Some Results on Backward Stochastic Differential Equations of Fractional Order

Abstract

Our aim in this paper is to deal with a new type differential equation so-called Caputo fractional backward stochastic differential equations (for short Caputo fBSDEs) and study the global existence and uniqueness of an adapted solution to Caputo fBSDEs of order α ∈ ( 1 2 , 1) whose coefficients satisfy Lipschitz condition by applying fundamental lemma which plays a crucial role in the theory of Caputo fBSDEs. The interesting point here is to use a new weighted norm in square-integrable measurable function space for fundamental lemma and therefore for the whole paper. For this class of systems, we then show the coincidence between the notion of stochastic Volterra integral equation and mild solution.