Determination of the solution of a stochastic parabolic equation by the terminal value

Abstract

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