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

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