ANALYSIS OF HAMILTON-JACOBI-BELLMAN EQUATIONS ARISING IN STOCHASTIC SINGULAR CONTROL

摘要

We study the partial differential equation max {Lu - f, H(Du)} = 0 where u is the unknown function, L is a second-order elliptic operator,f is a given smooth function and H is a convex function. This is a model equation for Hamilton-Jacobi-Bellman equations arising in stochastic singular control. We establish the existence of a unique viscosity solution of the Dirichlet problem that has a Hoelder continuous gradient. We also show that if H is uniformly convex, the gradient of this solution is Lipschitz continuous.