|
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 et al. (2014) to BSDEs, the existence and the uniqueness of the solution ( Y t , ξ , Z t , ξ , H t , ξ ) , ( Y t , x , P ξ , Z t , x , P ξ , H t , x , P ξ ) of the split equations are proved. The first and the second order derivatives of the process ( Y t , x , P ξ , Z t , x , P ξ , H t , x , P ξ ) with respect to x , the derivative of the process ( Y t , x , P ξ , Z t , x , P ξ , H t , x , P ξ ) with respect to the measure P ξ , and the derivative of the process ( ∂ μ Y t , x , P ξ ( y ) , ∂ μ Z t , x , P ξ ( y ) , ∂ μ H t , x , P ξ ( 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 ξ ) ≔ Y t t , x , P ξ 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.
|
