Nonlinear SDEs driven by Lévy processes and related PDEs

In this paper we study general nonlinear stochastic differential equations, where the usual Brownian motion is replaced by a Lévy process. We also suppose that the coefficient multiplying the increments of this process is merely Lipschitz continuous and not necessarily linear in the time-marginals of the solution as is the case in the classical McKean-Vlasov model. We first study existence, uniqueness and particle approximations for these stochastic differential equations. When the driving process is a pure jump Lévy process with a smooth but unbounded Lévy measure, we develop a stochastic calculus of variations to prove that the time-marginals of the solutions are absolutely continuous with respect to the Lebesgue measure. In the case of a symmetric stable driving process, we deduce the existence of a function solution to a nonlinear integro-differential equation involving the fractional Laplacian. This paper studies the following nonlinear stochastic differential equation: { Xt = X0 + ∫ t 0 σ(Xs− , Ps)dZs, t ∈ [0, T ], ∀s ∈ [0, T ], Ps denotes the probability distribution of Xs. (1) We assume that X0 is a random variable with values in Rk, distributed according to m, (Zt)t≤T a Lévy process with values in Rd, independent of X0, and σ : Rk×P(Rk)→ Rk×d, where P(Rk) denotes the set of probability measures on Rk. Notice that the classical McKean-Vlasov model, studied for instance in [22], is obtained as a special case of (1) by choosing σ linear in the second variable and Zt = (t, Bt), with Bt being a (d− 1)-dimensional standard Brownian motion. The first section of the paper is devoted to the existence problem and particle approximations for (1). Initially, we address the case of square integrable both, the initial condition X0, and the Lévy process (Zt)t≤T . Under these assumptions the existence and uniqueness problem for (1) can be handled exactly as in the Brownian case Zt = (t, Bt). The nonlinear stochastic differential equation (1) admits a unique solution as soon as σ is Lipschitz continuous on R×P2(R) endowed with the product of the canonical metric on Rk and the Vaserstein metric d on the set P2(R) of probability measures with finite second order moments. This assumption is much weaker than the assumptions imposed on σ in the classical McKean-Vlasov model, where it is also supposed to be linear in its second variable, that is, σ(x, ν) = ∫ Rk ς(x, y)ν(dy), for a Lipschitz continuous function ς : Rk × Rk → Rk×d. Then, replacing the nonlinearity by the ∗CERMICS, École des Ponts, ParisTech, 6-8 avenue Blaise Pascal, Cité Descartes, Champs sur Marne, 77455 Marne la Vallée Cedex 2, e-mail:[email protected] †CMAP, Ecole Polytechnique, CNRS, route de Saclay, 91128 Palaiseau Cedex e-mail: [email protected] ‡Department of Statistics and Center for Stochastic and Chaotic Processes in Science and Technology, Case Western Reserve University, Cleveland, OH 44106, e-mail: [email protected]