Shige Peng

Adapted solution of a backward stochastic differential equation

Abstract Let Wt; t ϵ [0, 1] be a standard k-dimensional Weiner process defined on a probability space ( Ω, F, P ), and let Ft denote its natural filtration. Given a F1 measurable d-dimensional random vector X, we look for an adapted pair of processes {x(t), y(t); t ϵ [0, 1]} with values in Rd and Rd×k respectively, which solves an equation of the form: x(t) + ∫ t 1 f(s, x(s), y(s)) ds + ∫ t 1 [g(s, x(s)) + y(s)] dW s = X. A linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process. We also generalize our results to the following equation: x(t) + ∫ t 1 f(s, x(s), y(s)) ds + ∫ t 1 g(s, x(s)) + y(s)) dW s = X under rather restrictive assumptions on g.

BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS IN FINANCE

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).

A general stochastic maximum principle for optimal control problems

The maximum principle for nonlinear stochastic optimal control problems in the general case is proved. The control domain need not be convex, and the diffusion coefficient can contain a control variable.

Fully Coupled Forward-Backward Stochastic Differential Equations and Applications to Optimal Control

Existence and uniqueness results of fully coupled forward-backward stochastic differential equations with an arbitrarily large time duration are obtained. Some stochastic Hamilton systems arising in stochastic optimal control systems and mathematical finance can be treated within our framework.

Solution of forward-backward stochastic differential equations

SummaryIn this paper, we study the existence and uniqueness of the solution to forward-backward stochastic differential equations without the nondegeneracy condition for the forward equation. Under a certain “monotonicity” condition, we prove the existence and uniqueness of the solution to forward-backward stochastic differential equations.

G-Expectation, G-Brownian Motion and Related Stochastic Calculus of Itô Type

We introduce a notion of nonlinear expectation --G--expectation-- generated by a nonlinear heat equation with infinitesimal generator G. We first discuss the notion of G-standard normal distribution. With this nonlinear distribution we can introduce our G-expectation under which the canonical process is a G--Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Itos type with respect to our G--Brownian motion and derive the related Itos formula. We have also give the existence and uniqueness of stochastic differential equation under our G-expectation. As compared with our previous framework of g-expectations, the theory of G-expectation is intrinsic in the sense that it is not based on a given (linear) probability space.

Backward stochastic differential equations and applications to optimal control

AbstractWe study the existence and uniqueness of the following kind of backward stochastic differential equation,

A Generalized dynamic programming principle and hamilton-jacobi-bellman equation

x(t) + \int_t^T {f(x(s),y(s),s)ds + \int_t^T {y(s)dW(s) = X,} }