A generalized Hamilton-Jacobi-Bellman equation

Abstract

We interpret the following fully nonlinear second order partial differential equation

$$\left\{ \begin{gathered}\partial _t u + \mathop {\inf }\limits_\alpha \left\{ {\mathcal{L}\left( {x, \alpha } \right)u + f\left( {x, u, \partial _x u\sigma \left( {x, \alpha } \right),\alpha } \right)} \right\} = 0,\left( {x, t} \right) \in D \times \left( {0, T} \right), \hfill \\for \left( {x, t} \right) \in D \times \left[ {0, T} \right];u\left( {x, T} \right) = g\left( x \right). \hfill \\\end{gathered} \right.$$

as the value function of certain optimal controlled diffusion problem. Where A ∈ ℝ ^k is control domain. xxxL(x, α) is a second order elliptic partial differential operator parametrized by the control variable α ∈ A ⊂ ℝ ^k. A particular case of this equation is when f=f(x, α). 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 as classical one. The cost function is described by a solution of certain backward stochastic differential equation.

partially supported by the Chinese National Natural Science Fundation.