Central limit results for jump diffusions with mean field interaction and a common factor

Stochastic Processes and their Applications

2019

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We consider weakly interacting diffusions on time varying random graphs. The system consists of a large number of nodes in which the state of each node is governed by a diffusion process that is influenced by the neighboring nodes. The collection of neighbors of a given node changes dynamically over time and is determined through a time evolving random graph process. A law of large numbers and a propagation of chaos result is established for a multi-type population setting where at each instant the interaction between nodes is given by an inhomogeneous random graph which may change over time. This result covers the setting in which the edge probabilities between any two nodes is allowed to decay to 0 as the size of the system grows. A central limit theorem is established for the single-type population case under stronger conditions on the edge probability function.

Section: Introductionmentioning

confidence: 99%

Weakly interacting particle systems on inhomogeneous random graphs

Bhamidi

Stochastic Processes and their Applications

2019

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“…The paper [12] studied LLN and [13] considered fluctuations about the LLN limit. In a setting where particle dynamics are given through jump-diffusions and the common factor is described by another jump-diffusion that is coupled with the particle dynamics, a CLT was recently obtained in [3]. The fluctuation limit theorems in [13,3], although allowing for a common factor, are limited to exchangeable populations.…”

Section: Introductionmentioning

confidence: 99%

“…In a setting where particle dynamics are given through jump-diffusions and the common factor is described by another jump-diffusion that is coupled with the particle dynamics, a CLT was recently obtained in [3]. The fluctuation limit theorems in [13,3], although allowing for a common factor, are limited to exchangeable populations. The goal of the current work is to study fluctuations for multi-type particle systems.…”

Section: Introductionmentioning

confidence: 99%

Some fluctuation results for weakly interacting multi-type particle systems

Stochastic Processes and their Applications

2016

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A collection of N -diffusing interacting particles where each particle belongs to one of K different populations is considered. Evolution equation for a particle from population k depends on the K empirical measures of particle states corresponding to the various populations and the form of this dependence may change from one population to another. In addition, the drift coefficients in the particle evolution equations may depend on a factor that is common to all particles and which is described through the solution of a stochastic differential equation coupled, through the empirical measures, with the N -particle dynamics. We are interested in the asymptotic behavior as N → ∞. Although the full system is not exchangeable, particles in the same population have an exchangeable distribution. Using this structure, one can prove using standard techniques a law of large numbers result and a propagation of chaos property. In the current work we study fluctuations about the law of large number limit. For the case where the common factor is absent the limit is given in terms of a Gaussian field whereas in the presence of a common factor it is characterized through a mixture of Gaussian distributions. We also obtain, as a corollary, new fluctuation results for disjoint sub-families of single type particle systems, i.e. when K = 1. Finally, we establish limit theorems for multi-type statistics of such weakly interacting particles, given in terms of multiple Wiener integrals.

“…The following result gives an alternative representation that is more intrinsic. 12) where the infimum is taken over all u .…”

Section: Remark 32 (I) Condition 31(b) Is Implied By Conditions 3mentioning

confidence: 99%

“…Many variations of such systems have also been studied. For example, in [27,28,12,13] limit theorems of the above form are established for a setting where there is a common noise process that influences the dynamics of each particle. LLN and POC in a setting with K-different subpopulations where particle evolution within each subpopulation is exchangeable, has been studied in [1] and a corresponding CLT has been established in [13].…”

Section: Introductionmentioning

confidence: 99%

Moderate deviation principles for weakly interacting particle systems

Probab. Theory Relat. Fields

2016

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Moderate deviation principles for empirical measure processes associated with weakly interacting Markov processes are established. Two families of models are considered: the first corresponds to a system of interacting diffusions whereas the second describes a collection of pure jump Markov processes with a countable state space. For both cases the moderate deviation principle is formulated in terms of a large deviation principle (LDP), with an appropriate speed function, for suitably centered and normalized empirical measure processes. For the first family of models the LDP is established in the path space of an appropriate Schwartz distribution space whereas for the second family the LDP is proved in the space of l2 (the Hilbert space of square summable sequences)-valued paths. Proofs rely on certain variational representations for exponential functionals of Brownian motions and Poisson random measures.AMS 2000 subject classifications: 60F10, 60K35, 60J75, 60J60.