Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 968093, 5 pages http://dx.doi.org/10.1155/2014/968093
Research Article Comparison Theorem for Nonlinear Path-Dependent Partial Differential Equations Falei Wang School of Mathematics, Shandong University, Jinan, Shandong 250100, China Correspondence should be addressed to Falei Wang;
[email protected] Received 27 March 2014; Accepted 19 April 2014; Published 29 April 2014 Academic Editor: Tongxing Li Copyright © 2014 Falei Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce a type of fully nonlinear path-dependent (parabolic) partial differential equation (PDE) in which the path 𝜔𝑡 on an interval [0, 𝑡] becomes the basic variable in the place of classical variables (𝑡, 𝑥) ∈ [0, 𝑇] × R𝑑 . 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 𝐺-expectation theory. In the 𝐺-expectation framework, the notion of 𝐺-Brownian motion and the corresponding stochastic calculus of Itˆo’s type were established. The key issue is that 𝐺-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 Fourni´e [5] introduced a new functional Itˆo formula which nontrivially generalized the classical one through new notion-path derivatives (see [6, 7] for more general and systematic research). It extends the Itˆo 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 Fourni´e [5].
2
Abstract and Applied Analysis
Let 𝑇 > 0 be fixed. For each 𝑡 ∈ [0, 𝑇], we denote by Λ 𝑡 the set of right continuous R𝑑 -valued functions on [0, 𝑡]. For each 𝜔𝑇 ∈ Λ 𝑇 the value of 𝜔𝑇 at time 𝑠 ∈ [0, 𝑇] is denoted by 𝜔(𝑠). Thus 𝜔𝑇 = 𝜔(𝑠)0≤𝑠≤𝑇 is a right continuous process on [0, 𝑇] and its value at time 𝑠 is 𝜔(𝑠). The path of 𝜔𝑇 up to time 𝑡 is denoted by 𝜔𝑡 ; that is, 𝜔𝑡 = 𝜔(𝑠)0≤𝑠≤𝑡 ∈ Λ 𝑡 . Denote Λ = ⋃𝑡∈[0,𝑇] Λ 𝑡 . We sometimes specifically write 𝜔𝑡 = 𝜔(𝑠)0≤𝑠 0, the functions defined by 𝑢1 := 𝑢1 − 𝛿/𝑡 are a subsolution of 𝐷𝑡 𝑢1 (𝜔𝑡 ) + 𝐺1 (𝜔𝑡 , 𝑢1 (𝜔𝑡 ) , 𝐷𝑥 𝑢1 (𝜔𝑡 ) , 𝐷𝑥𝑥 𝑢1 (𝜔𝑡 )) =
𝛿 , 𝑡2
(20)
4
Abstract and Applied Analysis
where 𝐺1 (𝜔𝑡 , 𝑟, 𝑝, 𝑋) := 𝐺1 (𝜔𝑡 , 𝑟 + 𝛿/𝑡, 𝑝, 𝑋). It is easy to check that the function 𝐺1 satisfies the same conditions as 𝐺1 . Since 𝑢1 ≤ 𝑢2 follows from 𝑢1 ≤ 𝑢2 in the limit 𝛿 → 0, it suffices to prove Theorem 14 under the additional assumption 𝐷𝑡 𝑢1 (𝜔𝑡 ) + 𝐺1 (𝜔𝑡 , 𝑢1 (𝜔𝑡 ) , 𝐷𝑥 𝑢1 (𝜔𝑡 ) , 𝐷𝑥𝑥 𝑢1 (𝜔𝑡 )) ≥ 𝑐, 𝑐= lim 𝑢 (𝜔𝑡 ) = −∞,
𝑡→0
𝛿 , 𝑇2
uniformly on [0, 𝑇) . (21)
We make the following assumption. (A) For each 𝜔𝑡 ∈ Λ, 𝑢, V ∈ R, 𝑝 ∈ R𝑑 , and 𝑄 ∈ S(𝑑) such that 𝑢 ≥ V, we have 𝐺 (𝜔𝑡 , 𝑢, 𝑝, 𝑄) ≤ 𝐺 (𝜔𝑡 , V, 𝑝, 𝑄) . −𝜆𝑡
(22)
−𝜆𝑡
Set 𝑢1 (𝜔𝑡 ) = 𝑢1 (𝜔𝑡 )𝑒 , 𝑢2 (𝜔𝑡 ) = 𝑢2 (𝜔𝑡 )𝑒 , for some 𝜆 < −𝐶. Then it is easy to check that 𝑢𝑖 (𝑖 = 1, 2) is a subsolution or supersolution of 𝐷𝑡 𝑢𝑖 (𝜔𝑡 ) + 𝐺𝑖 (𝜔𝑡 , 𝑢𝑖 (𝜔𝑡 ) , 𝐷𝑥 𝑢𝑖 (𝜔𝑡 ) , 𝐷𝑥𝑥 𝑢𝑖 (𝜔𝑡 )) = 0, (23) where, for each (𝜔𝑡 , 𝑢, 𝑝, 𝑄) ∈ Λ × R × R𝑑 × S(𝑑), 𝐺𝑖 (𝜔𝑡 , 𝑟, 𝑝, 𝑄) is given by 𝐺𝑖 (𝜔𝑡 , 𝑢, 𝑝, 𝑄) = 𝜆𝑢 + 𝑒−𝜆𝑡 𝐺𝑖 (𝜔𝑡 , 𝑒𝜆𝑡 𝑢, 𝑒𝜆𝑡 𝑝, 𝑒𝜆𝑡 𝑄) ,
(24)
which satisfies the assumption (A). Since 𝑢1 ≤ 𝑢2 implies 𝑢1 ≤ 𝑢2 , it suffices to prove Theorem 14 under the additional assumption (A). Without loss of generality, assume 𝑤𝛼 ≤ 𝐶2 . Suppose by the contrary that there exist 𝜔𝑡00 ∈ Λ, 𝑡0 < 𝑇, and √𝛼 < 𝑐/4(𝐶 ∨ 𝐶2 ) ∧ 1, such that 𝑚0 := 𝑤𝛼 (𝜔𝑡00 ) > 0.
(25)
Then, by Lemma 15, there exists 𝜔𝑡 ∈ Λ such that 𝑤𝛼 (𝜔𝑡 ) = sup 𝑤𝛼 (𝜔𝑡 ⊗ 𝛾𝑡 ) ≥ 𝑚0 . 𝛾𝑡 ∈Λ,𝑡≥𝑡
(26)
Consequently, 𝐷𝑡 𝑢1 (𝜔𝑡 ) − 𝐷𝑡 𝑢2 (𝜔𝑡 ) ≤ 0, 𝐷𝑥 𝑢1 (𝜔𝑡 ) − 𝐷𝑥 𝑢2 (𝜔𝑡 ) = 2𝛼𝜔 (𝑡) ,
(27)
𝐷𝑥𝑥 𝑢1 (𝜔𝑡 ) − 𝐷𝑥𝑥 𝑢2 (𝜔𝑡 ) ≤ 2𝛼. 2
Since 𝛼𝜔 (𝑡) ≤ 𝐶2 , we have 𝛼𝜔(𝑡) ≤ 𝐶√𝛼. Then 0 0. Denote 𝑢(𝜔𝑠 ) := V1 (𝑠, 𝜔(𝑠), 𝜔(𝑡))1𝑡≤𝑠 + V2 (𝑠, 𝜔(𝑠))1𝑠
