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Convex Analysisの二,三の進展について

Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Stochastic Equations in Infinite Dimensions

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

We are concerned with different properties of backward stochastic differential equations and their applications to finance. These equations, first introduced by Pardoux and Peng (1990), are useful for the theory of contingent claim valuation, especially cases with constraints and for the theory of recursive utilities, introduced by Duffie and Epstein (1992a, 1992b).

Backward Stochastic Differential Equations in Finance

We study here quasi-linear parabolic equations with quadratic growth in $\mathbb {R}^{n}$. These parabolic equations are at the core of the theory of PDE; see Friedman (Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs, 1964) [6], Ladyzhenskaya et al. (Translations of Mathematical Monographs. AMS, 1968) [4] for details. However, for the applications to physics and mechanics, one deals mostly with boundary value problems. The boundary is often taken to be bounded and the solution is bounded. This brings an important simplification. On the other hand, stochastic control theory leads mostly to problems in $\mathbb {R}^{n}$. Moreover, the functions are unbounded and the Hamiltonian may have quadratic growth. There may be conflicts which prevent solutions to exist. In stochastic control theory, a very important development deals with BSDE (Backward Stochastic Differential Equations). There is a huge interaction with parabolic PDE in $\mathbb {R}^{n}$. This is why, although we do not deal with BSDE in this paper, we use many ideas from Briand and Hu (Probab Theory Relat Fields 141(3–4):543–567, 2008) [1], Da Lio and Ley (SIAM J Control Optim 45(1):74–106, 2006) [2], Karoui et al. (Backward stochastic differential equations and applications, Princeton BSDE Lecture Notes, 2009) [3], Kobylanski (Ann Probab 28(2):558–602, 2000) [5]. Our presentation provided here is slightly innovative.

Parabolic Equations with Quadratic Growth in $\mathbb {R}^{n}$

We consider a class of discretionary stopping problems within the $G$-framework. We first establish the well-definedness of the stopping problem under the $G$-expectation, by showing the quasi-continuity of the stopped process. We then prove a verification theorem for $G$-optimal stopping problem. One corollary is a direct proof for the well-known fact that the $G$-optimal stopping problem is the same as the classical optimal stopping problem with appropriate parameters, when the payoff function is concave or convex.

A Note on $G$- Optimal Stopping Problems

In this paper, we study rough path properties of stochastic integrals of Ito's type and Stratonovich's type with respect to $G$-Brownian motion. The roughness of $G$-Brownian Motion is estimated and then the pathwise Norris lemma in $G$-framework is obtained.

Stochastic Calculus with respect to G-Brownian Motion Viewed through Rough Paths

We formulate and solve the martingale problem in a nonlinear expectation space. Unlike the classical work of Stroock and Varadhan (1969) where the linear operator in the associated PDE is naturally defined from the corresponding diffusion process, the main difficulty in the nonlinear setting is to identify an appropriate class of nonlinear operators for the associated fully nonlinear PDEs. Based on the analysis of the martingale problem, we introduce the notion of weak solution for stochastic differential equations under nonlinear expectations and obtain an existence theorem under the Holder continuity condition of the coefficients. The approach to establish the existence of weak solutions generalizes the classical Girsanov transformation method in that it no longer requires the two (probability) measures to be absolutely continuous.

Shige Peng

Papers

Parabolic Equations with Quadratic Growth in \(\mathbb {R}^{n}\)

A Note on $G$- Optimal Stopping Problems

Stochastic Calculus with respect to G-Brownian Motion Viewed through Rough Paths

Martingale Problems under Nonlinear Expectation