Mean-field backward stochastic differential equations and applications

2018/01/10 by Nacira Agram, Agram, Nacira, Yaozhong Hu +3 · 1 citation
Economics, Econometrics and Finance · Social Sciences · Decision Sciences · #Stochastic processes and financial applications #Insurance, Mortality, Demography, Risk Management #Probability and Risk Models

paper · pdf · doi:10.48550/arxiv.1801.03349

Abstract

In this paper we study the mean-field backward stochastic differential equations (mean-field bsde) of the form dY(t) =-f(t,Y(t),Z(t),K(t, . ),E[φ(Y(t),Z(t),K(t,.))])dt+Z(t)dB(t) +∫_R₀K(t,ζ)N(dt,dζ), where B is a Brownian motion, N is the compensated Poisson random measure. Under some mild conditions, we prove the existence and uniqueness of the solution triplet (Y,Z,K). It is commonly believed that there is no comparison theorem for general mean-field bsde. However, we prove a comparison theorem for a subclass of these equations. When the mean-field bsde is linear, we give an explicit formula for the first component Y(t) of the solution triplet. Our results are applied to solve a mean-field recursive utility optimization problem in finance.

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