Stochastic Variational formulas for solutions to linear diffusion equations

TL;DR

This paper derives a stochastic variational formula for solutions to a linear diffusion equation, linking it to stochastic control problems with singular terminal data, and analyzes the zero noise limit convergence.

Contribution

It introduces a novel stochastic variational formula for the diffusion equation's solution with singular terminal data, extending control theory methods.

Findings

Proves convergence of the stochastic control cost function in the zero noise limit.

Establishes a variational formula for the logarithm of the diffusion solution.

Connects solutions of diffusion equations to stochastic control with singular terminal conditions.

Abstract

This paper is concerned with solutions to a one dimensional linear diffusion equation and their relation to some problems in stochastic control theory. A stochastic variational formula is obtained for the logarithm of the solution to the diffusion equation, with terminal data which is the characteristic function of a set. In this case the terminal data for the control problem is singular, and hence standard theory does not apply. The variational formula is used to prove convergence in the zero noise limit of the cost function for the stochastic control problem and its first derivatives, to the corresponding quantities for a classical control problem.