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

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 aproior assumption are present. Numerical verification of the regularization, including numerical algorithm and examples are also illustrated.