arXiv:2109.11461v2 [math.PR] 29 Sep 2021 An adapted solution of a fractional backward stochastic differential equation Arzu Ahmadova * and Nazim I. Mahmudov † Department of Mathematics, Eastern Mediterranean University, Mersin 10, 99628, T.R. Northern Cyprus 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 unique- ness 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. Keywords: Fractional backward stochastic differential equations, backward stochastic nonlinear Volterra integral equation, existence and uniqueness, adapted process, weighted norm 1 Introduction Fractional stochastic differential equations (FSDEs) which are a generalization of differential equations by the use of fractional and stochastic calculus are more popular due to their applications in mathematical modelling and finance [1, 2, 3]. Recently, FSDEs are intensively applied to model mathematical problems in finance [4, 5], dynamics of complex systems in engineering [6] and other areas [7, 8]. Several results have been investigated on the qualitative theory and applications of fractional stochastic differential equations (FSDEs) [9]-[16]. Studying backward stochastic differential equations (BSDEs) has necessary applications in stochastic optimal control, stochastic differential game, probabilistic formula for the solutions of quasilinear partial differential equations and financial markets. The adapted solution for a linear BSDE which arises as the adjoint process for a stochastic control problem was first investigated by Bismut [17] in 1973, then by Bensousssan [18], and while Pardoux and Peng [19] first studied the result for the existence and uniqueness of an adapted solution to a continuous general non-linear BSDE which is a terminal value problem for an Itˆ o type stochastic differential equation under the uniform Lipschitz conditions of the following form:  dY (t)= h(t,Y (t),Z (t))dt + Z (t)dW (t),t ∈ [0,T ], Y (T )= ξ. They established existence and uniqueness of an adapted solution using Bihari’s inequality which is the most important generalization of the Gronwall-Bellman inequality. Since then, the theory of BSDE became a powerful tool in many fields such as mathematical finance, optimal control, semi-linear and quasi-linear partial differential equations [20, 21, 22, 23]. Later Peng and Pardoux developed the theory and applications of continuous BSDEs in a series of papers [19, 24, 25, 26, 27] * Corresponding author. Email: arzu.ahmadova@emu.edu.tr † Email: nazim.mahmudov@emu.edu.tr 1