Fixed-point theorems and Picard iteration methods for McKean-Vlasov mean-field SDEs with jumps

We consider a prototype class of Lévy-driven SDEs with McKean-Vlasov (mean-field) interaction in the drift. The coefficient is assumed to be affine in the state-variable and only measurable in the law. We study the equivalent functional fixed-point equation for the unknown time-dependent coefficients of the associated Markovian SDE. By proving a contraction property for the functional map in a suitable normed space, we infer existence and uniqueness results for the MK-V SDE, and derive a discretized Picard iteration method that approximates the law of the solution. Numerical illustrations show the effectiveness of the method, which appears to be appropriate to handle multi-dimensional settings. We finally describe possible extensions and generalizations to more general settings. This talk is based on joint work with Ankush Agarwal.