Article1992

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

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Stochastic processes and financial applications Nonlinear Partial Differential Equations Adaptive Dynamic Programming Control

Abstract

We interpret the following fully nonlinear second-order partial differential equation as the value function of a certain optimal controlled diffusion problem, where is a second order elliptic partial differential operator parametrized by the control variable αϵA: with Here σ,b, and c are functions defined on with values respectively in and is a real function defined on . A particular case of this equation is when . In this case, the equation is the well-known Hamilton-Jacobi-Bellman equation. The problem is formulated as follows: The state equation of the control problem is a classical one. The cost function is described by an adapted solution of a certain backward stochastic differential equation. The paper discusses Bellman's dynamic programming principle for this problem The value function is proved to be a viscosity solution of the above possibly degenerate fully nonlinear equation

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References

1990Adapted solution of a backward stochastic differential equation 1980Controlled Diffusion Processes 1990A General Stochastic Maximum Principle for Optimal Control Problems 1991Probabilistic interpretation for systems of quasilinear parabolic partial differential equations 1992Asset Pricing with Stochastic Differential Utility 1978An Introductory Approach to Duality in Optimal Stochastic Control 1972Necessary Conditions for Continuous Parameter Stochastic Optimization Problems 1992Stochastic Hamilton–Jacobi–Bellman Equations

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