A Fourier-based Picard-iteration approach for a class of McKean–Vlasov SDEs with Lévy jumps. Issue 4 (19th May 2021)

• Record Type: Journal Article
• Authors: Agarwal, Ankush; Pagliarani, Stefano
• Is Part Of: Stochastics. Volume 93:Issue 4(2021)
• Page Start: 592
• Page End: 624
• Publication Date: 2021-05-19
• Subjects: Nonlinear stochastic differential equations -- Lévy processes -- McKean–Vlasov model -- Picard iteration -- fourier transform; 65C30 -- 65T99 -- 65Q20; Stochastic processes -- Periodicals; Probabilities -- Periodicals; 519.2
• DOI: 10.1080/17442508.2020.1771337
• Languages: English
• ISSNs: 1744-2508

Abstract

Abstract : We consider a prototype class of Lévy-driven stochastic differential equations (SDEs) with McKean–Vlasov (MK–V) interaction in the drift coefficient. It is assumed that the drift coefficient is affine in the state variable, and only measurable in the law of the solution. We study the equivalent functional fixed-point equation for the unknown time-dependent coefficients of the associated linear 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 scheme that approximates the law of the solution through its characteristic function. Numerical illustrations show the effectiveness of our method, which appears to be appropriate to handle the multi-dimensional setting.