On terminal value problems for bi-parabolic equations driven by Wiener process and fractional Brownian motions

Abstract

In this paper, we study two terminal value problems (TVPs) for stochastic bi-parabolic equations perturbed by standard Brownian motion and fractional Brownian motion with Hurst parameter $h \in (\frac{1}{2}, 1)$ separately. For each problem, we provide a representation for the mild solution and find the space where the existence of the solution is guaranteed. Additionally, we show clearly that the solution of each problem is not stable, which leads to the ill-posedness of each problem. Finally, we propose two regularization results for both considered problems by using the filter regularization method.

Keywords

Bi-parabolic equation standard Brownian motion fractional Brownian motion terminal value problem ill-posedness

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