# Market completion with derivative securities

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## Abstract

Let \(S^{F}\) be a ℙ-martingale representing the price of a primitive asset in an incomplete market framework. We present easily verifiable conditions on the model coefficients which guarantee the completeness of the market in which in addition to the primitive asset, one may also trade a derivative contract \(S^{B}\). Both \(S^{F}\) and \(S^{B}\) are defined in terms of the solution \(X\) to a two-dimensional stochastic differential equation: \(S^{F}_{t} = f(X_{t})\) and \(S^{B}_{t}:=\mathbb{E}[g(X_{1}) | \mathcal{F}_{t}]\). From a purely mathematical point of view, we prove that every local martingale under ℙ can be represented as a stochastic integral with respect to the ℙ-martingale \(S :=(S^{F}, S^{B})\). Notably, in contrast to recent results on the endogenous completeness of equilibria markets, our conditions allow the Jacobian matrix of \((f,g)\) to be singular everywhere on \(\mathbb{R}^{2}\). Hence they cover as a special case the prominent example of a stochastic volatility model being completed with a European call (or put) option.

## Keywords

Completeness Derivatives Integral representation Diffusion Martingales Parabolic equations Analytic functions Jacobian determinant## Mathematics Subject Classification (2010)

60G44 60H05 91G20 35K15 35K90## JEL Classification

G10## 1 Introduction

*discounted*prices of liquidly traded securities in a financial market and with the property that \(S\) is a (vector) martingale under the measure ℙ. The model is said to be

*complete*if any contingent claim payoff can be obtained as the terminal value of a self-financing trading strategy. The second fundamental theorem of asset pricing (cf. [8]) allows us to restate the completeness property in purely mathematical terms as follows: every martingale \(M=(M_{t})\) admits an integral representation with respect to \(S\), that is,

The process \(S\) may for example describe the prices of stocks or option contracts, which nowadays are often traded as liquidly as their underlyings. Depending on the application one has in mind, the construction of \(S\) differs significantly. In general there are three possibilities to consider. Given its initial value, \(S\) may be defined in a *forward* form, in terms of its predictable characteristics under the measure ℙ. In this case, the verification of the completeness property is straightforward. For example, if \(S\) is a driftless diffusion process under the measure ℙ with volatility matrix process \(\sigma =(\sigma_{t})\), then the market is complete if and only if \(\sigma \) has full rank \(\mathrm{d}\mathbb{P} \times \mathrm{d}t\) almost surely (cf. [10, Theorem 1.6.6]). Alternatively, \(S\) can be defined in a *backward* form, as the conditional expectation under ℙ of its given terminal value. Finally, some components of \(S\) may be defined in a forward form and others in a backward form leading to a *forward–backward* setup.

In principle, the proof of our main result, Theorem 2.2 below, generalizes to the \(d\)-dimensional case. The reason we present the two-dimensional case only is twofold: first, the structural conditions on the coefficients \(\sigma^{F}\) and \(\psi \) become very complex in higher dimensions; second, using our current methods, an extension to higher dimensions would require additional regularity of \(\psi \) and in particular exclude the payoff functions of call and put options, which are only once weakly differentiable.

For our analysis, we assume that \(\sigma^{F}\) and \(\psi \) are specified in terms of a solution \(X\) to a two-dimensional stochastic differential equation with drift vector \(b=b(t,x)\) and volatility matrix \(\sigma = \sigma (t,x)\). With respect to the space variable, our conditions are quite classical: \(b=b(t,\cdot )\) is once continuously differentiable and \(\sigma = \sigma (t,\cdot )\) is twice continuously differentiable and possesses a bounded inverse. Further, the functions themselves and their derivatives are bounded. With respect to time, our conditions are quite exacting: \(b=b(\cdot ,x)\) and \(\sigma = \sigma ( \cdot ,x)\) have to be real analytic on \((0,1)\).

Our results extend and rigorously prove ideas on the completion of markets with derivative securities first formulated in [16] and [4].

The paper [16] is concerned with the specific case of stochastic volatility models. The main result in this paper requires the derivative payoff function to be a convex function of the stock price only and, unless given by the special case of a European call or put option, to be twice continuously differentiable. Perhaps most limiting from the point of view of applicability, it is required that the volatility risk premium is such that the drift coefficient of the volatility process under the equivalent martingale measure does not depend on the stock price. Moreover, also the correlation between the asset price and its (stochastic) volatility process and the volatility of the volatility process must not depend on the stock price.

In [4], the setup is not restricted to the two-dimensional case. However, the key conditions in that paper are not placed on model primitives, but on the conditional expectation \(\mathbb{E}[(S^{F}_{1},\psi )|\mathcal{F}_{t}] = (v^{1},v^{2})(t,X _{t})\). In particular, \(v=(v^{1},v^{2})\) is *assumed* to be (jointly) real analytic in the time and space variables, and in the main theorem of the paper, the Jacobian matrix (with respect to \(x\)) of \(v=v(t,x)\) is assumed to be nonzero on some open subset of \((0,1) \times \mathbb{R}^{2}\).

*all*security prices are defined in backward form, whereas the problem of market completion with derivative securities requires

*some*security prices to be defined in forward form, leading to a forward–backward setup; second, in an equilibrium setting, the martingale measure for the price process \(S\) is determined endogenously in terms of the utility functions of individual agents, whereas in our setting the measure ℙ is given exogenously. The distinguishing feature of the present paper concerns an important technical assumption placed on the Jacobian matrix of the vector of terminal payoffs. If \(S^{F}_{1} = f(X_{1})\) and \(\psi = g(X_{1})\), the aforementioned results require the Jacobian matrix

Due to the differences in the financial setup described above, we do not include any equilibrium-related examples in this paper. In principle, however, our structural condition on the model primitives could be easily applied in an equilibrium setting, where it would replace the assumption that the Jacobian matrix of terminal dividend payments has full rank, and therefore could be used to establish the endogenous completeness of Radner equilibria.

At first sight, it may appear that the most restrictive condition, limiting the applicability of our result, is the boundedness assumption on the coefficients of the diffusion \(X\). This assumption stems from the theory of elliptic and parabolic partial differential equations, which plays an essential part in our proofs. However, we demonstrate in Sect. 6 how we can still accommodate popular models from financial mathematics such as geometric Brownian motion or mean-reverting processes by means of suitable changes of variables.

### Notation and basic concepts

*Hölder-continuous*on \([0,1]\), that is, there exist constants \(N>0\) and \(\delta >0\) such that

*analytic*on \((0,1)\), that is, for every \(u\in (0,1)\), there exist \(\epsilon (u)>0\) and a family \(\{A_{n}(u)\}\) of elements in \(\mathbf{X}\) such that

\(\mathbf{L}^{p}_{\mathrm{loc}}(U)\) denotes the Lebesgue space of locally \(p\)-integrable, real-valued functions \(h\) on \(U\): for every bounded, open subset \(V\) of \(U\), \(h\in \mathbf{L}^{p}(U)\); \(\mathbf{L}^{p}_{ \mathrm{loc}}:=\mathbf{L}^{p}_{\mathrm{loc}}(\mathbb{R}^{2})\).

\(\mathbf{L}^{p}(U)\) (for \(p\geq 1\)) is the Lebesgue space of Lebesgue-measurable, real-valued functions \(h\) on \(U\) with the norm \(\|h\|_{\mathbf{L}^{p}(U)}:=(\int_{U} |h|^{p}\, \mathrm{d}x )^{1/p}\); \(\mathbf{L}^{p} :=\mathbf{L}^{p}(\mathbb{R}^{2})\).

\(\mathbf{L}^{\infty }(U)\) is the Lebesgue space of essentially bounded, real-valued functions \(h\) on \(U\) with the norm \(\|h\|_{\mathbf{L}^{ \infty }(U)}:=\text{ess} \sup_{U}|h|\); \(\mathbf{L}^{\infty } := \mathbf{L}^{\infty }(\mathbb{R}^{2})\).

\(\mathbf{C}(U)\) is the Banach space of all bounded and continuous real-valued functions \(h\) on \(U\) with the norm \(\|h\|_{\mathbf{C}(U)}:= \sup_{U}|h|\); \(\mathbf{C}:=\mathbf{C}(\mathbb{R}^{2})\).

As is common, for \(p\geq 1\), we denote by \(p'\) the conjugate exponent of \(p\), defined by \(p' :=p/(p-1)\) for \(1< p<\infty \), \(p':=\infty \) if \(p=1\), and \(p':=1\) if \(p=\infty \).

With these definitions in mind we define the following spaces:

\(\mathbf{W}^{m}_{p,0}(U)\) (for \(m\in \{0,1,\ldots \}\) and \(p\geq 1\)) is the Banach space obtained by taking the closure of \(\mathbf{C}^{ \infty }_{0}(U)\) in the space \(\mathbf{W}^{m}_{p}(U)\).

Our notation is in agreement with standard notation from linear algebra. Given two vectors \(x\), \(y\) in \(\mathbb{R}^{d}\), \(xy\) denotes the scalar product and \(|x|:=\sqrt{xx}\). Given a matrix \(M\in \mathbb{R}^{m \times n}\) with \(m\) rows and \(n\) columns, \(Mx\) denotes its product with the column vector \(x\), \(M^{\star }\) its transpose and \(\|M\|_{F} :=\sqrt{ \mathrm{tr}(MM^{\star })}\). For an \(n\times n\) matrix \(M\), we denote the determinant of \(M\) either by \(|M|\) or by \(\det {M}\). Let \(\ell = (\ell _{1},\ldots ,\ell_{k})\) denote a multi-index complying with the condition \(1\leq \ell_{1}<\cdots <\ell_{k}\leq d\). Given \(n\times n\) matrices \(M\), \(C^{1},\ldots ,C^{k}\), we write \(M(\ell ; C^{1},\ldots ,C ^{k})\) for the matrix that is obtained from \(M\) by replacing the \(\ell_{p}\)th column of \(M\) by the \(\ell_{p}\)th column of \(C^{p}\) for \(p=1,\ldots ,k\), while keeping the remaining columns unchanged; if \(k>n\), \(M(\ell ; C^{1},\ldots ,C^{k}):=0\). Let \(A\) be an operator on a Banach space \(\mathbf{X}\) and \(M\) an \(n\times n\) matrix such that \(m^{ij}\) is in the domain of \(A\), \(i,j=1,\ldots ,n\). We write \(AM\) for the entrywise application of the operator \(A\); so \(AM :=(Am^{ij})_{i,j=1, \ldots ,n}\).

Throughout the text, \(N>0\) denotes a constant whose value may vary from line to line.

## 2 Main result: forward–backward martingale representation

- (A1)The maps \(t\mapsto b^{j}(t,\cdot )\) and \(t\mapsto \sigma^{ij}(t, \cdot )\) from \([0,1]\) to \(\mathbf{C}\) are Hölder-continuous and their restriction to \((0,1)\) is analytic. The map \(t\mapsto \sigma ^{ij}(t,\cdot )\) is continuous from \([0,1]\) to \(\mathbf{C}^{2}\) and the map \(t\mapsto b^{j}(t,\cdot )\) is continuous from \([0,1]\) to \(\mathbf{C}^{1}\). The matrix \(\sigma \) is invertible and there exists a constant \(N>0\) such that$$ \|\sigma^{-1}(t,x)\|_{F}\leq N, \quad (t,x) \in [0,1]\times \mathbb{R}^{2}. $$(2.1)

### Remark 2.1

- (A2)The map \(t\mapsto r(t,\cdot )\) is Hölder-continuous as a map from \([0,1]\) to \(\mathbf{C}\), continuous as a map from \([0,1]\) to \(\mathbf{C}^{1}\), analytic as a map from \((0,1)\) to \(\mathbf{C}\). The function \(r\) is nonnegative, i.e.,$$ r(t,x)\geq 0, \quad (t,x)\in [0,1]\times \mathbb{R}^{2}. $$

- (A3)
The map \(t\mapsto e^{-N|\cdot |}\partial_{x^{j}x^{k}}f(t, \cdot )\) from \((0,1)\) to \(\mathbf{L}^{\infty }\) is analytic, the map \(t\mapsto e^{-N|\cdot |}\partial_{x^{j}}f(t,\cdot )\) from \([0,1]\) to \(\mathbf{L}^{\infty }\) is continuously differentiable, and the map \(t\mapsto e^{-N|\cdot |}\partial_{x^{j}x^{k}x^{\ell }}f(t, \cdot )\) from \([0,1]\) to \(\mathbf{L}^{\infty }\) is continuous.

^{1}

- (A4)Either the Jacobian matrix \(J[f,g](1,\cdot )\) has full rank almost everywhere on \(\mathbb{R}^{2}\), or for every bounded, open set \(K\) in \(\mathbb{R}^{2}\), there exists a function \(\varphi =\varphi (x)\) belonging to \(\mathbf{W}^{1}_{p,0}(K)\) for some \(p\geq 1\) such that \(\mathcal{B}_{K}[g,\varphi ;1]\neq 0\) and$$ \biggl| \frac{\partial g}{\partial x^{j}}(x)\biggr| \leq e^{N(1+|x|)}, \quad x\in \mathbb{R}^{2},\ j=1,2. $$

The main result of the paper is

### Theorem 2.2

(Forward–backward martingale representation)

*Suppose that*(A1)–(A4)

*hold*.

*Then the solution*\((S^{F},S^{B},Z)\)

*to the forward–backward stochastic differential equation*

*is well defined*.

*Moreover*,

*every martingale*\(M\)

*under*ℙ

*is a stochastic integral with respect to the two*-

*dimensional*ℙ-

*martingale*\(S = (S^{F},S^{B})\),

*that is*, (1.1)

*holds and the market model is complete under*ℙ.

### Remark 2.3

From a purely theoretical point of view, Theorem 2.2 asserts that under (A1)–(A4), the volatility process of the forward–backward stochastic differential equation (2.2) has full rank \(\mathrm{d}\mathbb{P} \times \mathrm{d}t\) almost surely.

### Remark 2.4

The proof of Theorem 2.2 is given in Sect. 5 and relies on specific smoothness and integrability properties of the solution to a parabolic equation, which we obtain in Sect. 3 and on the invertibility of a Jacobian matrix, which we study in Sect. 4.

## 3 Regularity of the solution to the associated parabolic equation

- (B1)The maps \(t\mapsto a^{jk}(t,\cdot )\), \(t\mapsto b^{j}(t,\cdot )\), \(t\mapsto c(t,\cdot )\) from \([0,1]\) to \(\mathbf{C}\) are Hölder-continuous and their restriction to \((0,1)\) is analytic. The map \(t\mapsto a^{jk}(t,\cdot )\) is continuous from \([0,1]\) to \(\mathbf{C}^{2}\) and the maps \(t\mapsto b^{j}(t,\cdot )\), \(t\mapsto c(t, \cdot )\) are continuous from \([0,1]\) to \(\mathbf{C}^{1}\). The matrix \(a\) is symmetric, \(a^{ij} = a^{ji}\), and uniformly elliptic, i.e., there exists \(N>0\) such thatand the function \(c\) is nonpositive, i.e.,$$ ya(t,x)y\geq \frac{1}{N^{2}}|y|^{2},\quad (t,x) \in [0,1]\times \mathbb{R}^{2}, y\in \mathbb{R}^{2}, $$$$ c(t,x) \leq 0, \quad (t,x)\in [0,1]\times \mathbb{R}^{2}. $$

- (B2)
The function \(g\) belongs to \(\mathbf{W}^{1}_{p}\).

### Theorem 3.1

*Suppose that conditions*(B1)

*and*(B2)

*hold*.

*Then there exists a unique measurable function*\(v=v(t,x)\)

*on*\([0,1]\times \mathbb{R}^{2}\)

*such that*

- 1.
\(t\mapsto v(t,\cdot )\)

*is a continuous map from*\([0,1]\)*to*\(\mathbf{W}^{1}_{p}\); - 2.
\(t\mapsto v(t,\cdot )\)

*is an analytic map from*\((0,1)\)*to*\(\mathbf{W} ^{2}_{p}\); - 3.
\(t\mapsto v(t,\cdot )\)

*is a*\(p\)-*integrable map from*\([0,1)\)*to*\(\mathbf{W}^{3}_{p}\); - 4.
\(t\mapsto \partial_{t} v(t,\cdot )\)

*is a*\(p\)-*integrable map from*\([0,1)\)*to*\(\mathbf{W}^{1}_{p}\);

*and such that*\(v=v(t,x)\)

*solves the homogeneous Cauchy problem*

### Proof

By assumption (B1), we know that for each \(t\in [0,1]\) and \(j,k=1,2\), the function \(a^{jk}(t,\cdot )\) is in \(\mathbf{C}^{2}\). In particular, the first-order partial derivatives of \(a^{jk}\) with respect to \(x\) are bounded and therefore the matrix \(a\) is uniformly continuous with respect to \(x\). Under the assumptions (B1) and (B2), the assertions of items 1 and 2 are immediately obtained upon making the time change \(t\to 1-t\) in [11, Theorem 3.1].

In addition, Theorem 3.1 in [11] tells us that \(t\mapsto v(t,\cdot )\) is a continuously differentiable map from \([0,1)\) to \(\mathbf{L}^{p}\) and a continuous map from \([0,1)\) to \(\mathbf{W}^{2}_{p}\), which implies that \(v=v(t,x)\) belongs to \(\mathbf{W}^{1,2}_{p}([0,1)\times \mathbb{R}^{2})\). Therefore, given the symmetry and uniform ellipticity of the matrix function \(a=a(t,x)\), the uniform continuity of \(a(t,\cdot )\), the fact that each function \(a^{jk}(t,\cdot )\), \(b^{j}(t,\cdot )\), \(c(t,\cdot )\) belongs to \(\mathbf{C}^{1}\) and the nonnegativity of \(c=c(t,x)\), we may use [14, Corollary 5.2.4] to deduce that \(v=v(t,x)\) in fact belongs to \(\mathbf{W}^{1,3}_{p}([0,1)\times \mathbb{R}^{2})\). The regularity in items 3 and 4 follows immediately. □

- (B3)There exists a constant \(N\geq 0\) such that$$ e^{-N|\cdot |}\frac{\partial g}{\partial x^{j}} (\cdot ) \in \mathbf{L}^{\infty },\quad j=1,2. $$

### Corollary 3.2

*Suppose conditions*(B1)

*and*(B2)

*hold*.

*Let*\(\phi =\phi (x)\)

*satisfy condition*(3.4).

*Then there exist a unique continuous function*\(v=v(t,x)\)

*on*\([0,1]\times \mathbb{R}^{2}\)

*and a constant*\(N\geq 0\)

*such that for every*\(p\geq 1\),

- 1.
\(t\mapsto e^{-N\phi (\cdot )}v(t,\cdot )\)

*is a continuous map from*\([0,1]\)*to*\(\mathbf{W}^{1}_{p}\); - 2.
\(t\mapsto e^{-N\phi (\cdot )}v(t,\cdot )\)

*is an analytic map from*\((0,1)\)*to*\(\mathbf{W}^{2}_{p}\); - 3.
\(t\mapsto e^{-N\phi (\cdot )}v(t,\cdot )\)

*is a*\(p\)-*integrable map from*\([0,1)\)*to*\(\mathbf{W}^{3}_{p}\); - 4.
\(t\mapsto e^{-N\phi (\cdot )}\partial_{t}v(t,\cdot )\)

*is a*\(p\)-*integrable map from*\([0,1)\)*to*\(\mathbf{W}^{1}_{p}\);

*and such that*\(v=v(t,x)\)

*solves the Cauchy problem*(3.2)

*and*(3.3).

### Proof

Defining \(v:=e^{N\phi -Ct}\tilde{v}\), we observe that \(\tilde{v}\) solves (3.6) and (3.7) if and only if \(v\) solves the Cauchy problem (3.2) and (3.3). For \(p>1\), the regularity of \(\tilde{v} = e^{-N\phi +Ct}v\) implies items 1–4 in the corollary. The proof is completed by noting that the case \(p=1\) follows trivially from the case \(p> 1\) by taking the constant \(N\) slightly larger. □

### Corollary 3.3

*Suppose that conditions*(B1)

*and*(B3)

*hold*.

*Let*\(v=v(t,x)\)

*be the function generated by Corollary*3.2

*and let*\(v_{j}\)

*be defined as in*(3.8).

*Then*\(v _{j} = v_{j}(t,x)\)

*solves the nonhomogeneous partial differential equation*

### Proof

From Corollary 3.2, we know that the function \(v=v(t,x)\) is three times weakly differentiable with respect to \(x\) and that the derivative with respect to \(t\) of the same function is once weakly differentiable with respect to \(x\). Given condition (B1), we also know that the coefficients of the operator \(\mathcal{G}\) are once continuously differentiable with respect to \(x\). Hence we may differentiate the parabolic partial differential equation (3.2) with respect to \(x^{j}\), \(j=1,2\), which shows that \(v_{j} = v_{j}(t,x)\) satisfies (3.10). □

## 4 Invertibility of the Jacobian matrix

- (B4)
The map \(t\mapsto e^{-N|\cdot |}\partial_{x^{j}x^{k}}f(t,\cdot )\) from \((0,1)\) to \(\mathbf{L}^{\infty }\) is analytic, the map \(t \mapsto e^{-N|\cdot |}\partial_{x^{j}}f(t,\cdot )\) from \([0,1]\) to \(\mathbf{L}^{\infty }\) is continuously differentiable, and the map \(t \mapsto e^{-N|\cdot |}\partial_{x^{j}x^{k}x^{\ell }}f(t,\cdot )\) from \([0,1]\) to \(\mathbf{L}^{\infty }\) is continuous.

- (B5)
Either the Jacobian matrix \(J[f,g](1,\cdot )\) has full rank almost everywhere on \(\mathbb{R}^{2}\), or for every open, bounded set \(K\) in \(\mathbb{R}^{2}\), there exists a test function \(\varphi =\varphi (x)\) belonging to \(\mathbf{W}^{1}_{p,0}(K)\) for some \(p\geq 1\) such that \(\mathcal{A}_{K}[g,\varphi ;1]\neq 0\).

The following theorem is the main result of this section and will eventually allow us to prove the martingale representation stated in Theorem 2.2.

### Theorem 4.1

*Suppose conditions* (B1) *and* (B3)*–*(B5) *are in place*. *Let* \(v= v(t,x)\) *be the function which is furnished by Corollary* 3.2. *Then the Jacobian matrix function* \(J[f,v] = J[f,v](t,x)\) *has full rank almost everywhere with respect to Lebesgue measure on* \([0,1]\times \mathbb{R}^{2}\).

Before we can prove Theorem 4.1, we first need to establish several lemmas below.

Let \(\mathbf{X}\) and \(\mathbf{Y}\) be Banach spaces, \(\mathbf{E}\) an open subset of \(\mathbf{X}\) and consider a map \(h: \mathbf{E}\to \mathbf{Y}\). If it exists, we denote by \(D^{k}h(x)\) the \(k\)th Fréchet derivative of \(h\) at the point \(x\in \mathbf{E}\); as is well known, this constitutes a \(k\)-linear map on the \(k\)-fold product \(\mathbf{X}\times \cdots \times \mathbf{X}\). Accordingly, for \(x^{1},\ldots ,x^{k}\in \mathbf{X}\), we denote by \(D^{k}h(x^{1}, \ldots ,x^{k})\) the \(k\)th Fréchet differential.

### Lemma 4.2

*Given matrices*\(M,C,C^{1},C^{2}\in \mathbb{R}^{2\times 2}\),

*the first and second order Fréchet differentials of the determinant map at*\(M\)

*are given by*

### Proof

The expressions are special cases of equations (4) and (6) in [3]. □

### Lemma 4.3

*Let*\(f =f(t,x),v =v(t,x) : [0,1]\times \mathbb{R}^{2} \to \mathbb{R}\)

*be measurable functions which on*\((0,1)\times \mathbb{R}^{2}\)

*are once weakly differentiable with respect to*\(t\),

*three times weakly differentiable with respect to*\(x\)

*and once weakly differentiable with respect to*\(t\)

*and*\(x\),

*and let*\(\mathcal{G}(t)\)

*and*\(\mathcal{G}_{j}(t)\)

*be the operators defined in*(3.1)

*and*(3.9),

*respectively*.

*Define*\(f_{j} :=\partial_{x^{j}}f\), \(v_{j} :=\partial_{x^{j}}v\), \(j=1,2\),

*and assume that*\(v_{j}\)

*satisfies the partial differential equation*

*Then the determinant function*\(w = w(t,x)\)

*defined on*\([0,1]\times \mathbb{R}^{2}\)

*by*\(w :=|J[f,v]|\)

*satisfies the nonhomogeneous partial differential equation*

### Proof

### Lemma 4.4

*Let*\(\gamma^{j},\eta : [0,1]\times \mathbb{R}^{2} \to \mathbb{R}\), \(j=1,2\),

*be measurable functions such that for*\(p>1\),

*the maps*\(t\mapsto \gamma^{j}(t,\cdot ),t\mapsto \eta (t,\cdot )\)

*from*\([0,1]\)

*to*\(\mathbf{L}^{p}_{\mathrm{loc}}\)

*are continuous*.

*Let*\(K\)

*be an open*,

*bounded set in*\(\mathbb{R}^{2}\)

*and*\(\varphi =\varphi (x)\)

*a test function belonging to*\(\mathbf{W}^{1}_{p',0}(K)\).

*Then for each*\(t\in [0,1]\),

*the pairing*

*is a bounded*,

*linear functional on*\(\mathbf{W}^{1}_{p',0}(K)\).

*Moreover*,

*the map*\(t\mapsto \tilde{\mathcal{A}}_{K}(\cdot ;t)\)

*is continuous as a map from*\([0,1]\)

*to*\(\mathbf{W}^{-1}_{p}(K)\).

### Proof

### Proof of Theorem 4.1

Let us now assume that for every open, bounded set \(K\) in \(\mathbb{R} ^{2}\) there exists a test function \(\varphi = \varphi (x)\) belonging to \(\mathbf{W}^{1}_{p',0}(K)\) such that \(\mathcal{A}_{K}[g,\varphi ;1] \neq 0\). From Corollary 3.2 and (B4), we know that the functions \(f=f(t,x)\) and \(v=v(t,x)\) satisfy the differentiability hypothesis of Lemma 4.3, and from Corollary 3.3 that \(v_{j}\) satisfies the partial differential equation (4.1). It follows from (4.2) that if \(w(t,x) = 0\) for all \((t,x)\in (0,1) \times H\), then also \(\mathcal{P}(t)v=0\) for all \((t,x)\in (0,1) \times H\).

## 5 Proof of Theorem 2.2

From here onwards we adopt the notation introduced in Sect. 2 and assume that conditions (A1)–(A4) are in place.

We fix a function \(\phi =\phi (x)\) on \(\mathbb{R}^{2}\) satisfying (3.4) and recall that \(\mathcal{L}^{X}(t)\), \(t\in [0,1]\), is the infinitesimal generator of the process \(X\).

### Lemma 5.1

*There exist a unique continuous function*\(v=v(t,x)\)

*on*\([0,1]\times \mathbb{R}^{2}\)

*and a constant*\(N\geq 0\)

*such that the following hold*:

- 1.
*For every*\(p\geq 1\),- (a)
\(t\mapsto e^{-N\phi (\cdot )}v(t,\cdot )\)

*is a continuous map from*\([0,1]\)*to*\(\mathbf{W}^{1}_{p}\); - (b)
\(t\mapsto e^{-N\phi (\cdot )}v(t,\cdot )\)

*is an analytic map from*\((0,1)\)*to*\(\mathbf{W}^{2}_{p}\); - (c)
\(t\mapsto e^{-N\phi (\cdot )}v(t,\cdot )\)

*is a*\(p\)-*integrable map from*\([0,1)\)*to*\(\mathbf{W}^{3}_{p}\); - (d)
\(t\mapsto e^{-N\phi (\cdot )}\partial_{t}v(t,\cdot )\)

*is a*\(p\)-*integrable map from*\([0,1)\)*to*\(\mathbf{W}^{1}_{p}\).

- (a)
- 2.
*The function*\(v=v(t,x)\)*solves the homogeneous Cauchy problem*$$\begin{aligned} \frac{\partial v}{\partial t} + (\mathcal{L}^{X}(t) -r)v &= 0, \quad t \in [0,1), \end{aligned}$$(5.1)$$\begin{aligned} v(1,\cdot ) &= g. \end{aligned}$$(5.2) - 3.
*The Jacobian matrix function*\(J[f,v]=J[f,v](t,x)\)*has full rank almost everywhere with respect to Lebesgue measure on*\([0,1]\times \mathbb{R}^{2}\).

Hereafter we denote by \(v=v(t,x)\) the function defined in Lemma 5.1.

### Proof

Observe that (A1)–(A4) imply (B1) and (B3)–(B5) on the corresponding coefficients in Theorem 4.1. The assertions for \(v\) and \(J[f,v]\) now follow directly from Theorem 4.1. □

### Lemma 5.2

*The martingale*

*is well defined and has the representation*

*Moreover*,

*for*\(t\in (0,1)\),

### Proof

Assume that the process \(S^{B}\) is actually *defined* by (5.3). From the continuity of \(v\) on \([0,1]\times \mathbb{R}^{2}\), it then follows that \(S^{B}\) is in fact a continuous process on \([0,1]\), and from the expression (5.2) for \(v(1,\cdot )\) that \(S^{B}_{1}=\psi \). Hence, to complete the proof, it remains to show that \(S^{B}\) given by (5.3) is a martingale under the measure ℙ.

From Lemma 5.1, we know that the map \(t\mapsto e^{-N \phi (\cdot )}v(t,\cdot )\) is analytic as a map from \((0,1)\) to \(\mathbf{W}^{2}_{p}\); in particular, it is continuously differentiable. This allows us to use a variant of the Itô formula due to Krylov (cf. [13, Sect. 2.10, Theorem 1]) and accounting for (5.1), we immediately obtain (5.4) from (5.3).

## 6 Example: a class of stochastic volatility models

- (C1)There exist constants \(N,D,\rho ,\epsilon >0\) such that for all \(y\in \mathbb{R}\) with \(\nu (y)>N\) and \(\sigma^{j}(y)>N\), the derivative \(\mathrm{d}\nu /\mathrm{d}y(y) \neq 0\) almost everywhere on ℝ and the functions \(\nu \), \(\sigma^{j}\) and \(\mu (p, \cdot )\) are infinitely differentiable and satisfyThe function \(\mu =\mu (p,y)\) has first and second continuous derivatives in \(p\) and \(y\), and \(y(e^{p})^{\ell } \partial_{y}^{k} \partial_{p}^{\ell }\mu \in \mathbf{L}^{\infty }\), \(\ell =0,1\), \(k=1,2\), \(\ell +k\leq 2\).$$ \biggl| \frac{\partial^{k}\mu }{\partial y^{k}}(p,y)\biggr| + \biggl| \frac{ \partial^{k}\nu }{\partial y^{k}}(y)\biggr| + \biggl| \frac{\partial ^{k}\sigma^{j}}{\partial y^{k}}(y)\biggr| \leq \frac{Dk!}{(\rho + \epsilon |y|)^{k}}, \quad (p,y)\in \mathbb{R}\times \mathbb{R}. $$

We are now ready to state the main result of this section.

### Theorem 6.1

*Suppose that condition* (C1) *is satisfied*. *Then the* \((P,V)\)-*market defined by* (6.1) *is complete*.

### Remark 6.2

We draw attention to the fact that in (C1), the quite specific assumptions on the space regularity of the coefficients of (6.1) are solely necessary because we are allowing \(P\) to evolve according to a geometric Brownian motion and \(Y\) to have mean reverting dynamics, both cases in which the coefficients are unbounded. This can be seen easily from the proof of Theorem 6.1 below. In the absence of this particular choice of dynamics, the verification of the assumptions of Theorem 2.2 is much simpler.

### Remark 6.3

Two specific examples of functions which satisfy the conditions on \(\nu \) and \(\sigma \) in (C1) are scaled and shifted versions of the \(\operatorname{arctan}\) and \(\operatorname{tanh}\) functions.

### Proof of Theorem 6.1

## Footnotes

## Notes

### Acknowledgements

It is a pleasure to thank Dmitry Kramkov for introducing me to the topic of market completion with derivative securities and for interesting discussions. I would like to thank Léonard Monsaingeon and Peter Takác for discussions relating to the presented work and Johannes Ruf for comments on an early version of this paper.

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