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

Article type: Research Article

Authors: Tuan, Nguyen Huy^{a; b} | Caraballo, Tomás^c | Thach, Tran Ngoc^{a; b; *}

Affiliations: [a] Department of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam | [b] Vietnam National University, Ho Chi Minh City, Vietnam | [c] Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, C/ Tarfia s/n, 41012 – Sevilla, Spain

Correspondence: [*] Corresponding author. E-mail: ngocthachtnt@gmail.com.

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∈(12,1) 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

DOI: 10.3233/ASY-201637

Journal: Asymptotic Analysis, vol. 123, no. 3-4, pp. 335-366, 2021