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

Consider a measurable function $\sigma :\mathbb {R} ^{d}\times {\mathcal {P}}(\mathbb {R} ^{d})\to {\mathcal {M}}_{d}(\mathbb {R} )$ where ${\mathcal {P}}(\mathbb {R} ^{d})$ is the space of probability distributions on $\mathbb {R} ^{d}$ equipped with the Wasserstein metric $W_{2}$ and ${\mathcal {M}}_{d}(\mathbb {R} )$ is the space of square matrices of dimension $d$ . Consider a measurable function $b:\mathbb {R} ^{d}\times {\mathcal {P}}(\mathbb {R} ^{d})\to \mathbb {R} ^{d}$ . Define $a(x,\mu ):=\sigma (x,\mu )\sigma (x,\mu )^{T}$ .

A stochastic process $(X_{t})_{t\geq 0}$ is a McKean–Vlasov process if it solves the following system:^[3]^[5]

$X_{0}$ has law $f_{0}$
$dX_{t}=\sigma (X_{t},\mu _{t})dB_{t}+b(X_{t},\mu _{t})dt$

where $\mu _{t}={\mathcal {L}}(X_{t})$ describes the law of $X$ and $B_{t}$ denotes a $d$ -dimensional Wiener process. This process is non-linear, in the sense that the associated Fokker-Planck equation for $\mu _{t}$ is a non-linear partial differential equation.^[5]^[6]

Existence of a solution

The following Theorem can be found in.^[4]

Existence of a solution — Suppose $b$ and $\sigma$ are globally Lipschitz, that is, there exists a constant $C>0$ such that:

$|b(x,\mu )-b(y,\nu )|+|\sigma (x,\mu )-\sigma (y,\nu )|\leq C(|x-y|+W_{2}(\mu ,\nu ))$

where $W_{2}$ is the Wasserstein metric.

Suppose $f_{0}$ has finite variance.

Then for any $T>0$ there is a unique strong solution to the McKean-Vlasov system of equations on $[0,T]$ . Furthermore, its law is the unique solution to the non-linear Fokker–Planck equation:

$\partial _{t}\mu _{t}(x)=-\nabla \cdot \{b(x,\mu _{t})\mu _{t}\}+{\frac {1}{2}}\sum \limits _{i,j=1}^{d}\partial _{x_{i}}\partial _{x_{j}}\{a_{ij}(x,\mu _{t})\mu _{t}\}$

Propagation of chaos

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 $(X_{t}^{i})_{1\leq i\leq N}$ .

Formally, define $(X^{i})_{1\leq i\leq N}$ to be the $d$ -dimensional solutions to:

$(X_{0}^{i})_{1\leq i\leq N}$ are i.i.d with law $f_{0}$
$dX_{t}^{i}=\sigma (X_{t}^{i},\mu _{X_{t}})dB_{t}^{i}+b(X_{t}^{i},\mu _{X_{t}})dt$

where the $(B^{i})_{1\leq i\leq N}$ are i.i.d Brownian motion, and $\mu _{X_{t}}$ is the empirical measure associated with $X_{t}$ defined by $\mu _{X_{t}}:={\frac {1}{N}}\sum \limits _{1\leq i\leq N}\delta _{X_{t}^{i}}$ where $\delta$ is the Dirac measure.

Propagation of chaos is the property that, as the number of particles $N\to +\infty$ , the interaction between any two particles vanishes, and the random empirical measure $\mu _{X_{t}}$ is replaced by the deterministic distribution $\mu _{t}$ .

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

Applications

Mean-field theory
Mean-field game theory^[5]
Random matrices: including Dyson's model on eigenvalue dynamics for random symmetric matrices and the Wigner semicircle distribution^[6]

References

^ 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.
^ 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 ^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.
^ ^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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Definition

Existence of a solution

Propagation of chaos

Applications

References