Article2009

Backward Stochastic Differential Equation Driven by Fractional Brownian Motion

80 citations0/0 files storedSIAM Journal on Control and Optimization

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Stochastic processes and financial applications Nonlinear Differential Equations Analysis Fuzzy Systems and Optimization

Abstract

We study general linear and nonlinear backward stochastic differential equations driven by fractional Brownian motions. The existence and uniqueness of the solutions are obtained under some mild assumptions. In the nonlinear case we obtain an inequality of the type similar to in the classical backward stochastic differential equations. This leads to a fixed point principle. An important tool is the quasi-conditional expectation introduced in [Y. Hu and B. Øksendal, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 6 (2003), pp. 1–32]. We also give a detailed study on this new “expectation.”

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References

1990Adapted solution of a backward stochastic differential equation 1997Backward Stochastic Differential Equations in Finance 2000Stochastic Calculus for Fractional Brownian Motion I. Theory 2003FRACTIONAL WHITE NOISE CALCULUS AND APPLICATIONS TO FINANCE 2000Integration questions related to fractional Brownian motion 2002Filtration-consistent nonlinear expectations and related g-expectations 2001Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion 2002Regularization of differential equations by fractional noise

Cited by

2016Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion 2013Fractional Backward Stochastic Differential Equations and Fractional Backward Variational Inequalities

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