Weak approximation of martingale representations

Weak approximation of martingale representations

Rama CONT and Yi LU
November 2014. Revision: September 2015.
Abstract

We present a systematic method for computing explicit approximations to martingale representations for a large class of Brownian functionals. The approximations are obtained by obtained by computing a directional derivative of the weak Euler scheme and yield a consistent estimator for the integrand in the martingale representation formula for any square-integrable functional of the solution of an SDE with path-dependent coefficients. Explicit convergence rates are derived for functionals which are Lipschitz-continuous in the supremum norm. Our results require neither the Markov property, nor any differentiability conditions on the functional or the coefficients of the stochastic differential equations involved.

1 Introduction

Let be a standard -dimensional Brownian motion defined on a probability space and its (-completed) natural filtration. Then, for any square-integrable -measurable random variable , or equivalently, any square-integrable -martingale , there exists a unique -predictable process with such that:

(1)

The classical proof of this representation result (see e.g. [33]) is non-constructive. However in many applications, such as stochastic control or mathematical finance, one is interested in an explicit expression for , which represents an optimal control or a hedging strategy. Expressions for the integrand have been derived using a variety of methods and assumptions, using Markovian techniques [10, 13, 15, 23, 30], integration by parts [2] or Malliavin calculus [1, 3, 21, 24, 27, 28, 18]. Some of these methods are limited to the case where is a Markov process; others require differentiability of in the Fréchet or Malliavin sense [3, 28, 18, 17] or an explicit form for the density [2]. Almost all of these methods invariably involve an approximation step, either through the solution of an auxiliary partial differential equation (PDE) or the simulation of an auxiliary stochastic differential equation.

A systematic approach to obtaining martingale representation formulae has been proposed in [7], using the Functional Itô calculus [12, 6, 5]: it was shown in [7, Theorem 5.9] that for any square-integrable -martingale ,

where is the weak vertical derivative of with respect to , constructed as an limit of pathwise directional derivatives. This approach does not rely on any Markov property nor on the Gaussian structure of the Wiener space and is applicable to functionals of a large class of Itô processes.

In the present work we build on this approach to propose a general framework for computing explicit approximations to the integrand in a general setting in which is allowed to be a functional of the solution of a stochastic differential equation (SDE) with path-dependent coefficients:

(2)

where designates the path stopped at and

are continuous non-anticipative functionals. For any square-integrable variable of the form where is a continuous functional, we construct an explicit sequence of approximations for the integrand in (1). These approximations are constructed as vertical derivatives, in the sense of the functional Itô calculus[7], of the weak Euler approximation of the martingale , obtained by replacing by the corresponding Euler scheme :

where is the concatenation of paths at and is the Dupire derivative [12, 4], a directional derivative defined as a pathwise limit of finite-diference approximations. and thus readily computable path-by-path in a simulation setting.

The main results of the paper are the following. We first show the existence and continuity of these pathwise derivatives in Theorem 4.1. The convergence of the approximations to the integrand in (1) is shown in Proposition 5.1. Under a Lipschitz assumption on , we provide in Theorem an error estimate for the approximation error. The proposed approximations are easy to compute and readily integrated in commonly used numerical schemes for SDEs.

Our approach requires neither the Markov property of the underlying processes nor the differentiability of coefficients, and is thus applicable to functionals of a large class of semimartingales. By contrast to methods based on Malliavin calculus [1, 3, 21, 24, 28, 18], it does not require Malliavin differentiability of the terminal variable nor does it involve any choice of ’Malliavin weights’, a delicate step in these methods.

Ideas based on Functional Itô calculus have also been recently used by Leão and Ohashi [26] for weak approximation of Wiener functionals, using a space-filtration discretization scheme. However, unlike the approach proposed in [26], our approach is based on a Euler approximation on a fixed time grid, rather than the random time grid used in [26], which involves a sequence of first passage times. Our approach is thus much easier to implement and analyze and is readily integrated in commonly used numerical schemes for approximations of SDEs, which are typically based on fixed time grids.

Outline

We first recall some key concepts and results from the Functional Itô calculus in section 2. Section 3 provides some estimates for the path-dependent SDE (2) and studies some properties of the Euler approximation for this SDE. In Section 4 we show that the weak Euler approximation (Definition 9) may be used to approximate any square-integrable martingale adapted to the filtration of by a sequence of smooth functionals of , in the sense of the functional Itô calculus. Moreover, we provide explicit expressions for the functional derivatives of these approximations. Section 5 analyzes the convergence of this approximation and provides error estimates in Theorem 5.1. Finally, in Section 6 we compare our approximation method with those based on Malliavin calculus.

Notations: In the sequel, we shall denote by the set of all matrices with real coefficients. We simply denote and . For , we shall denote by the transpose of , and the Frobenius norm of . For , is the scalar product on .

Let . We denote by the space of functions defined on with values in which are right continuous with left limits (càdlàg). For a path and , we denote by:

  • the value of at time ,

  • its left limit at ,

  • the path of stopped at

  • the supremum norm.

We note that and are elements of . For a càdlàg stochastic process , we shall similarly denote and .

2 Functional Itô calculus

The Functional Itô calculus [4] is a functional calculus which extends the Itô calculus to path-dependent functionals of stochastic processes. It was first introduced in a pathwise setting [6, 5, 12] using a notion of pathwise derivative for functionals on the space of right-continuous functions with left limits, and extended in [7] to a weak calculus applicable to all square-integrable martingales, which has a natural connection to the martingale representation theorem. We recall here some key concepts and results of this approach, following [4].

Let be the canonical process on , and be the filtration generated by . We are interested in non-anticipative functionals of , that is, functionals such that

(3)

The process then only depends on the path of up to and is -adapted.

It is convenient to define such functionals on the space of stopped paths [4]: a stopped path is an equivalence class in for the following equivalence relation:

(4)

The space of stopped paths is defined as the quotient of by the equivalence relation (4):

We denote the subset of consisting of continuous stopped paths. We endow this set with a metric space structure by defining the following distance:

is then a complete metric space. Any functional verifying the non-anticipativeness condition (3) can be equivalently viewed as a functional on :

Definition 1.

A non-anticipative functional on is a measurable map
: on the space of stopped paths.

Using the metric structure of , we denote by the set of continuous maps . Some weaker notions of continuity for non-anticipative functionals turn out to be useful [6]:

Definition 2.

A non-anticipative functional is said to be:

  • continuous at fixed times if for any , is continuous with respect to the uniform norm in , i.e. , , , ,

  • left-continuous if , , such that ,

    We denote by the set of left-continuous functionals. Similarly, we can define the set of right-continuous functionals.

We also introduce a notion of local boundedness for functionals.

Definition 3.

A non-anticipative functional is said to be boundedness-preserving if for every compact subset of , , such that:

We denote by the set of boundedness-preserving functionals.

We now recall some notions of differentiability for functionals following [7, 4]. For and , we define the vertical perturbation of as the càdlàg path obtained by shifting the path by after :

Definition 4.

A non-anticipative functional is said to be:

  • horizontally differentiable at if

    exists. If exists for all , then the non-anticipative functional is called the horizontal derivative of .

  • vertically differentiable at if the map:

    is differentiable at . Its gradient at is called the vertical derivative of at :

    with

    where is the canonical basis of . If is vertically differentiable at all , defines a non-anticipative map called the vertical derivative of .

We may repeat the same operation on and define similarly , , . This leads us to define the the following classes of smooth functionals:

Definition 5 (Smooth functionals).

We define as the set of non-anticipative functionals which are

  • horizontally differentiable with continuous at fixed times;

  • times vertically differentiable with for ;

  • .

We denote

Many examples of functionals may fail to be globally smooth, but their derivatives may still be well behaved except at certain stopping times, which motivates the following definition [4]:

Definition 6.

A non-anticipative functional is said to be locally regular of class if there exists an increasing sequence of stopping times with and , and a sequence of functionals such that:

We recall now the functional Itô formula for non-anticipative functionals of a continuous semimartingale [7, Theorem 4.1]:

Proposition 2.1 ([6, 7]).

Let be a continuous semimartingale defined on a probability space . For any non-anticipative functional and any , we have:

Actually the same functional Itô formula may also be obtained for functionals whose vertical derivatives are right-continuous rather than left-continuous. We denote by the set of non-anticipative functionals satisfying:

  • is horizontally differentiable with continuous at fixed times;

  • is twice vertically differentiable with and ;

  • ;

The localization is more delicate in this case, and we are not able to state a local version of the functional Itô formula by simply replacing by in Definition 6 (see Remark in [16]). However if the stopping times are deterministic, then the functional Itô formula is still valid (Proposition and Remark in [16]).

Definition 7.

A non-anticipative functional is said to be locally regular of class if there exists an increasing sequence of deterministic times with and , and a sequence of functionals such that:

Proposition 2.2 ([7]).

Let be a continuous semimartingale defined on a probability space . For any non-anticipative functional and any , we have:

Finally we present briefly the martingale representation formula established in [7]. Let be a continuous -valued martingale defined on a probability space with absolutely continuous quadratic variation:

where is a -valued process. Denote by the natural filtration of and the set of -adapted processes which admit a functional representation in :

(5)

If is non-singular almost everywhere, i.e. , -a.e., then for any , the predictable process

is uniquely defined up to an evanescent set, independently of the choice of in the representation (5). This process is called the vertical derivative of with respect to . For martingales which are smooth functionals of , the operator yields the integrand in the martingale representation theorem:

Corollary 2.1.

If is a square-integrable martingale, then

Consider now the case where is a square-integrable martingale. Let be the space of square-integrable -martingales with initial value zero, equipped with the norm . Cont & Fournié [7, Theorem 5.8] show that the operator admits a unique continuous extension to a weak derivative which satisfies the following martingale representation formula:

Proposition 2.3 ([7]).

For any square-integrable -martingale , we have:

This weak vertical derivative coincides with the pathwise vertical derivative when admits a locally regular functional representation, i.e. with . For a general square-integrable martingale , the weak derivative is not directly computable through a pathwise perturbation. An approximation procedure is thus necessary for computing . The result of [7] guarantees the existence of such approximations; in the sequel we propose explicit, and computable, constructions of such approximations.

3 Euler approximations for path-dependent SDEs

Let be a standard d-dimensional Brownian motion defined on a probability space and its (-completed) natural filtration. We consider the following stochastic differential equation with path-dependent coefficients (2):

where are non-anticipative maps, assumed to be Lipschitz continuous with respect to the following distance defined on :

Assumption 1 (Lipschitz continuity of coefficients).

are Lipschitz continuous:

Remark 3.1.

This Lipschitz condition with respect to the distance is weaker than a Lipschitz condition with respect to the distance introduced in the previous section: it allows for a Hölder smoothness of degree 1/2 in the variable.

Under Assumption 1, (2) has a unique strong, -adapted, solution .

Proposition 3.1.

Under Assumption 1, there exists a unique -adapted process satisfying (2). Moreover for , we have:

(6)

for some constant depending on and .

Remark 3.2.

Assumption 1 might seem to be quite strong. Indeed, the previous proposition still holds under weaker conditions. For example, the Hölder condition with respect to differences in can be replaced by the weaker condition: where denotes the path which takes constant value . However, this assumption is necessary for the convergence of the Euler approximation described later in this section, and especially the results concerning its rate of convergence.

Proof.

Existence and uniqueness of a strong solution follows from [32] (Theorem 7 , Chapter 5): see [4, Section 5]. Let us prove (6). Using the Burkholder-Davis-Gundy inequality and Hölder’s inequality, we have:

And we conclude by Gronwall’s inequality. ∎

In the following, we always assume that Assumption 1 holds. The strong solution of equation (2) is then a semimartingale and defines a non-anticipative functional given by the Itô map associated to (2).

3.1 Euler approximations as non-anticipative functionals

We now consider an Euler approximation for the SDE (2) and study its properties as a non-anticipative functional. Let . The Euler approximation of on the grid is defined as follows:

Definition 8.

[Euler scheme] For , we denote by the piecewise constant Euler approximation for (2) computed along the path , defined as follows: is constant in each interval , with and

(7)

where and by convention .

When computed along the path of the Brownian motion is simply the piecewise constant Euler-Maruyama scheme [31] for the stochastic differential equation (2).

By definition, the path depends only on a finite number of increments of : , , . We can thus define a map

such that for

(8)

By a slight abuse of notation, we denote the path stopped at .

The map is then locally Lipschitz continuous, as shown by the following lemma.

Lemma 3.1.

For every , there exists a constant such that for any

Proof.

As the two paths and are stepwise constant by construction, it suffices to prove the inequality at times . We prove by induction that:

(9)

with some constant which depends only on , (and ).

For , this is clearly the case as . Assume that (9) is verified for some , consider now , we have:

and

Thus

(The constant may differ from one line to another).
And consequently we have:

for some different constant depending only on , and (and ). And we conclude by induction. ∎

3.2 Strong convergence

To simplify the notations, will be noted simply in the following. The following result, which gives a uniform estimate of the discretization error, extends similar results known in the Markovian case [14, 31, 22] to the path-dependent SDE (2) (see also [20]):

Proposition 3.2.

Under Assumption 1 we have the following estimate in for the strong error of the piecewise constant Euler-Maruyama scheme:

with a constant depending only on , , and .

Proof.

The idea is to construct a ’Brownian interpolation’ of :

where is the largest subdivision point which is smaller or equal to .

Clearly is a continuous semimartingale and can be controlled by the sum of the two following terms:

(10)

We start with the term . Using the Burkholder-Davis-Gundy inequality and Hölder’s inequality, we have

We have used as is piecewise constant.

Consider now the second term