Mean-field backward stochastic differential equations and applications

Abstract

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.