Mean-field forward and backward SDEs with jumps. Associated nonlocal quasi-linear integral-PDEs

Abstract

In this paper we consider a mean-field backward stochastic differential equation (BSDE) driven by a Brownian motion and an independent Poisson random measure. Translating the splitting method introduced by Buckdahn, Li, Peng and Rainer [6] to BSDEs, the existence and the uniqueness of the solution $(Y^{t,\xi },Z^{t,\xi },H^{t,\xi })$, $(Y^{t,x,P_{\xi }},Z^{t,x,P_{\xi }},H^{t,x,P_{\xi }})$ of the split equations are proved. The first and the second order derivatives of the process $(Y^{t,x,P_{\xi }},Z^{t,x,P_{\xi }},H^{t,x,P_{\xi }})$ with respect to $x$, the derivative of the process $(Y^{t,x,P_{\xi }},Z^{t,x,P_{\xi }},H^{t,x,P_{\xi }})$ with respect to the measure $P_{\xi }$, and the derivative of the process $({\partial }_{\mu }{Y}^{t,x,P_{\xi }}(y),{\partial }_{\mu }{Z}^{t,x,P_{\xi }}(y),{\partial }_{\mu }{H}^{t,x,P_{\xi }}(y))$ with respect to $y$ are studied under appropriate regularity assumptions on the coefficients, respectively. These derivatives turn out to be bounded and continuous in $L^2$. The proof of the continuity of the second order derivatives is particularly involved and requires subtle estimates. This regularity ensures that the value function $V(t,x,P_{\xi }):=Y_t^{t,x,P_{\xi }}$ is regular and allows to show with the help of a new Itô formula that it is the unique classical solution of the related nonlocal quasi-linear integral-partial differential equation (PDE) of mean-field type.

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References in corpus (1)

• The Master Equation for Large Population Equilibriums