Abstract :
This paper is devoted to the study of a second order Hamilton]Jacobi equation
in infinite dimensions with a locally Lipschitz continuous Hamiltonian H, related
with a stochastic optimal control problem driven by white noise. In an earlier paper
the author studied the case of globally Lipschitz continuous Hamiltonian H. This
paper is concerned with the case of locally Lipschitz continuous H that include
e.g., the quadratic case, which appears very frequently in the applications. Existence,
uniqueness, and C2 regularity of a local solution follow by methods given
previously. The heart of the work is the proof of a priori estimates of the C1-norm
of the local solution. This has been done by studying properties of nonlinear
transition semigroups and by a careful application of them to our problem. The
results have been applied to a stochastic optimal control problem proving existence
of feedback optimal controls.
