Viability for Coupled SDEs Driven by Fractional Brownian Motion

Abstract

In this paper, we are concerned with a class of coupled multidimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $$H\in (1/2,1)$$ . Using pathwise approach and based on Perov’s fixed point theorem, we prove that the existence and uniqueness of solution for the equations considered under some local Lipschitz conditions. Subsequently, by establishing some priori estimates, we obtain a viability result for the stochastic systems under investigation. The sufficient and necessary condition is also an alternative global existence result for the fractional differential equations with restrictions on the state. Finally, by direct and inverse images for fractional stochastic tangent sets, we establish the deterministic necessary and sufficient conditions which control that the solution for the coupled stochastic systems under investigation evolves in some particular sets.