Itô-Taylor Schemes for Solving Mean-Field Stochastic Differential Equations

Abstract

This paper is devoted to numerical methods for mean-field stochastic differential equations (MSDEs). We first develop the mean-field Itô formula and mean-field Itô-Taylor expansion. Then based on the new formula and expansion, we propose the Itô-Taylor schemes of strong order γ and weak order η for MSDEs, and theoretically obtain the convergence rate γ of the strong Itô-Taylor scheme, which can be seen as an extension of the well-known fundamental strong convergence theorem to the mean-field SDE setting. Finally some numerical examples are given to verify our theoretical results.

A Fourier-based Picard-iteration approach for a class of McKean–Vlasov SDEs with Lévy jumps

“…It is usually shown that, in the limit N → ∞, the empirical law of a Markovian system with N interacting particles, converges to the law of the solution to the MK-V SDE under study, which can be then approximated via timediscretization and simulation. Following the work by Sznitman in [21], where the first propagation of chaos result was proved, many authors have contributed to this stream of literature, see, for example, [2], [23], [11], and [20], by supposing different forms and regularity assumptions on the MK-V SDE coefficients. Although a very powerful approximating tool which is applicable in many settings, the simulation of large-particle systems can be computationally very expensive.…”

“…Szpruch et al [22] provided an alternative iterative particle representation that can be combined with multilevel Monte Carlo techniques in order to simulate the solutions. Sun et al [20] developed Itô-Taylor schemes of Euler-and Milstein-type for numerically estimating the solution of MK-V SDEs with Lipschitz regular coefficients and square-integrable initial law. Gobet and Pagliarani [8] recently developed analytical approximations of the transition density of the solutions by extending a perturbation technique that was previously developed for standard SDEs.…”

A Fourier-based Picard-iteration approach for a class of McKean-Vlasov SDEs with Lévy jumps

“…It is usually shown that, in the limit N → ∞, the empirical law of a Markovian system with N interacting particles, converges to the law of the solution to the MK-V SDE under study, which can be then approximated via time-discretization and simulation. Following the work by Sznitman in [14], where the first propagation of chaos result was proved, many authors have contributed to this stream of literature, see, for example, [2], [16], [8], and [13]. Although a very powerful approximating tool, the simulation of large-particle systems can be computationally very expensive.…”

“…[15] provided an alternative iterative particle representation that can be combined with Multilevel Monte Carlo techniques in order to simulate the solutions. In [13], Sun et. al.…”

Numerical methods for mean-field stochastic differential equations with jumps

“…In this work, we aim to propose the general Itô-Taylor schemes for solving MSDEJs. The authors studied the mean-field Itô formula and proposed the general Itô-Taylor schemes for MSDEs in [35]. By using the mean-field Itô formula, we first develop the Itô formula for MSDEJs, then based on which, we construct the Itô-Taylor expansion for MSDEJs and further propose the Itô-Taylor schemes of strong order γ and weak order η for solving MSDEJs.…”