Journal of Theoretical Probability

, Volume 28, Issue 1, pp 337–395

# 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

## Notes

### Acknowledgments

The authors wish to express their thanks to Rainer Buckdahn, Yaozhong Hu, Shige Peng, and Aurel Răşcanu for their useful suggestions and discussions.

## References

1. 1.
Alòs, E., Mazet, O., Nualart, D.: Stochastic calculus with respect to fractional Brownian motion with Hurst parameter lesser than $$\frac{1}{2}$$. Stoch. Process. Appl. 86, 121–139 (2000)
2. 2.
Alòs, E., Mazet, O., Nualart, D.: Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29, 766–801 (2001)
3. 3.
Alòs, E., Nualart, D.: Stochastic integration with respect to the fractional Brownian motion. Stochastics 75, 129–152 (2003)Google Scholar
4. 4.
Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff International Publishing, Leiden (1976)
5. 5.
Bender, C.: Explicit solutions of a class of linear fractional BSDEs. Syst. Control Lett. 54, 671–680 (2005)
6. 6.
Biagini, F., Hu, Y., Øksendal, B., Sulem, A.: A stochastic maximal principle for processes driven by fractional Brownian motion. Stoch. Process. Appl. 100, 233–253 (2002)
7. 7.
Biagini, F., Hu, Y., Øksendal, B., Zhang, T.: Stochastic Calculus for Fractional Brownian Motion and Applications. Springer, London (2006)Google Scholar
8. 8.
Decreusefond, L., Üstünel, A.S.: Stochastic analysis of the fractional Brownian motion. Potential Anal. 10, 177–214 (1998)
9. 9.
Duncan, T.E., Hu, Y., Pasik-Duncan, B.: Stochastic calculus for fractional Brownian motion I. Theory. SIAM J. Control Optim. 38, 582–612 (2000)
10. 10.
Hu, Y.: Integral transformations and anticipative calculus for fractional Brownian motions. Memoirs of the American Mathematical Society, vol.175 (2005)Google Scholar
11. 11.
Hu, Y., Øksendal, B.: Fractional white noise calculus and applications to finance. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6, 1–32 (2003)
12. 12.
Hu, Y., Peng, S.: Backward stochastic differential equation driven by fractional Brownian motion. SIAM J. Control Optim. 48, 1675–1700 (2009)
13. 13.
Ma, J., Protter, P., Yong, J.: Solving forward–backward stochastic differential equations explicitly—a four step scheme. Probab. Theor. Relat. Fields 98, 339–359 (1994)
14. 14.
Memin, J., Mishura, Y., Valkeila, E.: Inequalities for the moments of Wiener integrals with respect to fractional Brownian motions. Stat. Probab. Lett. 55, 421–430 (2001)
15. 15.
Mishura, Y.: Stochastic Calculus for Fractional Brownian Motion and Related Processes. Springer, Berlin (2007)Google Scholar
16. 16.
Nualart, D.: The Malliavin Calculus and Related Topics, 2nd edn. Springer, Berlin (2006)
17. 17.
Nualart, D., Pardoux, E.: Stochastic calculus with anticipating integrands. Probab. Theor. Relat. Fields 78, 535–581 (1988)
18. 18.
Pardoux, E., Peng, S.: Backward stochastic differential equations and quasilinear parabolic partial differential equations. In: Rozovski, B.L., Sowers, R.B. (eds.) Stochastic PDE and Their Applications. LNCIS 176, pp. 200–217. Springer, Berlin (1992)Google Scholar
19. 19.
Pardoux, E., Răşcanu, A.: Backward stochastic differential equations with subdifferential operators and related variational inequalities. Stoch. Process. Appl. 76, 191–215 (1998)
20. 20.
Pardoux, E., Răşcanu, A.: Stochastic differential equations, Backward SDEs, Partial differential equations (to appear)Google Scholar
21. 21.
Young, L.C.: An inequality of the Höder type connected with Stieltjes integration. Acta Math. 67, 251–282 (1936)