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

## TL;DR

This paper establishes the existence and uniqueness of mild solutions to the 1-D dynamic programming equation in stochastic optimal control with multiplicative noise, using nonlinear semigroup theory, and extends results to higher dimensions.

## Contribution

It introduces a novel approach using nonlinear semigroup theory to analyze the dynamic programming equation in stochastic control, including multidimensional cases.

## Key findings

- Unique mild solution in 1D for the dynamic programming equation
- Solution regularity in $C([0,T];W^{1,inity})$ and $ ext{second derivative}$ in $C([0,T];L^1)$
- Extension of results to n-dimensional stochastic 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 $\varphi\in C([0,T];W^{1,\infty}(\mathbb{R}))$ with $\varphi_{xx}\in C([0,T];L^1(\mathbb{R}))$. The $n$-dimensional case is also investigated.