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Applied Mathematics and Optimization

, Volume 27, Issue 2, pp 125–144 | Cite as

Backward stochastic differential equations and applications to optimal control

  • Shige Peng
Article

Abstract

We study the existence and uniqueness of the following kind of backward stochastic differential equation,
$$x(t) + \int_t^T {f(x(s),y(s),s)ds + \int_t^T {y(s)dW(s) = X,} }$$
under local Lipschitz condition, where (Ω, ℱ,P, W(·), ℱt) is a standard Wiener process, for any given (x, y),f(x, y, ·) is an ℱt-adapted process, andX is ℱt-measurable. The problem is to look for an adapted pair (x(·),y(·)) that solves the above equation. A generalized matrix Riccati equation of that type is also investigated. A new form of stochastic maximum principle is obtained.

Key words

Backward stochastic differential equations Controlled diffusion processes Stochastic maximum principle Matrix-valued random Riccati equations 

AMS classification

60H 93E 

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Copyright information

© Springer-Verlag New York Inc. 1993

Authors and Affiliations

  • Shige Peng
    • 1
  1. 1.Department of MathematicsShandong UniversityJinan, ShandongPeople's Republic of China

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