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 [math]\displaystyle{ \sigma:\R^d \times \mathcal{P}(\R^d)\to \mathcal{M}_{d}(\R) }[/math] where [math]\displaystyle{ \mathcal{P}(\R^d) }[/math] is the space of probability distributions on [math]\displaystyle{ \R^d }[/math] equipped with the Wasserstein metric [math]\displaystyle{ W_2 }[/math] and [math]\displaystyle{ \mathcal{M}_{d}(\R) }[/math] is the space of square matrices of dimension [math]\displaystyle{ d }[/math]. Consider a measurable function [math]\displaystyle{ b:\R^d\times \mathcal{P}(\R^d)\to \mathcal{M}_d(\R) }[/math]. Define [math]\displaystyle{ a(x,\mu) := \sigma(x,\mu)\sigma(x,\mu)^T }[/math].
A stochastic process [math]\displaystyle{ (X_t)_{t\geq 0} }[/math] is a McKean–Vlasov process if it solves the following system:[3][5]
- [math]\displaystyle{ X_0 }[/math] has law [math]\displaystyle{ f_0 }[/math]
- [math]\displaystyle{ dX_t = a(X_t, \mu_t) dB_t + b(X_t, \mu_t) dt }[/math]
where [math]\displaystyle{ \mu_t = \mathcal{L}(X_t) }[/math] describes the law of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ dB }[/math] denotes the Wiener process. This process is non-linear, in the sense that the dynamics of [math]\displaystyle{ \mu_t }[/math] do not depend linearly on [math]\displaystyle{ \mu_t }[/math].[5][6]
Existence of a solution
The following Theorem can be found in.[4]
Existence of a solution — Suppose [math]\displaystyle{ b }[/math] and [math]\displaystyle{ \sigma }[/math] are globally Lipschitz, that is, there exists a constant [math]\displaystyle{ C\gt 0 }[/math] such that:
[math]\displaystyle{ |b(x,\mu)-b(y,\nu)| + |\sigma(x,\mu)-\sigma(y,\nu)| \leq C(|x-y|+W_2(\mu,\nu)) }[/math]
where [math]\displaystyle{ W_2 }[/math] is the Wasserstein metric.
Suppose [math]\displaystyle{ f_0 }[/math] has finite variance.
Then for any [math]\displaystyle{ T\gt 0 }[/math] there is a unique strong solution to the McKean-Vlasov system of equations on [math]\displaystyle{ [0,T] }[/math]. Furthermore, its law is the unique solution to the non-linear Fokker–Planck equation:
[math]\displaystyle{ \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\} }[/math]
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 [math]\displaystyle{ (X_t^i)_{1\leq i\leq N} }[/math].
Formally, define [math]\displaystyle{ (X^i)_{1\leq i\leq N} }[/math] to be the [math]\displaystyle{ d }[/math]-dimensional solutions to:
- [math]\displaystyle{ (X_0^i)_{1\leq i\leq N} }[/math] are i.i.d with law [math]\displaystyle{ f_0 }[/math]
- [math]\displaystyle{ dX_t^i = a(X_t^i, \mu_{X_t}) dB_t^i + b(X_t^i, \mu_{X_t}) dt }[/math]
where the [math]\displaystyle{ (B^i)_{1\leq i\leq N} }[/math] are i.i.d Brownian motion, and [math]\displaystyle{ \mu_{X_t} }[/math] is the empirical measure associated with [math]\displaystyle{ X_t }[/math] defined by [math]\displaystyle{ \mu_{X_t} := \frac{1}{N}\sum\limits_{1\leq i\leq N} \delta_{X_t^i} }[/math] where [math]\displaystyle{ \delta }[/math] is the Dirac measure.
Propagation of chaos is the property that, as the number of particles [math]\displaystyle{ N\to +\infty }[/math], the interaction between any two particles vanishes, and the random empirical measure [math]\displaystyle{ \mu_{X_t} }[/math] is replaced by the deterministic distribution [math]\displaystyle{ \mu_t }[/math].
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. http://tel.archives-ouvertes.fr/docs/00/65/87/66/PDF/tachet.pdf.
- ↑ 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.
- ↑ 3.0 3.1 McKean, H. P. (1966). "A Class of Markov Processes Associated with Nonlinear Parabolic Equations". Proc. Natl. Acad. Sci. USA 56 (6): 1907–1911. doi:10.1073/pnas.56.6.1907. PMID 16591437. Bibcode: 1966PNAS...56.1907M.
- ↑ 4.0 4.1 4.2 4.3 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. doi:10.3934/krm.2022017. ISSN 1937-5093. http://dx.doi.org/10.3934/krm.2022017.
- ↑ 5.0 5.1 5.2 Carmona, Rene; Delarue, Francois; Lachapelle, Aime. "Control of McKean-Vlasov Dynamics versus Mean Field Games". https://carmona.princeton.edu/download/mfg/cdl.pdf.
- ↑ 6.0 6.1 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. https://projecteuclid.org/journals/annals-of-probability/volume-22/issue-1/Dynamics-of-the-McKean-Vlasov-Equation/10.1214/aop/1176988866.full.
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