Mean-field SDEs with jumps and nonlocal integral-PDEs

Recently Buckdahn et al. (Mean-field stochastic differential equations and associated PDEs, arXiv:1407.1215 , 2014 ) studied a mean-field stochastic differential equation (SDE), whose coefficients depend on both the solution process and also its law, and whose solution process ( X s t , x , P ξ , X s t , ξ = X s t , x , P ξ | x = ξ ) , s ∈ [ t , T ] , ( t , x ) ∈ [ 0 , T ] × R d , ξ ∈ L 2 ( F t , R d ) , admits the flow property. This flow property is the key for the study of the associated nonlocal partial differential equation (PDE). In this work we extend these studies in a non-trivial manner to mean-field SDEs which, in addition to the driving Brownian motion, are governed by a compensated Poisson random measure. We show that under suitable regularity assumptions on the coefficients of the SDE, the solution X t , x , P ξ is twice differentiable with respect to x and its law. We establish the associated nonlocal integral-PDE, and we show that V ( t , x , P ξ ) = E [ Φ ( X T t , x , P ξ , P X T t , ξ ) ] is the unique classical solution V : [ 0 , T ] × R d × P 2 ( R d ) → R of this nonlocal integral-PDE with terminal condition Φ .