Keywords

1 Introduction

Consider a complete probability space on which lives an n-dimensional Wiener process W. Let \(\underline{\mathcal {F}} = (\mathcal {F}_t)_{0\le t \le T}\) denote the augmentation under of the filtration generated by W until the constant terminal time \(T<\infty \). One of the main results of Itô calculus is the martingale representation theorem which in the present setting is as follows: Let M be a RCLL local martingale relative to , then there exists a progressively measurable n-dimensional process \(\varphi \) such that

$$\begin{aligned} M(t) = M(0) + \int _0^t\varphi (s)'dW(s), 0 \le t \le T, \text { and } \int _0^T|\varphi (t)|^2dt <\infty \; \text {a.s.} \end{aligned}$$

In particular,  M has continuous sample paths a.s.

Considerable effort has in the literature been made in order to find explicit formulas for the integrand \(\varphi \), i.e. in order to find constructive representations of martingales, mainly using Malliavin calculus, see e.g. [8, 15, 16, 20] and the references therein. The recently developed functional Itô calculus includes a new type of constructive representation of square integrable martingales due to Cont and Fournié see e.g. [1, 3,4,5]. The main result of the present paper is an extension of this result to local martingales.

The organization of the paper is as follows. Section 9.2 is based on [1] and contains a brief and heuristic account of the relevant parts of functional Itô calculus including the constructive martingale representation theorem for square integrable martingales. Section 9.3 contains the local martingale extension of this theorem and a simple example.

2 Constructive Representation of Square Integrable Martingales

Denote an n-dimensional sample path by \(\omega \). Denote a sample path stopped at t by \(\omega _t\), i.e. let \(\omega _t(s) = \omega (t \wedge s), 0\le s \le T\). Consider a real-valued functional of sample paths \(F(t,\omega )\) which is non-anticipative (essentially meaning that \(F(t,\omega ) =F(t,\omega _t)\)). The horizontal derivative at \((t,\omega )\) is defined by

$$\begin{aligned} \mathcal {D} F(t,\omega ) = \lim _{h \searrow 0}\frac{F(t+h,\omega _t)-F(t,\omega _t)}{h}. \end{aligned}$$

The vertical derivative at \((t,\omega )\) is defined by

$$\nabla _\omega F(t,\omega ) = (\partial _iF(t,\omega ), i=1,...,n)',$$

where

$$\begin{aligned} \partial _iF(t,\omega ) = \lim _{h\rightarrow 0} \frac{F(t,\omega _t+ he_i I_{[t,T]})-F(t,\omega _t)}{h}. \end{aligned}$$

Higher order vertical derivatives are obtained by vertically differentiating vertical derivatives.

One of the main results of functional Itô calculus is the functional Itô formula, which is just the standard Itô formula with the usual time and space derivatives replaced by the horizontal and vertical derivatives. If the functional F is sufficiently regular (regarding e.g. continuity and boundedness of its derivatives), which we write as , then the functional Itô formula holds, see [1, ch. 5,6]. We remark that [12] contains another version of this result.

Using the functional Itô formula it easy to see that if Z is a martingale satisfying

(9.1)

then, for every \(t\in [0,T]\),

$$\begin{aligned} Z(t) = Z(0) + \int _0^t\nabla _\omega F(s,W_s)'dW(s) \;\; a.s. \end{aligned}$$

We may therefore define the vertical derivative with respect to the process W of a martingale Z satisfying (9.1) as the -a.e. unique process \(\nabla _WZ\) given by

$$\begin{aligned} \nabla _WZ(t) = \nabla _\omega F(t,W_t), 0 \le t \le T. \end{aligned}$$
(9.2)

Let \(\mathcal {C}_b^{1,2}(W)\) be the space of processes Z which allow the representation in (9.1). Let \(\mathcal {L}^2(W)\) be the space of progressively measurable processes \(\varphi \) satisfying the condition \(E[\int _0^T\varphi (s)'\varphi (s)ds]<\infty \). Let \(\mathcal {M}^2(W)\) be the space of square integrable martingales with initial value 0. Let \(D(W) = \mathcal {C}_b^{1,2}(W) \cap \mathcal {M}^2(W)\).

It can be shown that \(\{\nabla _WZ: Z \in D(W)\}\) is dense in \(\mathcal {L}^2(W)\) and that D(W) is dense in \(\mathcal {M}^2(W)\) [1, ch. 7]. Using this it is possible to show that the vertical derivative operator \(\nabla _W(\cdot )\) admits a unique extension to \(\mathcal {M}^2(W)\), in the following sense: For \(Y \in \mathcal {M}^2(W)\) the (weak) vertical derivative \(\nabla _W Y\) is the unique element in \(\mathcal {L}^2(W)\) satisfying

$$\begin{aligned} E[Y(T)Z(T)] = E\left[ \int _0^T\nabla _WY(t)'\nabla _WZ(t)dt\right] \end{aligned}$$
(9.3)

for every \(Z\in D(W)\), where \(\nabla _WZ\) is defined in (9.2). The constructive martingale representation theorem ([1, ch. 7]) follows:

Theorem 9.1

(Cont and Fournié) For any square integrable martingale Y relative to and every \(t\in [0,T]\),

$$\begin{aligned} Y(t) = Y(0) + \int _0^t\nabla _{W}Y(s)'dW(s)\;\; a.s. \end{aligned}$$

3 Constructive Representation of Local Martingales

This section contains an extension of the vertical derivative \(\nabla _W(\cdot )\) and the constructive martingale representation in Theorem 9.1 to local martingales. Let \(\mathcal {M}^{{\tiny {\text {loc}}}}(W)\) denote the space of local martingales relative to with initial value zero and RCLL sample paths. In Theorem 9.2 we extend the vertical derivative to \(\mathcal {M}^{{\tiny {\text {loc}}}}(W)\). Using this extension we can formulate the constructive martingale representation theorem also for local martingales, see Theorem 9.3.

Before extending the definition of the vertical derivative to \(\mathcal {M}^{{\tiny {\text {loc}}}}(W)\) we recall the definition of a local martingale.

Definition 9.1

M is said to be a local martingale if there exists a sequence of non-decreasing stopping times \(\{\theta _n\}\) with \(\lim _{n \rightarrow \infty } \theta _n= \infty \) a.s. such that the stopped local martingale \(M(\cdot \wedge \theta _n)\) is a martingale for each \(n\ge 1\).

Theorem 9.2

(Definition of \(\nabla _W(\cdot )\) on \(\mathcal {M}^{{\tiny {\text {loc}}}}(W)\) )

  • There exists a progressively measurable -a.e. unique extension of the vertical derivative \(\nabla _W(\cdot )\) from \(\mathcal {M}^2(W)\) to \(\mathcal {M}^{{\tiny {\text {loc}}}}(W)\), such that, for \(M \in \mathcal {M}^{{\tiny {\text {loc}}}}(W)\),

    $$\begin{aligned} \begin{array}{rcl} M(t) &{} =&{} \int _0^t\nabla _WM(s)'dW(s), 0 \le t \le T, \text{ and } \\ &{}&{} \int _0^T|\nabla _WM(t)|^2dt <\infty \text{ a.s. } \end{array} \end{aligned}$$
    (9.4)

  • Specifically, for \(M \in \mathcal {M}^{{\tiny {\text {loc}}}}(W)\) the vertical derivative \(\nabla _WM\) is defined as the progressively measurable -a.e. unique process satisfying

    (9.5)

    where \(\nabla _WM_n\) is the vertical derivative of \(M_n:= M(\cdot \wedge \tau _n) \in \mathcal {M}^2(W)\) and \(\tau _n\) is given by

    $$\begin{aligned} \tau _n = \theta _n \wedge \inf \{s\in [0,T]:|M(s)| \ge n\}\wedge T \end{aligned}$$
    (9.6)

    where \(\{\theta _n\}\) is an arbitrary sequence of stopping times of the kind described in Definition 9.1.

Remark 9.1

Note that if M in Theorem 9.2 satisfies

$$\begin{aligned} M(t) = \int _0^t\gamma (s)'dW(s), 0 \le t\le T \;\text{ a.s. } \end{aligned}$$

for some process \(\gamma \), then \(\gamma = \nabla _WM\) -a.e. It follows that the extended vertical derivative \(\nabla _WM\) defined in Theorem 9.2 does not depend (modulo possibly on a null set ) on the particulars of the chosen stopping times \(\{\theta _n\}\).

Proof

The martingale representation theorem implies that, for \(M \in \mathcal {M}^{{\tiny {\text {loc}}}}(W)\), there exists a progressively measurable process \(\varphi \) satisfying

$$\begin{aligned} M(t) =\int _0^t\varphi (s)'dW(s), 0 \le t \le T, \text { and } \int _0^T|\varphi (t)|^2dt <\infty \; \text {a.s.} \end{aligned}$$
(9.7)

Therefore, if we can prove that

(9.8)

then it follows that there exists a progressively measurable process, denote it by \(\nabla _WM\), which is -a.e. uniquely defined by (9.5) and satisfies

which in turn implies that the integrals of \(\nabla _WM\) and \(\varphi \) coincide in the way that (9.7) implies (9.4). All we have to do is therefore to prove that (9.8) holds.

Let us recall some results about stopping times and martingales. The stopped local martingale \(M(\cdot \wedge \theta _n)\) is a martingale for each n, by Definition 9.1. Stopped RCLL martingales are martingales. The minimum of two stopping times is a stopping time and the hitting time

$$\begin{aligned} \inf \{s\in [0,T]:|M(s)| \ge n\} \end{aligned}$$

is, for each n, in the present setting, a stopping time. Using these results we obtain that \(M(\cdot \wedge \theta _n \wedge \inf \{s\in [0,T]:|M(s)| \ge n\}\wedge T) = M(\cdot \wedge \tau _n)\) is a martingale, for each n. Moreover, M is by the standard martingale representation result a.s. continuous. Hence, we may define a sequence of, a.s. continuous, martingales \(\{M_n\}\) by

$$\begin{aligned} M_n = M(\cdot \wedge \tau _n) = \int _0^{\cdot \wedge \tau _n}\varphi (s)'dW(s) \; \text {a.s.} \end{aligned}$$
(9.9)

where the last equality follows from (9.7). Now, use the definition of \(\tau _n\) in (9.6) to see that

$$\begin{aligned} |M_n(t)| = \left| \int _0^{t \wedge \tau _n}\varphi (s)'dW(s)\right| \le n \;\text {a.s.} \end{aligned}$$

for any t and n, and that in particular \(M_n\) is, for each n, a square integrable martingale. Moreover, (9.9) implies that \(M_n\) satisfies

$$\begin{aligned} M_n(t) = \int _0^tI_{\{s \le \tau _n\}}\varphi (s)'dW(s), 0 \le t \le T \;\text {a.s.} \end{aligned}$$
(9.10)

Since each \(M_n\) is a square integrable martingale we may use Theorem 9.1 on \(M_n\), which together with (9.10) implies that

$$\begin{aligned} M_n(t)= & {} \int _0^t\nabla _WM_n(s)'dW(s) \\= & {} \int _0^tI_{\{s\le \tau _n\}}\varphi (s)'dW(s), 0 \le t \le T \;\text {a.s.} \nonumber \end{aligned}$$
(9.11)

where \(\nabla _WM_n\) is the vertical derivative of \(M_n\) with respect to W (defined in (9.3)) and where we also used the continuity of the Itô integrals. The equality of the two Itô integrals in (9.11) implies that

(9.12)

The local martingale property of M implies that \(\lim _{n \rightarrow \infty } \theta _n= \infty \;a.s.\) Using this and the definition of \(\tau _n\) in (9.6) we conclude that for almost every \(\omega \in \varOmega \) and each \(t \in [0,T]\) there exists an \(N(\omega ,t)\) such that

$$\begin{aligned} n\ge N(\omega ,t) \Rightarrow \sup _{0\le s \le t}|M(\omega ,s)|\le n \text { and } t\le \theta _n(\omega ) \Rightarrow t \le \tau _n(\omega ). \end{aligned}$$
(9.13)

It follows from (9.12) and (9.13) that there exists an \(N(\omega ,t)\) such that

which means that (9.8) holds. \(\square \)

If M is a RCLL local martingale then \(M-M(0) \in \mathcal {M}^{{\tiny {\text {loc}}}}(W)\), which implies that \(\nabla _W(M-M(0))\) is defined in Theorem 9.2. This observation allows us to extend the definition of the vertical derivative to RCLL local martingales not necessarily starting at zero in the following obvious way.

Definition 9.2

The vertical derivative of a local martingale M relative to with RCLL sample paths is defined as the progressively measurable unique process \(\nabla _WM\) satisfying

$$\begin{aligned} \nabla _WM(t) = \nabla _W(M-M(0))(t), 0 \le t \le T, \end{aligned}$$
(9.14)

where \(\nabla _W(M-M(0))(t)\) is defined in Theorem 9.2.

The following result is an immediate consequence of Theorem 9.2 and Definition 9.2.

Theorem 9.3

If M is a local martingale relative to with RCLL sample paths, then

$$\begin{aligned} M(t)= & {} M(0) + \int _0^t\nabla _WM(s)'dW(s), 0 \le t \le T, \text{ and }\\&\int _0^T|\nabla _WM(t)|^2dt<\infty \;a.s., \end{aligned}$$

where \(\nabla _WM(s)\) is defined in Definition 9.2.

Let us try to clarify the theory by studying a simple example. It is straightforward to extend the results above to the case when the Wiener process W is replaced by an adapted process X given by

$$\begin{aligned} X(t)=X(0) + \int _0^t\sigma (s)dW(s), \end{aligned}$$
(9.15)

where \(\sigma \) is a matrix-valued adapted process satisfying suitable assumptions, mainly invertibility, see also [1, 4]. Thus, a local martingale M can be represented as

$$\begin{aligned} M(t)-M(0)= \int _0^t\nabla _W M(s)'dW(s) = \int _0^t\nabla _X M(s)'dX(s), \end{aligned}$$

and the relationship between the vertical derivatives with respect to W and X is \(\nabla _W M(t)' = (\nabla _X M(t)')\sigma (t)\), cf. (9.15). As example consider the one-dimensional case and let X with \(X(0)=0\) be given by (9.15) under the assumption that \(\sigma (s)\) is a deterministic function of time and let M be given by \(M(t)= F(t,X_t)\) where F is the non-anticipative functional \(F(t,\omega ) = \omega ^3(t)- 3\int _0^t\omega (s)\sigma ^2(s)ds\), i.e. let M be the local martingale defined by

$$\begin{aligned} M(t)= X^3(t)- 3\int _0^tX(s)\sigma ^2(s)ds. \end{aligned}$$

In this case the vertical derivative simplifies to the standard derivative, that is, \(\nabla F_\omega (t,\omega ) = 3\omega ^2(t)\), see also [1, 4] (we remark that the horizontal derivative is \(\mathcal {D} F(t,\omega ) = -3\omega (t)\sigma ^2(t)\)). In this case, \(\nabla _X M(t) = 3X^2(t)\) and

$$\begin{aligned} M(t) = \int _0^t3X^2(s)dX(s) = \int _0^t3 X^2(s) \sigma (s)dW(s), \end{aligned}$$

which we remark is easily found using the standard Itô formula. Note that this also means that \(\nabla _W M(t) = 3X^2(t)\sigma (t) = \nabla _X M(t)\sigma (t)\).

Concluding Remarks

Many of the applications that rely on martingale representation are within mathematical finance. A particular application that may benefit from the local martingale extension of the present paper is optimal investment theory, in which the discounted (using the state price density) optimal wealth process is a (not necessarily square integrable) martingale, see e.g. [9, ch. 3], see also [13]. In particular, using functional Itô calculus it is possible to derive an explicit formula for the optimal portfolio in terms of the vertical derivative of the discounted optimal wealth process, see also [14]. Similar explicit formulas for optimal portfolios based on the Malliavin calculus approach to constructive martingale representation have, under restrictive assumptions, been studied extensively, see e.g. [2, 6, 7, 10, 11, 17,18,19]. The general connection between Malliavin calculus and functional Itô calculus is studied in e.g. [1, 4].