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Backward stochastic differential equations and applications to optimal control

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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.

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Authors and Affiliations

  1. Department of Mathematics, Shandong University, 250100, Jinan, Shandong, People's Republic of China

    Shige Peng

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  1. Shige Peng
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Additional information

Communicated by D. Ocone

This work was partially supported by the Chinese National Natural Science Foundation and SEDC Foundation for Young Academics.

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Peng, S. Backward stochastic differential equations and applications to optimal control. Appl Math Optim 27, 125–144 (1993). https://doi.org/10.1007/BF01195978

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  • DOI: https://doi.org/10.1007/BF01195978

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