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. The equations are a model for Vlasov equation and were first studied by Henry McKean in 1966. 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.

Definition
Consider a measurable function $$\sigma:\R^d \times \mathcal{P}(\R^d)\to \mathcal{M}_{d}(\R)$$ where $$\mathcal{P}(\R^d)$$ is the space of probability distributions on $$\R^d$$ equipped with the Wasserstein metric $$W_2$$ and $$\mathcal{M}_{d}(\R)$$ is the space of square matrices of dimension $$d$$. Consider a measurable function $$b:\R^d\times \mathcal{P}(\R^d)\to \R^d$$. 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:


 * $$X_0$$ 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.

Existence of a solution
The following Theorem can be found in.

Propagation of chaos
The McKean-Vlasov process is an example of propagation of chaos. 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, the mean-field process just defined will converge to the corresponding McKean-Vlasov process.

Applications

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