Large deviations of mean-field stochastic differential equations with jumps

Cai, Yujie; Huang, Jianhui; Maroulas, Vasileios

doi:10.1016/j.spl.2014.08.010

Cited by 11 publications

(3 citation statements)

References 24 publications

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“…A very general weak convergence approach that is sufficient to prove the uniform Laplace principle was introduced by Budhiraja, Dupuis, and Maroulas [4]. Since then, the ULP has been used by many authors including Budhiraja and Biswas [1], Wu [23], and Cai, Huang and Maroulas [5].…”

Section: Introductionmentioning

confidence: 99%

Equivalences and counterexamples between several definitions of the uniform large deviations principle

Salins¹

2019

Probab. Surveys

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This paper explores the equivalences between four definitions of uniform large deviations principles and uniform Laplace principles found in the literature. Counterexamples are presented to illustrate the differences between these definitions and specific conditions are described under which these definitions are equivalent to each other. A fifth definition called the equicontinuous uniform Laplace principle (EULP) is proposed and proven to be equivalent to Freidlin and Wentzell's definition of a uniform large deviations principle. Sufficient conditions that imply a measurable function of infinite dimensional Wiener process satisfies an EULP using the variational methods of Budhiraja, Dupuis and Maroulas are presented. This theory is applied to prove that a family of Hilbert space valued stochastic equations exposed to multiplicative noise satisfy a uniform large deviations principle that is uniform over all initial conditions in bounded subsets of the Hilbert space. This is an improvement over previous weak convergence methods which can only prove uniformity over compact sets.

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Section: Introductionmentioning

confidence: 99%

Equivalences and counterexamples between several definitions of the uniform large deviations principle

Salins¹

2019

Probab. Surveys

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“…The case in which the driving process has jumps has been much less studied, however, with the foundations set by [21], several papers on the topic have appeared in recent years, see e.g. [1,2,4,6,20,25,31,35]. By analogy to the relation between McKean-Vlasov SDEs with Brownian noise and nonlinear Fokker-Planck equations [3,19], McKean-Vlasov SDEs with jumps can be related to nonlocal integral-PDEs [20,21,25].…”

Section: Introductionmentioning

confidence: 99%

“…They have also found applications in areas such as financial mathematics [4] and neuronal networks [31]. Such equations, regardless of the driving noise, are known in the literature under numerous different names, including McKean-Vlasov SDEs [2,5,13,18,14], mean-field SDEs [6,20,25,35], distribution dependent SDEs (DDSDEs) [3,39] and nonlinear SDEs [21]. The choice of the name depends on the preferred interpretation of the process (X t ) t≥0 and different names are often used interchangeably.…”

Section: Introductionmentioning

confidence: 99%

Exponential ergodicity for SDEs and McKean-Vlasov processes with Lévy noise

Liang¹,

Majka²,

Wang

2019

Preprint

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We study stochastic differential equations (SDEs) of McKean-Vlasov type with distribution dependent drifts and driven by pure jump Lévy processes. We prove a uniform in time propagation of chaos result, providing quantitative bounds on convergence rate of interacting particle systems with Lévy noise to the corresponding McKean-Vlasov SDE. By applying techniques that combine couplings, appropriately constructed L 1 -Wasserstein distances and Lyapunov functions, we show exponential convergence of solutions of such SDEs to their stationary distributions. Our methods allow us to obtain results that are novel even for a broad class of Lévy-driven SDEs with distribution independent coefficients.

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Nonlocal stochastic functional differential equations driven by G‐Brownian motion and mean random dynamical systems

Chen

Yang

2020

Math Methods in App Sciences

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In this paper, we consider a class of nonlocal stochastic functional differential equations driven by G‐Brownian motion (GNSFDEs) at phase space whose coefficients are dependent on the p‐th moment. Existence and uniqueness of solutions for GNSFDEs are investigated by virtue of theory of nonlinear expectation. Furthermore, we show that the solution map of GNSFDEs can generate a p‐mean random dynamical system in nonlinear expectation framework. In addition, two examples are provided to illustrate the effectiveness of the obtained results

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Large deviations of mean-field stochastic differential equations with jumps

Cited by 11 publications

References 24 publications

Equivalences and counterexamples between several definitions of the uniform large deviations principle

Equivalences and counterexamples between several definitions of the uniform large deviations principle

Exponential ergodicity for SDEs and McKean-Vlasov processes with Lévy noise

Nonlocal stochastic functional differential equations driven by G‐Brownian motion and mean random dynamical systems

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