumption, \(\{\mu _j, j \in J\} = \mathcal {I} \cap \mathcal {M}\). Because the set \(\{\delta _j, j \in J\}\) is Borel due to [8, Theorem 8.3.7] and \(\nu \mapsto \nu \circ X_0^{-1}\) is continuous, \(\mathcal {I}\) is Borel. The set \(\mathcal {M}\) is Borel due to [1, Theorem 15.13]. We conclude that \(\{\mu _j, j \in J\}\) is Borel. Let \(\Phi :\{\mu _j, j \in J\} \rightarrow J\) be defined by \(\Phi (\mu _j) = j\) for all \(j \in J\). We note that \(\Phi \) is a continuous injection. Thus, the inverse map \(\Phi ^{-1}\) is Borel due to Kuratovski’s theorem ([8, Proposition 8.3.5]). This means that \(j \mapsto \mu _j(G)\) is Borel for all \(G \in \mathcal {D}\).
Step 2 Because \(\mu _n \sim \mu \), we have \(\mu _n(X_0 = j_0) = 1\). As in the proof of Theorem 4.4, we see that for all \(g \in A^*\) the process \(K^g_{\cdot \wedge \rho }\) is a \(\mu _n\)-martingale.
Step 3 For every \(t \in \mathbb {R}_+\) we denote by \(\theta _t :D(\mathbb {R}_+, J) \rightarrow D(\mathbb {R}_+, J)\) the shift operator given by \(\theta _t \omega (s) = \omega (t + s)\). Recalling that \(\rho \) is bounded, we deduce from [30, Lemma III.2.44] that
Hence, we can associate to each \(G \in \mathcal {D}\) a (not necessarily unique) \(G' \in \mathcal {D}^o_{\rho } \otimes \mathcal {D}\) such that
We define
It follows from [30, Lemma III.2.47] that \(\nu \) is a probability measure, i.e. that \(\nu \) is defined unambiguously. Our goal is to show that \(\nu \) solves the martingale problem \((A^*, L^*, j_0, \infty )\). Providing an intuition, \(\nu \) is the law of
in case \(X^1\) is sampled according to \(\mu _n\) and \(X^2\) is sampled according to \(\mu _{j}\) with \(j = X^1_{\rho (X^1)}\). In other words, we extend \(\mu _n\) to a solution of the global martingale problem. For \(G \in \mathcal {D}^o_0\) we can choose \(G' = G \times D (\mathbb {R}_+, J)\). Consequently,
Let \(\psi \) be a bounded \({\mathbf {D}}^o\)-stopping time and fix \(m \in \mathbb {N}\). For \(\omega , \alpha \in D(\mathbb {R}_+, J)\) and \(t \in \mathbb {R}_+\) we set
and
Due to [18, Theorem IV.103] the map \(V\) is \(\mathcal {D}^o_\rho \otimes \mathcal {D}\)-measurable such that \(V(\omega , \cdot )\) is a \({\mathbf {D}}^o\)-stopping time for all \(\omega \in D(\mathbb {R}_+, J)\). Furthermore, it is evident from the definition that
for \(\omega \in D (\mathbb {R}_+, J)\). For all \(\omega \in \{\rho < \psi \wedge \rho _m\} \in \mathcal {D}_\rho ^o\) and \(\alpha \in D(\mathbb {R}_+, J)\) with \(\alpha (0) = \omega (\rho (\omega ))\) we have \( V(\omega , \alpha ) \le \rho _m(\alpha ). \) Note further that for \(\omega \in \{\rho < \psi \wedge \rho _m\}\)
Because \(K^g_{\cdot \wedge \rho }\) is a \(\mu _n\)-martingale, we have
due to the optional stopping theorem. Therefore, we obtain
again due to the optional stopping theorem (recall that \(V(\omega , \cdot )\) is bounded and that \(K^g_{\cdot \wedge \rho _m}\) is a \(\mu _j\)-martingale for all \(j \in J\)). We conclude from [47, Proposition II.1.4] that \(K^g_{\cdot \wedge \rho _m}\) is a \(\nu \)-martingale and hence that under \(\nu \) the coordinate process \((X_t)_{t \ge 0}\) solves the martingale problem \((A^*, L^*, j_0, \infty )\). The uniqueness assumption implies that \(\nu = \mu ^*\). Because also for \(G \in \mathcal {D}^o_\rho \) we can choose \(G' = G \times D(\mathbb {R}_+, J)\), we obtain that \( \mu ^* (G) = \nu (G) = \mu _n(G). \) The proof is complete. \(\square \)
Notes
In [25] the MLMM has been called minimal martingale measure. Because we distinguish between ELMMs and EMMs we adjust the terminology.
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Appendix A: Independence of Markov chains and Brownian motion
Appendix A: Independence of Markov chains and Brownian motion
Let \(0 <T \le \infty \) be a time horizon and let \((\Omega , \mathcal {F}, \mathbf {F}, \mathbb {P})\) be a complete probability space with right-continuous and complete filtration \(\mathbf {F}= (\mathcal {F}_t)_{t \in [0, T]}\), which supports a one-dimensional Brownian motion \(W = (W_t)_{t \in [0, T]}\) and a Feller–Markov chain \(\xi = (\xi _t)_{t \in [0, T]}\). We suppose that the initial value \(\xi _0\) of \(\xi \) is deterministic. Recall our convention that \(W\) is a Brownian motion for \(\mathbf {F}\) and that \(\xi \) is a Markov chain for \(\mathbf {F}\).
Lemma A.1
\(\sigma (W_t, t \in [0, T])\) and \(\sigma (\xi _t, t \in [0, T])\) are independent.
Proof
Let \(f \in C_0(J)\) be such that \(Q f \in C_0(J)\), where \(Q\) is the \(Q\)-matrix of \(\xi \). We set
Let \(g \in C^2_b(\mathbb {R})\) such that \(\inf _{x \in \mathbb {R}} g(x) > 0\) and set
Itô’s formula yields that \(K\) is a martingale. The process \(M\) is a martingale by Dynkin’s formula. Thus, integration by parts yields that
Because \(\xi \) has only finitely many jumps in a finite time interval, the process \(M\) is of finite variation over finite time intervals. Thus, since \(K\) has continuous paths, it follows that a.s. \([M, K] =0\). In view of (A.1), we conclude that \(MK\) is a martingale. Now, recalling Example 4.2, we can argue as in Step 3 of the proof of Theorem 4.3 to obtain the claimed independence. \(\square \)
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Criens, D. No arbitrage in continuous financial markets. Math Finan Econ 14, 461–506 (2020). https://doi.org/10.1007/s11579-020-00262-1
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DOI: https://doi.org/10.1007/s11579-020-00262-1
Keywords
- No arbitrage
- Financial bubble
- Minimal martingale measure
- Itô process
- Switching diffusion
- Stochastic exponential