Nonlinear SDEs driven by L\'evy processes and related PDEs

By Benjamin Jourdain, S. M'el'eard, Wojbor A. Woyczynski. 2007

148 citations recorded by the source corpus.

In this paper we study general nonlinear stochastic differential equations, where the usual Brownian motion is replaced by a L\'evy 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 L\'evy process with a smooth but unbounded L\'evy 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.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b47e4b6318ae3ed79352f186ee8b56cf541677f8

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Citations in the corpus

Robust Markowitz mean‐variance portfolio selection under ambiguous covariance matrix (2016, unknown): “, 0, δ0) follows from standard arguments (recall that Σ ∈ VΘ is bounded) as in [33] or [19], and we have the estimate:”
A Probabilistic Approach to Mean Field Games with Major and Minor Players (2014, background): “See for example the elegant treatment in [27] and a more recent account in [17].”
A mean-field stochastic maximum principle via Malliavin calculus (2012, background): “Recently, there has been increasing interest in the study of this type of stochastic control problems; see for example [1,11,15].”
Bellman equation and viscosity solutions for mean-field stochastic control problem (2015, unknown): “…history with the pioneering works by Kac [26] and McKean [33], and later on with papers in the general framework of propagation of chaos, see e.g. [ 25,36], the optimal control of McKean − Vlasov dynamics is a rather new problem, which attracts an increasing interest since the emergence of the MFG…”
Mean Field Games and Systemic Risk (2013, unknown): “See for example [21] or [16].”
Probabilistic Analysis of Mean-Field Games (2012, methodology): “Here, we use convergence estimates which are part of the standard theory of propagation of chaos (see for example [26, 16]) and the Lipschitz continuity and linear growth the FBSDE value function u which we prove earlier in the paper.”
Forward-Backward Stochastic Differential Equations and Controlled McKean Vlasov Dynamics (2013, background): “See, for example, [13, 19, 23] for existence and uniqueness results.”