Generic Strong Well-Posedness for McKean--Vlasov SDEs with Jumps in a Krylov-Stable Coefficient Space

Abstract

We prove that strong well-posedness is generic for McKean--Vlasov stochastic differential equations with jumps in a complete coefficient space that allows genuinely discontinuous state dependence. For every $T>0$, there exists a dense residual subset $\mathfrak R_T$, independent of the initial state, such that every coefficient $a\in\mathfrak R_T$ generates a unique strong solution for every deterministic initial condition $x\in\mathbb R^d$. Moreover, the solution map $x\mapsto X^{a,x}$ is continuous in $\mathcal S_T^2$, and every approximation by spatially Lipschitz coefficients converges locally uniformly in $x$ to the same canonical solution family. The coefficients are uniformly bounded, uniformly non-degenerate in the diffusion component, and Lipschitz continuous in the law with respect to the $2$-Wasserstein distance, while their state dependence may be merely measurable. The ambient space is obtained by completing a spatially Lipschitz core under a Krylov-stable metric based on local $L^{2(d+1)}$ coefficient errors. A stopped conditional Krylov--Melnikov occupation estimate converts convergence in this metric into occupation convergence along non-degenerate It\^o--L\'evy trajectories. The key probabilistic ingredient is a set-valued stability theorem: near a core coefficient, every strong solution of every nearby solvable equation remains close to the unique core solution, without any uniqueness assumption on the nearby equation. Together with closedness of the strong-solution relation, this permits a Baire-category argument on the completed coefficient space. We also give an intrinsic spatial-translation criterion showing that the completion contains natural classes with essential spatial discontinuities. A projective-limit argument yields the corresponding global-time generic well-posedness result.