Fractional Backward Stochastic Differential Equations and Fractional Backward Variational Inequalities

By Lucian Maticiuc, Tianyang Nie. Journal of theoretical probability, 2011

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In the framework of fractional stochastic calculus, we study the existence and the uniqueness of the solution for a backward stochastic differential equation, formally written as: {−dY(t)=f(t,η(t),Y(t),Z(t))dt−Z(t)δBH(t),t∈[0,T],Y(T)=ξ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{l} -\hbox {d}Y(t)= f(t,\eta (t),Y(t),Z(t))\hbox {d}t-Z(t)\delta B^{H}\left( t\right) ,\quad t\in [0,T],\\ Y(T)=\xi , \end{array} \right. \end{aligned}$$\end{document}where η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} is a stochastic process given by η(t)=η(0)+∫0tσ(s)δBH(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta (t)=\eta (0) +\int _{0}^{t}\sigma (s) \delta B^{H}(s)$$\end{document}, t∈[0,T]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in [0,T]$$\end{document}, and BH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^{H}$$\end{document} is a fractional Brownian motion with Hurst parameter greater than 1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/2$$\end{document}. The stochastic integral used in above equation is the divergence-type integral. Based on Hu and Peng’s paper, Backward stochastic differential equation driven by fractional Brownian motion, SIAM J Control Optim (2009), we develop a rigorous approach for this equation. Moreover, we study the existence of the solution for the multivalued backward stochastic differential equation {−dY(t)+∂φ(Y(t))dt∋f(t,η(t),Y(t),Z(t))dt−Z(t)δBH(t),t∈[0,T],Y(T)=ξ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{l} -\hbox {d}Y(t)+\partial \varphi (Y(t))\hbox {d}t\ni f(t,\eta (t),Y(t),Z(t))\hbox {d}t-Z(t)\delta B^{H}\left( t\right) ,\quad t\in [0,T],\\ Y(T)=\xi , \end{array}\right. \end{aligned}$$\end{document}where ∂φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \varphi $$\end{document} is a multivalued operator of subdifferential type associated with the convex function φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document}.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Finance

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