Mild solutions to the dynamic programming equation for stochastic optimal control problems

Abstract

We show via the nonlinear semigroup theory in $L^1(\mathbb{R})$ that the $1$-D dynamic programming equation associated with a stochastic optimal control problem with multiplicative noise has a unique mild solution $\phi \in C([0,T];W^{1,\infty }(\mathbb{R}))$ with ${\phi }_{xx}\in C([0,T];L^1(\mathbb{R}))$. The $n$-dimensional case is also investigated.