Nonexplosion Criteria for Solutions of SDE with Fractional Brownian Motion

Abstract

We obtain sufficient conditions for nonoccurrence of explosion for solutions of one-dimensional stochastic differential equation (SDE) driven by fractional Brownian motion (fBm) with Hurst parameter H ∈ (1/2, 1); SDE is interpreted as a stochastic integral equation of the fractional Ito-integral type. In the case where explosion cannot occur, every solution of SDE exists in the future. One of our result corresponds to an extension of the Wintner theorem on continuation of solutions of ordinary differential equations. We shall obtain a priori bounds for nonexplosive solution and also give an example of explosive solutions.

Keywords:

• Continuation of solutions
• Explosion time
• Fractional Brownian motion
• Fractional Ito-integral
• Stochastic differential equation

Mathematics Subject Classification:

• 60H10
• 60H05
• 60H20

Acknowledgments

I would like to thank the referee for his suggestions and remarks on the set of processes in Hu [6, p. 57] which helped me in considering the fractional Ito-integral of φ-differentiable random variables.