Comparison Theorem for Nonlinear Path-Dependent Partial Differential Equations

Abstract

We introduce a type of fully nonlinear path-dependent (parabolic) partial differential equation (PDE) in which the path on an interval [0, ] becomes the basic variable in the place of classical variables . Then we study the comparison theorem of fully nonlinear PPDE and give some of its applications.

1. Introduction

Motivated by uncertainty problems, risk measures, and the superhedging in finance mathematics, Peng [1] systemically established -expectation theory. In the -expectation framework, the notion of -Brownian motion and the corresponding stochastic calculus of Itô’s type were established. The key issue is that -diffusion processes are connected to a large class of fully nonlinear PDEs in the Markov case studied by Peng [2] and Soner et al. [3].

Recently, Dupire [4] and Cont and Fournié [5] introduced a new functional Itô formula which nontrivially generalized the classical one through new notion-path derivatives (see [6, 7] for more general and systematic research). It extends the Itô stochastic calculus to functionals of a given process. It provides an excellent tool for the study of path-dependence. In fact, they showed that a smooth path functional solves a linear path-dependent PDE if its composition with a Brownian motion generates a martingale, which provided a functional extension of the classical Feynman-Kac formula. Moreover, by virtue of a backward stochastic differential equation approach, we can obtain the uniqueness of the smooth solution to the semilinear path-dependent PDE (see also [8–10] and the references therein). However, these methods are mainly based on stochastic calculus.

The objective of this paper is to study fully nonlinear PPDE. We refer to Krylov [11] and Wang [12] for the classical fully nonlinear PDE (see also [13–15]). Peng [16] introduced an approach of frozenness of the main course of the paths where the maximization takes place. This approach is based on techniques of PDE and can be directly applied to treat fully nonlinear path-dependent PDE. The advantage of this PDE approach is that one can treat the solution locally (path by path), whereas stochastic calculus is mainly a global approach. In this paper, we will use this method to obtain a comparison principle of fully nonlinear PPDE. In particular, some properties of the solution to fully nonlinear PPDE are also obtained. We claim that these ideas carry over to much more general frameworks, such as the case of viscosity solution to PPDE. Moreover, this method can have direct applications to stochastic analysis, for example, martingales under a fully nonlinear expectation, stochastic optimal control problems, stochastic games, nonlinear pricing and risk measuring, and backward stochastic differential equations. These more technical details are left to future work and will be presented in forthcoming papers.

The paper is organized as follows. In Section 2, we present some existing results in the theory of Dupire’s path derivatives that we will use in this paper. In Section 3, we obtain the comparison theorem of fully nonlinear PPDE and give some of its applications.

2. Preliminaries

In this section, we give an overview of the definitions concerning path derivatives. The following notations are mainly from Dupire [4] and Cont and Fournié [5].

Let be fixed. For each , we denote by the set of right continuous -valued functions on . For each the value of at time is denoted by . Thus is a right continuous process on and its value at time is . The path of up to time is denoted by ; that is, . Denote . We sometimes specifically write to indicate the terminal position of which often plays a special role in this framework. For each and , denote by the value of at and which is also an element of .

Now consider the function of path; that is, . This function can be also regarded as a family of real valued functions: Denote , for , .

We introduce the distance on . Let and denote the inner product and norm in . For each and , denote

It is obvious that is a Banach space with respect to . Since is not a linear space, is not a norm.

Definition 1 (continuous). A function is said to be -continuous at , if for any there exists such that for each with we have . is said to be -continuous if it is -continuous at each .

Remark 2. In our framework we often regard as a function of , , and ; that is, . Thus, for a fixed , is regarded as a function of .

Definition 3. Given and , if there exists , such that then we say that is (vertically) differentiable at and denote . is said to be vertically differentiable in if exists for each . We can similarly define the Hessian . It is an -valued function defined on , where is the space of all symmetric matrices.

For each , with , set It is clear that and .

Definition 4. For a given if then we say that is (horizontally) differentiable in at and denote . is said to be horizontally differentiable in if exists for each .

Definition 5. Define as the set of functions defined on which are times horizontally and times vertically differentiable in , such that all these derivatives are -continuous.

Example 6. If with , then which is the classic derivative. In general, these derivatives also satisfy the classic properties: linearity, product, and chain rule.

3. Comparison Theorem for Fully Nonlinear PPDE

Now we introduce the following fully nonlinear path-dependent PDE: where and are continuous functions. Moreover, satisfies the following elliptic conditions: for each , , , , and, for each , , , there exists some constant such that

Definition 7. A function is called a -solution of the path-dependent PDE (8) if, for each , , equality (8) is satisfied. is called a subsolution (resp., supersolution) of (8) if the “” in (8) is replaced by “” (resp., “”).

Remark 8. The solution of classical PDE is a special case when , . Indeed, for each and , and thus is a classical solution of PDE.

The following result is the so-called comparison principle or comparison theorem of path-dependent PDE.

Theorem 9. Suppose is a -subsolution and is a -supersolution. Moreover, is bounded from above and is bounded from below. Then the maximum principle holds: if for all , then for each .

Remark 10. In the case when for some Lipschitz function on , the above result is the comparison theorem of semilinear path-dependent PDE, which is given by [9].

In order to prove Theorem 9, we will make use of the following definitions.

Definition 11. is said to be in , if it satisfies the following conditions: (i)for each fixed , is an upper semicontinuous function of ,(ii)for each with , .
We also denote . A function (resp., ) is called a -upper (resp., -lower) semicontinuous function. is called a -continuous function.

Definition 12. Define as the set of functions , such that, for each , , , exist.

Definition 13. A function is called a -solution of the path-dependent PDE (8) if for each , equality (8) is satisfied. is called a -subsolution (resp., -supersolution) of (8) if the “” in (8) is replaced by “” (resp., “”).

Theorem 14. Suppose is a -subsolution of PPDE (8) with and is a -supersolution of PPDE (8) with , where for each . Moreover, is bounded from above and is bounded from below. Then the maximum principle holds: if for all , then for each .

It is obvious that . Then Theorem 9 is a direct consequence of Theorem 14.

For each and , set

In order to prove Theorem 14, we need the following lemma, which is essentially from Peng [16, Lemma 6].

Lemma 15. If for some , then there exists , satisfying , , and , such that

Proof. For each and , we can find such that and for each with we get
Then the proof is immediate in light of Lemma 6 of Peng [16].

Lemma 16. Let and be given satisfying for all , . Then

Proof. Since for each , we conclude For each , and thus which is the desired result.

Now we are going to give the proof of Theorem 14.

Proof of Theorem 14. We first observe that, for , the functions defined by are a subsolution of where . It is easy to check that the function satisfies the same conditions as . Since follows from in the limit , it suffices to prove Theorem 14 under the additional assumption
We make the following assumption.
(A) For each , , , and such that , we have
Set , , for some . Then it is easy to check that is a subsolution or supersolution of where, for each , is given by which satisfies the assumption (A). Since implies , it suffices to prove Theorem 14 under the additional assumption (A).
Without loss of generality, assume . Suppose by the contrary that there exist , , and , such that
Then, by Lemma 15, there exists such that Consequently, Since , we have . Then This induces a contradiction and the proof is completed.

Corollary 17. The path-dependent PDE (8) has at most one bounded -solution.

Lemma 18. If is a bounded -subsolution of PPDE (8) and is a bounded function on , then, for each ,

Proof. Since is a -supersolution of PPDE (8), applying Theorem 14, we have the desired result.

Remark 19. If is the -solution of PPDE (8) for some bounded , then the above theorem is the classical comparison theorem of PDE of [17].

Example 20. Consider the following PPDE: where for some .
If for some bounded Lipschitz function and , then we can solve the PPDE (30) by the following method.
First consider the following system of fully nonlinear parabolic partial differential equations, defined on and parameterized by : and, then, another one defined on : From Krylov [11] and Wang [18], for each , , and , where and .
Denote ; we obtain . Applying Theorem 14, is the unique -solution of PPDE (30). Indeed, is the conditional -expectation of (see [19, 20]).

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the Graduate Independent Innovation Foundation (YZC12062) of Shandong University.