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]

Definition

McKean–Vlasov processes take the form^[3]^[4]

$dX_{t}=a(X_{t},\mu _{t})dB_{t}+b(X_{t},\mu _{t})dt$

where $\mu _{t}={\mathcal {L}}(X_{t})$ describes the law of X and dB denotes the Wiener process. That is the coefficients of the SDE depend on the marginal distribution of the process X. In general, the process can describe non-linear diffusion.^[4]^[5]

Applications

Mean-field theory
Mean-field game theory^[4]
Random matrices: including Dyson's model on eigenvalue dynamics for random symmetric matrices and the Wigner semicircle distribution^[5]

References

^ Des Combes, Rémi Tachet (2011). "Non-parametric model calibration in finance: Calibration non paramétrique de modèles en finance" (PDF). Archived from the original (PDF) on 2012-05-11. {{cite journal}}: Cite journal requires |journal= (help)
^ 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.
^ ^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.
^ ^a ^b ^c Carmona, Rene; Delarue, Francois; Lachapelle, Aime. "Control of McKean-Vlasov Dynamics versus Mean Field Games" (PDF). Princeton University.
^ ^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.

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McKean–Vlasov process

Definition

Applications

References

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