On the hedging of options on exploding exchange rates

Abstract

We study a novel pricing operator for complete, local martingale models. The new pricing operator guarantees put-call parity to hold for model prices and the value of a forward contract to match the buy-and-hold strategy, even if the underlying follows strict local martingale dynamics. More precisely, we discuss a change of numéraire (change of currency) technique when the underlying is only a local martingale, modelling for example an exchange rate. The new pricing operator assigns prices to contingent claims according to the minimal cost for superreplication strategies that succeed with probability one for both currencies as numéraire. Within this context, we interpret the lack of the martingale property of an exchange rate as a reflection of the possibility that the numéraire currency may devalue completely against the asset currency (hyperinflation).

Appendices

Appendix A: Local martingales on stochastic intervals

In this appendix, we provide some technical results for stochastic processes that satisfy the local martingale property up to a stopping time. Such stochastic processes appear throughout this paper.

Similarly to Perkowski and Ruf [30], we consider the time set $\mathcal{T}:= [0,\infty] \cup\{\mathfrak{T}\}$, where $\mathfrak{T}$ represents a time “beyond horizon”; the natural ordering is extended to $\mathcal{T}$ by $t < \mathfrak{T}$ for all t∈[0,∞]. For any $t \in\mathcal{T}$ and any sequence $(t_{i})_{i \in \mathbb {N}}$ with $t_{i} \in\mathcal{T}$ for all $i \in \mathbb {N}$, we write lim _i↑∞ t _i =t if either $t = \mathfrak{T}$ and $\inf_{i \geq j} t_{i} = \mathfrak{T}$ for some $j \in \mathbb {N}$, or if $t < \mathfrak{T}$, $\sup_{i \geq j} t_{i} < \mathfrak{T}$ and lim _{i↑∞;i≥j} t _i =t for some $j \in \mathbb {N}$.

Throughout this appendix, we fix a time horizon T∈(0,∞], an arbitrary stochastic basis $(\varOmega, \mathcal {F}_{T}, (\mathcal {F}_{t})_{t \in[0, T]}, \mathbb {P})$, and a process N=(N _t ) _t∈[0,T] taking values in [−∞,∞]. For a $\mathcal{T}$-valued random variable τ, we define the stopped stochastic process $N^{\tau}= (N_{t}^{\tau})_{t \in[0,T]} :=(N_{t \wedge\tau})_{t \in[0,T]}$. Throughout this appendix, we fix a stopping time τ, which is a map $\tau: \varOmega\rightarrow\mathcal{T}$ such that $\{\tau\leq t\} \in \mathcal {F}_{t}$ for all t∈[0,T]. If not specified further, all (in)equalities are interpreted in the $\mathbb {P}$-almost sure sense.

We start with a definition.

Definition A.1

(Local martingale on a stochastic interval)

We call N:

• (1) a local martingale on [0,τ] if there exists a nondecreasing sequence of stopping times $(\tau_{i})_{i \in \mathbb {N}}$ with lim _i↑∞ τ _i >τ∧T such that $N^{\tau_{i} \wedge\tau}$ is a martingale for all $i \in \mathbb {N}$;

• (2) a local martingale on [0,τ) if there exists a nondecreasing sequence of stopping times $(\tau_{i})_{i \in \mathbb {N}}$ with lim _i↑∞ τ _i =τ such that $N^{\tau_{i}}$ is a martingale for all $i \in \mathbb {N}$.

In particular, if T=τ=∞, then a local martingale on [0,τ) corresponds exactly to the usual notion of a local martingale. Observe that if N is a local martingale on [0,τ], then it is a local martingale on [0,τ). If the definition of a local martingale on [0,τ) required additionally the assumption that τ _i <τ for all $i \in \mathbb {N}$ (something that Definition A.1 does not require), this implication would in general not hold true; consider for example a compensated Poisson process and τ the time of its first jump. Observe also that if $\widetilde{\tau}$ is a stopping time with $\widetilde{\tau} \wedge T < \epsilon\vee\tau$ for all ϵ>0, then any local martingale on [0,τ) is also a local martingale on $[0,\widetilde{\tau}]$.

In the following, we repeatedly use the fact that

$$N^{{\eta_1} \vee\eta_2} = N^{{\eta_1} } + N^{ \eta_2} - N^{{\eta_1} \wedge\eta_2} $$

is a (local) martingale if $N^{{\eta_{1}} }$ and $N^{ \eta_{2}}$ are (local) martingales for some stopping times η ₁ and η ₂. The next lemma is useful in several of the proofs in this paper.

Lemma A.2

(Localization sequence for a local martingale on a stochastic interval)

The following two statements are equivalent:

• (a.1) N is a local martingale on [0,τ].

• (a.2) There exists a nondecreasing sequence of stopping times $(\tau_{i})_{i \in \mathbb {N}}$ such that lim _i↑∞ τ _i >τ∧T and that N is a local martingale on [0,τ _i ∧τ] for all $i \in \mathbb {N}$.

The following three statements are equivalent:

• (b.1) N is a local martingale on [0,τ).

• (b.2) There exists a nondecreasing sequence of stopping times $(\tau_{i})_{i \in \mathbb {N}}$ such that lim _i↑∞ τ _i =τ and that N is a local martingale on [0,τ _i ] for all $i \in \mathbb {N}$.

• (b.3) There exists a nondecreasing sequence of stopping times $(\tau_{i})_{i \in \mathbb {N}}$ such that lim _i↑∞ τ _i =τ and that N is a local martingale on [0,τ _i ) for all $i \in \mathbb {N}$.

Proof

For the first part, we only need to show the implication from (a.2) to (a.1). Thus, assume (a.2), which yields that there exists for all $i \in \mathbb {N}$ a stopping time η _i with $\mathbb {P}({\eta}_{i} \leq\tau_{i} \wedge\tau\wedge T) \leq2^{-i}$ such that $N^{{\eta}_{i} \wedge{\tau}_{i} \wedge\tau}$ is a martingale. For all $i \in \mathbb {N}$, define $\widetilde{\tau}_{i} = \max_{j \in\{1, \ldots, i\}} ({\eta}_{j} \wedge{\tau}_{j})$ and observe that $N^{\widetilde{\tau}_{i} \wedge\tau}$ is a martingale for all $i \in \mathbb {N}$ and that $\lim_{i \uparrow\infty} \widetilde {\tau}_{i} > \tau\wedge T$. This shows that (a.1) holds.

For the second part, we only need to show the implication from (b.3) to (b.1). Assume now (b.3). Then there exists a nondecreasing sequence of stopping times $({\eta}_{i})_{i \in \mathbb {N}}$ such that

$$\mathbb {P}\bigl( (\{\tau_i = \mathfrak{T}\} \cap\{ \eta_i < \mathfrak{T}\} ) \cup (\{\tau_i < \mathfrak{T}\} \cap \{\eta_i < (\tau_i - 2^{-i}) \wedge i \} ) \bigr) \leq2^{-i} $$

and $N^{\eta_{i}}$ is a martingale for all $i \in \mathbb {N}$. Define $\widetilde{\tau}_{i} := \tau\wedge\max_{j \in\{1, \ldots, i\}} \eta_{i}$ and observe that $N^{\widetilde{\tau}_{i}}$ is a martingale for all $i \in \mathbb {N}$ and that $\lim_{i \uparrow\infty} \widetilde{\tau}_{i} = \tau$, which yields (b.1). □

For the next lemma, observe that the random times

$$\begin{aligned} \rho_j := \inf \{t \in[0,T] : N_t^\tau> j \} \end{aligned}$$

(A.1)

with $\inf\emptyset:= \mathfrak{T}$ for all $j \in \mathbb {N}$ take values in $[0,\tau\wedge T] \cup\mathfrak{T}$ and are stopping times if the underlying filtration $(\mathcal {F}_{t})_{t \in[0,T]}$ is right-continuous and N(ω) is a right-continuous path for all ω∈Ω; see for example Problem 1.2.6 in Karatzas and Shreve [22].

Lemma A.3

(Localization sequence for a nonnegative local martingale)

Assume that the underlying filtration $(\mathcal {F}_{t})_{t \in[0,T]}$ is right-continuous and N(ω) is a right-continuous path taking values in [0,∞] for all ω∈Ω. Define the stopping times $(\rho_{j})_{j \in \mathbb {N}}$ as in (A.1) and ρ:=lim _j↑∞ ρ _j . Then the following statements hold:

• (i) If $N^{\rho_{j} \wedge\tau}$ is a supermartingale for all $j \in \mathbb {N}$ (in particular, if N ^τ is a supermartingale), then $\rho= \mathfrak{T}$.

• (ii) If $N^{\rho_{j} \wedge\tau_{i}^{(j)}}$ is a supermartingale for all $i,j \in \mathbb {N}$ for some nondecreasing sequences of stopping times $(\tau_{i}^{(j)})_{i \in \mathbb {N}}$ with $\lim_{i \uparrow\infty} \tau_{i}^{(j)} = \tau$ for all $j \in \mathbb {N}$, then ρ≥τ.

The following statements are equivalent:

• (a.1) N is a local martingale on [0,τ].

• (a.2) $N^{\rho_{j} \wedge\tau}$ is a uniformly integrable martingale for all $j \in \mathbb {N}$.

• (a.3) $N^{\rho_{j}}$ is a local martingale on [0,τ] for all $j \in \mathbb {N}$.

The following statements are equivalent:

• (b.1) N is a local martingale on [0,τ).

• (b.2) $N^{\rho_{j} \wedge\tau_{i}}$ is a uniformly integrable martingale for all $i,j \in \mathbb {N}$ for some nondecreasing sequence of stopping times $(\tau_{i})_{i \in \mathbb {N}}$ with lim _i↑∞ τ _i =τ.

• (b.3) $N^{\rho_{j}}$ is a local martingale on [0,τ) for all $j \in \mathbb {N}$.

Proof

Assume that $N^{\rho_{j} \wedge\tau}$ is a nonnegative supermartingale and note that $N^{\rho_{j} \wedge\tau}_{T} \geq j$ if ρ _j ≤τ∧T; thus $\mathbb {P}(\rho_{j} \leq\tau\wedge T) \leq N_{0} / j$ for all $j \in \mathbb {N}$, which yields (i). Next, assume that there exist sequences of stopping times $(\tau _{i}^{(j)})_{i \in \mathbb {N}}$ such that $N^{\rho_{j} \wedge\tau_{i}^{(j)}}$ is a supermartingale for all $i,j \in \mathbb {N}$. Fix a sequence (i _j ) _j∈N so that

$$\mathbb {P}\bigl( (\{\tau= \mathfrak{T}\} \cap \{\tau_{i_j}^{(j)} < \mathfrak {T} \} ) \cup (\{\tau< \mathfrak{T}\} \cap \{ \tau_{i_j}^{(j)} < (\tau- 2^{-j} ) \wedge j \} ) \bigr) \leq2^{-j}. $$

Then we have $N_{0} \geq \mathbb {E}[ N^{\rho_{j} \wedge\tau_{i_{j}}^{(j)}}_{T}] \geq j \mathbb {P}(\rho_{j} \leq\tau_{i_{j}}^{(j)} \wedge T)$ and thus

$$\mathbb {P}\bigl( (\{\tau= \mathfrak{T}\} \cap\{\rho_j < \mathfrak{T}\} ) \cup (\{\tau< \mathfrak{T}\} \cap \{ \rho_j < (\tau- 2^{-j} ) \wedge j \} ) \bigr) \leq\frac{N_0}{j} + 2^{-j} $$

for all $j \in \mathbb {N}$, which yields (ii).

Now assume (a.1) and observe that $\sup_{t \in[0,T]} N_{t}^{\rho_{j} \wedge\tau} \leq j + N_{T}^{\rho_{j} \wedge\tau}$ and that $N_{T}^{\rho_{j} \wedge\tau}$ is integrable for all $j\in \mathbb {N}$ since N ^τ is a supermartingale. This observation in conjunction with dominated convergence shows (a.2). Next, assume (a.3) and observe that $N^{\rho _{j}}$ is a supermartingale on [0,τ] for all $j \in \mathbb {N}$, and thus $\rho=\mathfrak{T}$ by (i). The first part of Lemma A.2 then yields (a.1).

Now assume (b.1), which gives the existence of a nondecreasing sequence of stopping times $(\tau_{i})_{i \in \mathbb {N}}$ with lim _i↑∞ τ _i =τ such that N is a local martingale on [0,τ _i ] for all $i \in \mathbb {N}$. Using the implication of (a.1) to (a.2) with τ replaced by τ _i for all $i \in \mathbb {N}$, we observe that (b.2) holds. Next, assume (b.3). Then (ii) yields that ρ≥τ and the second part of Lemma A.2 then yields (b.1). □

Note that the implication of (b.3) to (b.1) in Lemma A.3 with T=τ=∞ yields that any nonnegative right-continuous process N is automatically a local martingale (on [0,∞)) if $N^{\rho_{j}}$ is a local martingale (on [0,∞)) for all $j \in \mathbb {N}$. Furthermore, by (ii), $N^{\rho_{j}}$ is a supermartingale for all $j \in \mathbb {N}$ if and only if N is a supermartingale.

We call the stopping time τ foretellable if there exists a nondecreasing sequence of stopping times $(\tau_{i})_{i \in \mathbb {N}}$ such that lim _i↑∞ τ _i =τ (in particular, there exists some $i(\omega) \in \mathbb {N}$ with $\tau_{i(\omega)}(\omega) = \mathfrak{T}$ if $\tau(\omega) = \mathfrak{T}$) and τ _i ∧T<τ∨ϵ for all $i \in \mathbb {N}$ and ϵ>0. We then call $(\tau_{i})_{i \in \mathbb {N}}$ an announcing sequence for τ.

The following result illustrates that a nonnegative local martingale on a half-open stochastic interval (with respect to a foretellable stopping time) can be extended to one on a closed interval. For example, if N is defined by N _t :=1 _{t<τ} for all t∈[0,T], then N can be extended to a process M=(M _t ) _t∈[0,T] with M _t :=1 for all t∈[0,T], representing a local martingale on [0,T].

Proposition A.4

(Extension of local martingales on a stochastic interval)

Suppose that the assumptions of Lemma A.3 hold and assume that τ is foretellable and that N is a local martingale on [0,τ). Then there exists a unique local martingale M=(M _t ) _t∈[0,T] on [0,T] such that M=M ^τ, (M _t 1 _{t<τ}) _t∈[0,T]=(N _t 1 _{t<τ}) _t∈[0,T], M ₀=N ₀ and, moreover, lim _s↑τ(ω) M _s (ω)=M _t (ω) for all ω∈Ω with $\tau(\omega) \notin\{ 0,\mathfrak{T}\}$. The process M has nonnegative and right-continuous paths.

Proof

The uniqueness of M follows directly from its left-continuity at time τ. Let $(\tau_{i})_{i \in \mathbb {N}}$ denote an announcing sequence for τ and $(\widetilde{\tau}_{i})_{i \in \mathbb {N}}$ a nondecreasing sequence of stopping times such that $N^{\widetilde{\tau}_{i}}$ is a martingale for all $i \in \mathbb {N}$ and $\lim_{i \uparrow\infty} \widetilde{\tau}_{i} = \tau$. We assume without loss of generality that $\tau_{i} = \tau_{i} \wedge \widetilde{\tau}_{i}$ for all $i \in \mathbb {N}$. Observe that $N^{\tau_{i}}$ is a nonnegative supermartingale for all $i \in \mathbb {N}$. By imitating the proof of Theorem 1.3.15 in Karatzas and Shreve [22] based on Doob’s up- and down-crossing inequalities (replace therein n by τ _n for all $n \in \mathbb {N}$ and ∞ by τ), we obtain that $M_{t} := \lim_{i \uparrow\infty} N_{t}^{\tau_{i}}$ for all t∈[0,T] exists.

We need to show that M defined in this way is a local martingale on [0,T]. By Lemma A.3, it suffices to show that $M^{\widetilde{\rho}_{j}}$ is a martingale for all $j\in \mathbb {N}$, where $\widetilde{\rho}_{j} := \inf\{t \in[0,T] : {M}_{t} > j\}$ with $\inf\emptyset:= \mathfrak{T}$. Fix an arbitrary $j \in \mathbb {N}$ and observe that by dominated and monotone convergence

$$\begin{aligned} \mathbb {E}\bigl[M_T^{\widetilde{\rho}_j} \bigr] =& \mathbb {E}\Bigl[\lim _{i \uparrow\infty }N_T^{\tau_i} \mathbf {1}_{\{\tau_i< \widetilde{\rho}_j\}} \Bigr] + \mathbb {E}\Bigl[ \lim_{i \uparrow\infty} N_T^{\widetilde{\rho}_j} \mathbf {1}_{\{\tau_i \geq \widetilde{\rho}_j\}} \Bigr] = \lim_{i \uparrow\infty} \mathbb {E}\bigl[N_T^{{\widetilde{\rho}}_j \wedge\tau_i} \bigr], \\ =& N_0 = M_0, \end{aligned}$$

which yields the statement since by Fatou’s lemma, $M^{\widetilde{\rho }_{j}}$ is a supermartingale. □

We warn the reader that usually $M_{T}^{\tau}\neq N^{\tau}_{T}$, even if N is a martingale on [0,τ], since N need not be left-continuous at τ. We also refer the reader to the related Exercise IV.1.48 in Revuz and Yor [32], where the case of not necessarily nonnegative local martingales is treated.

Appendix B: Conditions on the filtration in Sects. 2–4

In this appendix, we discuss the technical assumptions on the underlying filtration that are necessary for the results in Sects. 2–4. Throughout this appendix, we fix a time horizon T∈(0,∞] and denote a set of states by Ω≠∅ and a filtration by $({\mathcal {F}}_{t})_{t \in[0,T]}$.

We refer to Appendix A for the definition of a stopping time. For any stopping time τ, we define

$${\mathcal {F}}_{\tau} := \bigl\{ A \in{ \mathcal {F}}_T \, \big|\, A \cap\{\tau\leq t\} \in{ \mathcal {F}}_t \text{ for all } t \in[0,T] \bigr\} $$

and

$${\mathcal {F}}_{\tau-} := \sigma \bigl( \bigl\{ A \cap\{\tau> t\}\, \big|\, A \in { \mathcal {F}}_t \text{ for some } t \in[0,T] \bigr\} \cup{ \mathcal {F}}_0^0 \bigr), $$

if $({\mathcal {F}}_{t})_{t \in[0,T]}$ is the right-continuous modification of a filtration $({\mathcal {F}}_{t}^{0})_{t \in[0,T]}$; see p. 156 in Föllmer [12].

In Sect. 2, we are constructing a probability measure on $(\varOmega, \mathcal {F}_{R-})$ for a certain stopping time R:=lim _i↑∞ R _i , where $(R_{i})_{i \in \mathbb {N}}$ is a sequence of nondecreasing stopping times, defined in Sect. 2. This construction is based on an extension theorem, more precisely on Theorem V.4.1 in Parthasarathy [29], and thus requires certain technical assumptions on the filtration $({\mathcal {F}})_{t \in[0,T]}$. Specifically, we require in Sects. 2–4 that

• (i) $({\mathcal {F}}_{t})_{t \in[0,T]}$ be the right-continuous modification of a filtration $({\mathcal {F}}_{t}^{0})_{t \in[0,T]}$, and

• (ii) $({\mathcal {F}}_{R_{i}-})_{i \in \mathbb {N}}$ be a standard system, as defined in Sect. 6 of Föllmer [12].

Furthermore, in Sect. 3, we require that

• (iii) any probability measure $\mathbb {P}$ on $(\varOmega, \mathcal {F}_{R-})$ can be extended to a probability measure $\widetilde{\mathbb {P}}$ on $(\varOmega , \mathcal {F}_{T})$.

A sufficient condition for requirement (ii) is that

$$(\widehat{\mathcal {F}}_{t})_{t \in[0,T]} := ({\mathcal {F}}_t \cap{ \mathcal {F}}_{R-})_{t \in[0,T]} $$

is the right-continuous modification of a standard system (RCMSS); see Remark 6.1.1 in Föllmer [12], applied to the filtration $(\mathcal{G}_{t})_{t \geq0}$ with $\mathcal{G}_{t} := {\widehat {\mathcal {F}}}_{1/(1-t) - 1}$ if T=∞, and $\mathcal{G}_{t} := {\widehat{\mathcal {F}}}_{t T}$ otherwise, for all t∈[0,1], and $\mathcal{G}_{t} = {\widehat{\mathcal {F}}}_{T}$ for all t>1. We remark that $(\widehat{\mathcal {F}}_{t})_{t \in[0,T]}$ then usually does not satisfy the “usual conditions” as it is not completed under some probability measure. Observe that if $({\mathcal {F}}_{t}^{0})_{t \in[0,T]}$ is a standard system, then so is $({\mathcal {F}}^{0}_{t} \cap{ \mathcal {F}}_{R-})_{t \in[0,T]}$.

In the following, we provide a canonical example for Ω and for a filtration $({\mathcal {F}}_{t})_{t \in[0,T]}$, such that $({\mathcal {F}}_{t} \cap{ \mathcal {F}}_{R-})_{t \in[0,T]}$ is an RCMSS. This example provides a sufficiently rich structure so that one might as well assume, throughout this paper, that the underlying filtered measurable space is of that form.

Towards this end, let E denote any locally compact space with countable base (for instance, $E=\mathbb {R}^{n}$ for some $n \in \mathbb {N}$), and let Ω denote the space of right-continuous paths ω:[0,T]→[0,∞]×E whose first component ω ⁽¹⁾ satisfies ω ⁽¹⁾(R(ω)+t)=∞ for all t≥0, and that have left limits on (0,R(ω)), where R(ω) denotes the first time that ω ⁽¹⁾=∞. Let $({\mathcal {F}}_{t}^{0})_{t \in[0,T]}$ denote the filtration generated by the paths and $(\mathcal {F}_{t})_{t \in [0,T]}$ its right-continuous modification. Then it follows as in Dellacherie [10], Meyer [27] and Example 6.3.2 of Föllmer [12] that $({\mathcal {F}}_{t} \cap{ \mathcal {F}}_{R-})_{t \in[0,T]}$ is an RCMSS. We identify the process X(ω), which appears in Sect. 2, with the first coordinate of ω.

Observe that in the canonical setup of the last paragraph, the extension of requirement (iii) always exists. To see this, define $\widetilde{\mathbb {P}}(A) := \mathbb {P}(\omega^{R-} \in A)$ for all $A \in \mathcal {F}_{T}$, where ω ^R−∈Ω is given for all ω∈Ω by

$$\omega^{R-}(t) := \omega(t) \mathbf {1}_{\{t < R(\omega)\}} + (\infty\times e) \mathbf {1}_{\{t \geq R(\omega)\}} $$

for some e∈E for all t∈[0,T]. This specific construction then yields one extension $\widetilde{\mathbb {P}}$ on $(\varOmega, \mathcal {F}_{T})$.

Appendix C: Proof of Proposition 2.3 and further statements concerning the change of measure in Sect. 2

In this appendix, we provide additional statements on the change of measure suggested in Sect. 2 and on the proof of Proposition 2.3. We refer to Appendix A for the definition of a stopping time.

Below, we shall rely on the next lemma:

Lemma C.1

(Convergence of stopping times)

Assume the setup of Theorem 2.1 and fix a stopping time τ. Then we have $\mathbb {Q}(S>\tau) = 0$ if and only if $\widehat{\mathbb {Q}}(R>\tau) = 0$.

Proof

Without loss of generality, we set x ₀=1. Then (2.1) yields that

$$\widehat{\mathbb {Q}}(R>\tau) = \widehat{\mathbb {Q}} (\{\tau\leq T\} \cap{\{R > \tau\wedge T \}} ) = \mathbb {E}^{{\mathbb {Q}}} \bigl[\mathbf {1}_{\{\tau\leq T\}} X^\tau _T \bigr] \leq \mathbb {E}^{{\mathbb {Q}}} \bigl[\mathbf {1}_{\{S > \tau\}} X^\tau_T \bigr], $$

which yields one direction of the statement. The other direction follows from (2.3) in the same manner. □

Next, we formulate a generalized version of Bayes’ formula. If X is a $\mathbb {Q}$-martingale, this formula is well known; see for example Lemma 3.5.3 in Karatzas and Shreve [22]. If X is a strictly positive continuous $\mathbb {Q}$-local martingale, Bayes’ formula has been derived in Ruf [33].

Proposition C.2

(Bayes’ formula)

Assume the setup of Theorem 2.1 and fix two stopping times ρ,τ with ρ≤τ $\mathbb {Q}$- and $\widehat{\mathbb {Q}}$-almost surely and an $\mathcal {F}_{\tau\wedge T}$-measurable random variable Z∈[0,∞]. Then we have the Bayes’ formula

$$\begin{aligned} \mathbb {E}^{\widehat{\mathbb {Q}}} \bigl[ Z \mathbf {1}_{\{R > \tau\wedge T\}}Y^\tau_T \, \big|\, \mathcal {F}_{\rho} \bigr] \mathbf {1}_{\{S > \rho\wedge T\}} &= \mathbb {E}^{\mathbb {Q}} \bigl[ Z \mathbf {1}_{\{S > \tau\wedge T\}} \, \big|\, \mathcal {F}_{\rho } \bigr] \mathbf {1}_{\{R>\rho\wedge T\}} Y^\rho_T \\ & = \mathbb {E}^{\mathbb {Q}} \bigl[ Z \mathbf {1}_{\{S>\tau\wedge T\}} \, \big|\, \mathcal {F}_{\rho } \bigr] \mathbf {1}_{\{R>\rho\wedge T\}} Y^\rho_T \mathbf {1}_{\{S>\rho\wedge T\}} . \end{aligned}$$

(C.1)

This equality holds $\mathbb {Q}$- and $\widehat{\mathbb {Q}}$-almost surely.

Proof

Without loss of generality, assume that x ₀=1. Then (C.1) holds $\widehat{\mathbb {Q}}$-almost surely since $\widehat{\mathbb {Q}}(S > \rho\wedge T) = 1$ and for all $A \in \mathcal {F}_{\rho}$,

$$\begin{aligned} \mathbb {E}^{\widehat{\mathbb {Q}}} \bigl[\mathbf {1}_A Z \mathbf {1}_{\{R>\tau\wedge T\}} Y^\tau_T \bigr] &= \mathbb {E}^{\mathbb {Q}} \bigl[\mathbf {1}_A Z \mathbf {1}_{\{S>\tau\wedge T\}} \bigr] = \mathbb {E}^{\mathbb {Q}} \bigl[\mathbf {1}_A \mathbb {E}^{\mathbb {Q}} [Z \mathbf {1}_{\{S>\tau\wedge T\}} \, |\, \mathcal {F}_\rho ] \bigr] \\ &= \mathbb {E}^{\widehat{\mathbb {Q}}} \bigl[\mathbf {1}_A \mathbb {E}^{\mathbb {Q}} [Z \mathbf {1}_{\{S>\tau\wedge T\} }\, |\,\mathcal {F}_\rho ] \mathbf {1}_{\{R>\rho\wedge T\}} Y^\rho_T \bigr]. \end{aligned}$$

Moreover, (C.1) holds $\mathbb {Q}$-almost surely since $\mathbb {Q}(R>\rho \wedge T) = 1$ and

$$\begin{aligned} \mathbb {E}^{\mathbb {Q}} \bigl[\mathbf {1}_A Z \mathbf {1}_{\{S>\tau\wedge T\}} Y_T^\rho \bigr] &= \mathbb {E}^{\widehat{\mathbb {Q}}} \bigl[ \mathbf {1}_A Z \mathbf {1}_{\{R > \tau\wedge T\}} Y^\rho_T Y^\tau _T \bigr] \\ &= \mathbb {E}^{\widehat{\mathbb {Q}}} \bigl[\mathbf {1}_A \mathbb {E}^{\widehat{\mathbb {Q}}} [ Z \mathbf {1}_{\{R > \tau\wedge T\}} Y^\tau_T \, |\, \mathcal {F}_\rho ] {Y^\rho_T} \bigr] \\ &= \mathbb {E}^{\mathbb {Q}} \bigl[\mathbf {1}_A \mathbb {E}^{\widehat{\mathbb {Q}}} [ Z \mathbf {1}_{\{R > \tau\wedge T\}} Y^\tau_T \, |\,\mathcal {F}_\rho ] \mathbf {1}_{\{S>\rho\wedge T\}} \bigr] \end{aligned}$$

for all $A \in \mathcal {F}_{\rho}$. □

Bayes’s formula yields a simple proof of Proposition 2.3:

Proof of Proposition 2.3

The statement in (i) is a corollary of Proposition C.2 if we replace τ by τ∧t and use $Z=N_{t}^{\tau}$ and ρ=τ∧s in (C.1) for all t∈[0,T] and s∈[0,t].

Assume now that (N _t 1 _{S>t}) _t∈[0,T] is a $\mathbb {Q}$-local martingale on [0,S). Then there exists a nondecreasing sequence of stopping times $(\tau_{i})_{i \in \mathbb {N}}$ with $\mathbb {Q}(\lim_{i \uparrow\infty} \tau_{i} = S) = 1$ and such that $(N^{\tau_{t}} \mathbf {1}_{\{S > \tau_{i} \wedge t\}})_{t \in [0,T]}$ is a $\mathbb {Q}$-martingale for all $i \in \mathbb {N}$. Now, (i) implies that $N^{\tau_{i}} Y^{\tau_{i}}$ is a $\widehat{\mathbb {Q}}$-martingale. An application of Lemma C.1 with τ:=lim _i↑∞ τ _i yields that NY is a $\widehat{\mathbb {Q}}$-local martingale on [0,R). The reverse direction follows in the same manner. This yields (ii).

Assume next that $(N_{t}^{S_{i}} \mathbf {1}_{\{S>S_{i} \wedge t\}})_{t \in [0,T]}$ and thus $(N_{t}^{S_{i}^{Y}} \mathbf {1}_{\{S>S_{i}^{Y} \wedge t\}})_{t \in [0,T]}$ are $\mathbb {Q}$-martingales for all $i \in \mathbb {N}$. Then the statement in (iii) follows from (i) and the fact that $\widehat{\mathbb {Q}}(\lim_{i \uparrow\infty} S^{Y}_{i} > T) = 1$ by (i) in Lemma A.3. □

We conclude this appendix by providing a Girsanov-type result. Towards this end, denote the quadratic covariation process of two $\mathbb {Q}$-semimartingales N ⁽¹⁾ and N ⁽²⁾ with càdlàg paths by [N ⁽¹⁾,N ⁽²⁾]=([N ⁽¹⁾,N ⁽²⁾] _t ) _t∈[0,T]. If X has càdlàg paths, the process $N^{S_{i}}$ is a $\mathbb {Q}$-semimartingale with càdlàg paths and $[N,X]^{S_{i}} := [N^{S_{i}},X]$ has $\mathbb {Q}$-integrable variation for all $i \in \mathbb {N}$, then the quadratic covariation process [N,X] has a compensator “up to time S”, that is, there exists a process 〈N,X〉=(〈N,X〉 _t ) _t∈[0,T] such that $\langle N,X\rangle^{S_{i}}$ is the compensator of $[N,X]^{S_{i}}$ for all $i \in \mathbb {N}$; see Theorem III.3.11 of Jacod and Shiryaev [20]. For any càdlàg stochastic process Z=(Z _t ) _t∈[0,T], we define Z _t−:=lim _s↑t Z _s for all t∈(0,T) and Z ₀₋:=Z ₀.

Proposition C.3

(Girsanov-type theorem)

Assume the setup of Theorem 2.1 and let N=(N _t ) _t∈[0,T] denote a progressively measurable stochastic process taking values in [−∞,∞] such that N _t =N _t 1 _{R>t} for all t∈[0,T] and such that $N^{S_{i}}$ is a $\mathbb {Q}$-semimartingale with càdlàg paths for all $i \in \mathbb {N}$. Suppose that X has càdlàg paths. We then have the following statements:

• (i) The process $N^{R_{i}}$ is a $\widehat{\mathbb {Q}}$-semimartingale with càdlàg paths for all $i \in \mathbb {N}$.

• (ii) If N is a $\mathbb {Q}$-local martingale on [0,S) (equivalently, on [0,R∧S)) and if $[N,X]^{S_{i}}$ has $\mathbb {Q}$-integrable variation for all $i \in \mathbb {N}$, then $\widetilde{N} =(\widetilde{N} _{t})_{t \in [0,T]}$ defined by

  $$\begin{aligned} \widetilde{N} _t := N_t - \int_0^{t} Y_{s-} \,\mathrm {d}\langle N,X \rangle _s\quad\textit{for all } t \in[0,T] \end{aligned}$$

  is a $\widehat{\mathbb {Q}}$-local martingale on [0,R) (equivalently, on the interval [0,R∧S)).

• (iii) If N is a $\mathbb {Q}$-local martingale on [0,S) (equivalently, on [0,R∧S)) and if we have $\mathbb {Q}(S>S_{i} \wedge T) = 1$ for all $i \in \mathbb {N}$, then $\widehat{N} =(\widehat{N} _{t})_{t \in [0,T]}$ defined by

  $$\begin{aligned} \widehat{N} _t := N_t - \int _0^{t \wedge S} Y_s \,\mathrm {d}[ N,X ]_s\quad \textit{for all } t \in[0,T] \end{aligned}$$

  is a $\widehat{\mathbb {Q}}$-local martingale on [0,R) (equivalently, on the interval [0,R∧S)).

Proof

The proof is based on a simple localization argument. Observe that

$$\mathrm {d}\widehat{\mathbb {Q}} \,\big|_{\mathcal {F}_{R_i} \cap \mathcal {F}_{R-}} = X^{R_i}_T \,\mathrm {d}{\mathbb {Q}} \,\big|_{\mathcal {F}_{R_i} \cap \mathcal {F}_{R-}}, $$

so that $\widehat{\mathbb {Q}}$ is absolutely continuous with respect to ${\mathbb {Q}}$ on $\mathcal {F}_{R_{i}} \cap \mathcal {F}_{R-}$ for all $i \in \mathbb {N}$. Thus (i) corresponds directly to Theorem III.3.13 in Jacod and Shiryaev [20]. By Theorems III.3.11 in Jacod and Shiryaev [20], the process $\widetilde{N}^{R_{i}}$ is a $\widehat{\mathbb {Q}}$-local martingale; thus $\widetilde{N}$ is a $\widehat{\mathbb {Q}}$-local martingale on [0,R _i ) for all $i \in \mathbb {N}$. Lemma A.2 then yields (ii). Similar reasoning yields that $\widehat{N}$ is a $\widehat{\mathbb {Q}}$-local martingale on [0,R) by applying Theorem 3 in Lenglart [24], after observing that the proof therein also works for probability spaces that do not satisfy the usual assumptions. □

Remark C.4

(Lack of martingale property in Proposition C.3)

One might wonder whether (ii) or (iii) of Proposition C.3 can be strengthened by replacing each “local martingale” by “martingale”. Example 2.5 illustrates that such a statement would be false, even in the case of X being a strictly positive, true $\mathbb {Q}$-martingale. To see this, replace $\widehat{\mathbb {Q}}$ by $\mathbb {Q}^{Z}$ and the processes N by X and X by Z in Proposition C.3. Then N is a true $\mathbb {Q}$-martingale, but $\widetilde{N} = \widehat{N} = N$ is only a strict $\mathbb {Q}^{Z}$-local martingale.

Appendix D: Proof of Lemma 4.1

In this appendix, we provide the

Proof of Lemma 4.1

The fact that (ii) implies (i) follows directly from (2.2) and (2.3) with Z=1 _A and τ=R _i ∧S _j for all $A \in \mathcal {F}_{R_{i} \wedge S_{j}}$ and $i,j \in \mathbb {N}$, since .

For the reverse direction, fix a stopping time τ and note that it suffices to show (2.1) for events $A \in \mathcal {F}_{\tau\wedge T}$ that satisfy

$$A = A \cap\{R > \tau\wedge T\} \cap\{S > \tau\wedge T\}, $$

since as Y is a -local martingale and thus cannot explode. Therefore, we may assume without loss of generality that $A \in \mathcal {F}_{(R \wedge S)-}$. Let $\widehat{ \mathbb {Q}^{\$}}$ denote the unique probability measure on $(\varOmega, \mathcal {F}_{R-})$ that was constructed in Theorem 2.1 with $\mathbb {Q}$ replaced by $\mathbb {Q}^{\$}$. We need to show the identity .

Since $\bigcup_{i,j \in \mathbb {N}} \mathcal {F}_{(R_{i} \wedge S_{j})-}$ is a π-system that generates $\mathcal {F}_{(R \wedge S)-}$, it suffices to show that for all $i,j \in \mathbb {N}$. Next, fix $i, j \in \mathbb {N}$ and note that by (i), $\widehat{ \mathbb {Q}^{\$}}$ and are equivalent on $\mathcal {F}_{(R_{i} \wedge S_{j})-}$. Therefore, the -martingale Z=(Z _t ) _t∈[0,T] with for all t∈[0,T] is well defined. We need to show that Z _T =1. Observe that the measure defined by is also equivalent to and the processes S ^€ are -local martingales; see also Proposition 2.3. Since was assumed to be unique among these measures, we conclude. □