A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation. (English) Zbl 0756.49015

Peng, Shige

Stochastics Stochastics Rep. 38, No. 2, 119-134 (1992).

A fully nonlinear second order partial differential equation is interpreted as the value function of a certain optimally controlled diffusion problem. 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. Bellman’s dynamic programming principle is discussed for this problem. The value function is proved to be a viscosity solution of the possibly degenerate fully nonlinear equation.

Reviewer: R.Gessing (Gliwice)

MSC:

49L20	Dynamic programming in optimal control and differential games
49L25	Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games

Keywords:

nonlinear second order partial differential equation; optimally controlled diffusion problem; viscosity solution