Stochastic optimal control problems with measurable coefficients and $L_d$-drift

TL;DR

This paper studies stochastic control problems with measurable coefficients, establishing the connection between value functions, PDEs, and semigroups, and demonstrating regularization effects of noise without requiring regularity assumptions.

Contribution

It provides new regularity estimates and a stochastic representation for value functions in control problems with measurable coefficients and $L_d$-drift, extending previous frameworks.

Findings

Value functions are $L_{d_0}$-viscosity solutions to Hamilton-Jacobi-Bellman equations.

The semigroup associated with finite horizon costs regularizes lower semicontinuity to local Hölder continuity.

The semigroup is an $L_{d+1}$-viscosity solution to a related PDE with established regularity estimates.

Abstract

We consider controlled stochastic differential equations (SDEs) with measurable coefficients, a uniformly elliptic diffusion coefficient and an $L_d$-drift. No space-regularity will be assumed for the coefficients. In this framework we investigate the relation of value functions, partial differential equations (PDEs) and operator semigroups. First, for a cost with infinite time horizon on a bounded domain, we identify the value function as $L_{d_0}$-viscosity solution to a Hamilton-Jacobi-Bellman equation and we establish quantitative regularity estimates. The constant $d_0 \in (d/2, d)$ only depends on the space dimension $d$, the ellipticity constants of the diffusion coefficient and the $L_d$-bound of the drift. To illustrate applications of these results, we provide a uniqueness theorem under an additional assumption on the diffusion coefficient, showing a stochastic representation, and we discuss stability of value functions. Second, we consider a cost with a finite time horizon, terminal and running terms. We show that the value function indexed over the terminal cost is a nonlinear semigroup on $C_b (\mathbb{R}^d)$ and we establish a regularization by noise effect, which shows that the semigroup regularizes lower semicontinuity to local H\"older continuity. Lastly, we relate the semigroup to a parabolic PDE, showing that it is an $L_{d + 1}$-viscosity solution, and we establish local in time and global in space quantitative regularity estimates. Our proofs for the regularity of the value functions, the $C_b$-Feller property of the semigroup and its regularization by noise effects are based on a strong Markov selection principle and analytic estimates for linear diffusions that were recently established by N. V. Krylov in a series of papers. We highlight that our method covers frameworks without uniqueness of the controlled SDEs, as well as the associated PDEs.