Journal of Theoretical Probability

, Volume 28, Issue 1, pp 337–395 | Cite as

Fractional Backward Stochastic Differential Equations and Fractional Backward Variational Inequalities

Article

Abstract

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:
$$\begin{aligned} \left\{ \begin{array}{l} -\hbox {d}Y(t)= f(t,\eta (t),Y(t),Z(t))\hbox {d}t-Z(t)\delta B^{H}\left( t\right) ,\quad t\in [0,T],\\ Y(T)=\xi , \end{array} \right. \end{aligned}$$
where \(\eta \) is a stochastic process given by \(\eta (t)=\eta (0) +\int _{0}^{t}\sigma (s) \delta B^{H}(s)\), \(t\in [0,T]\), and \(B^{H}\) is a fractional Brownian motion with Hurst parameter greater than \(1/2\). 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
$$\begin{aligned} \left\{ \begin{array}{l} -\hbox {d}Y(t)+\partial \varphi (Y(t))\hbox {d}t\ni f(t,\eta (t),Y(t),Z(t))\hbox {d}t-Z(t)\delta B^{H}\left( t\right) ,\quad t\in [0,T],\\ Y(T)=\xi , \end{array}\right. \end{aligned}$$
where \(\partial \varphi \) is a multivalued operator of subdifferential type associated with the convex function \(\varphi \).

Keywords

Backward stochastic differential equation Fractional Brownian motion Divergence-type integral Malliavin calculus Backward stochastic variational inequality  Subdifferential operator 

Mathematics Subject Classification (2010)

60H10 60G22 47J20 60H05 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Faculty of Mathematics“Alexandru Ioan Cuza” UniversityIasiRomania
  2. 2.Department of Mathematics“Gheorghe Asachi” Technical UniversityIasiRomania
  3. 3.School of MathematicsShandong UniversityJinanChina
  4. 4.Laboratoire de MathématiquesCNR-UMR 6205 Université de Bretagne OccidentaleBrest Cedex 3France
  5. 5.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia

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