Mean-field backward stochastic differential equations and applications.

In this paper we study the linear mean-field backward stochastic differential equations (mean-field BSDE) of the form (0.1) d Y (t) = − [ α 1 (t) Y (t) + β 1 (t) Z (t) + ∫ R 0 η 1 (t , ζ) K (t , ζ) ν (d ζ) + α 2 (t) E [ Y (t) ] + β 2 (t) E [ Z (t) ] + ∫ R 0 η 2 (t , ζ) E [ K (t , ζ) ] ν (d ζ) + γ (t) ] d t + Z (t) d B (t) + ∫ R 0 K (t , ζ) N ̃ (d t , d ζ) , t ∈ 0 , T , Y (T) = ξ. where (Y , Z , K) is the unknown solution triplet, B is a Brownian motion, N ̃ is a compensated Poisson random measure, independent of B. We prove the existence and uniqueness of the solution triplet (Y , Z , K) of such systems. Then we give an explicit formula for the first component Y (t) by using partial Malliavin derivatives. To illustrate our result we apply them to study a mean-field recursive utility optimization problem in finance. [ABSTRACT FROM AUTHOR]