On backward problems for stochastic fractional reaction equations with standard and fractional Brownian motion

Abstract

In this work, we study two final value problems for fractional reaction equation with standard Brownian motion W(t) and fractional Brownian motion BH(t), for H∈(14,12)∪(12,1). Firstly, the well-posedness of each problem is investigated under strongly choices of data. We aim to find spaces where we obtain the existence of a unique solution of each problem, and establish some regularity results. Next, since the first problem and the second problem when H∈(12,1) are ill-posed due to the lack of regularity of the terminal condition, a well-known regularization method called Fourier truncation is applied to construct regularized solutions. Furthermore, convergence results of regularized solutions are proposed.