McKean–Vlasov process

In probability theory, a McKean–Vlasov process is a stochastic process described by a stochastic differential equation where the coefficients of the diffusion depend on the distribution of the solution itself.[1][2] The equations are a model for Vlasov equation and were first studied by Henry McKean in 1966.[3] It is an example of propagation of chaos, in that it can be obtained as a limit of a mean-field system of interacting particles: as the number of particles tends to infinity, the interactions between any single particle and the rest of the pool will only depend on the particle itself.[4]

Definition

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Consider a measurable function   where   is the space of probability distributions on   equipped with the Wasserstein metric   and   is the space of square matrices of dimension  . Consider a measurable function  . Define  .

A stochastic process   is a McKean–Vlasov process if it solves the following system:[3][5]

  •   has law  
  •  

where   describes the law of   and   denotes a  -dimensional Wiener process. This process is non-linear, in the sense that the associated Fokker-Planck equation for   is a non-linear partial differential equation.[5][6]

Existence of a solution

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The following Theorem can be found in.[4]

Existence of a solution — Suppose   and   are globally Lipschitz, that is, there exists a constant   such that:

 

where   is the Wasserstein metric.

Suppose   has finite variance.

Then for any   there is a unique strong solution to the McKean-Vlasov system of equations on  . Furthermore, its law is the unique solution to the non-linear Fokker–Planck equation:

 

Propagation of chaos

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The McKean-Vlasov process is an example of propagation of chaos.[4] What this means is that many McKean-Vlasov process can be obtained as the limit of discrete systems of stochastic differential equations  .

Formally, define   to be the  -dimensional solutions to:

  •   are i.i.d with law  
  •  

where the   are i.i.d Brownian motion, and   is the empirical measure associated with   defined by   where   is the Dirac measure.

Propagation of chaos is the property that, as the number of particles  , the interaction between any two particles vanishes, and the random empirical measure   is replaced by the deterministic distribution  .

Under some regularity conditions,[4] the mean-field process just defined will converge to the corresponding McKean-Vlasov process.

Applications

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References

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  1. ^ Des Combes, Rémi Tachet (2011). Non-parametric model calibration in finance: Calibration non paramétrique de modèles en finance (PDF) (Doctoral dissertation). Archived from the original (PDF) on 2012-05-11.
  2. ^ Funaki, T. (1984). "A certain class of diffusion processes associated with nonlinear parabolic equations". Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete. 67 (3): 331–348. doi:10.1007/BF00535008. S2CID 121117634.
  3. ^ a b McKean, H. P. (1966). "A Class of Markov Processes Associated with Nonlinear Parabolic Equations". Proc. Natl. Acad. Sci. USA. 56 (6): 1907–1911. Bibcode:1966PNAS...56.1907M. doi:10.1073/pnas.56.6.1907. PMC 220210. PMID 16591437.
  4. ^ a b c d Chaintron, Louis-Pierre; Diez, Antoine (2022). "Propagation of chaos: A review of models, methods and applications. I. Models and methods". Kinetic and Related Models. 15 (6): 895. arXiv:2203.00446. doi:10.3934/krm.2022017. ISSN 1937-5093.
  5. ^ a b c Carmona, Rene; Delarue, Francois; Lachapelle, Aime. "Control of McKean-Vlasov Dynamics versus Mean Field Games" (PDF). Princeton University.
  6. ^ a b Chan, Terence (January 1994). "Dynamics of the McKean-Vlasov Equation". The Annals of Probability. 22 (1): 431–441. doi:10.1214/aop/1176988866. ISSN 0091-1798.