Determination the Solution of a Stochastic Parabolic Equation by the Terminal Value

TL;DR

This paper addresses the inverse problem of reconstructing the history of a stochastic diffusion process from its terminal value, establishing stability, proposing a Tikhonov-based regularization, and verifying it through numerical examples.

Contribution

The paper introduces a new Carleman estimate for stochastic parabolic equations and develops a regularization method with proven stability, existence, and error estimates.

Findings

Conditional stability established via Carleman estimate

A Tikhonov regularization method is proposed and analyzed

Numerical examples verify the effectiveness of the regularization

Abstract

This paper studies the inverse problem of determination the history for a stochastic diffusion process, by means of the value at the final time $T$. By establishing a new Carleman estimate, the conditional stability of the problem is proven. Based on the idea of Tikhonov method, a regularized solution is proposed. The analysis of the existence and uniqueness of the regularized solution, and proof for error estimate under an a-proior assumption are present. Numerical verification of the regularization, including numerical algorithm and examples are also illustrated.