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

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

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

where ${\displaystyle \mu _{t}={\mathcal {L}}(X_{t})}$ describes the law of ${\displaystyle X}$ and ${\displaystyle B_{t}}$ denotes a ${\displaystyle d}$-dimensional Wiener process. This process is non-linear, in the sense that the associated Fokker-Planck equation for ${\displaystyle \mu _{t}}$ 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 ${\displaystyle b}$ and ${\displaystyle \sigma }$ are globally Lipschitz, that is, there exists a constant ${\displaystyle C>0}$ such that:

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

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

Suppose ${\displaystyle f_{0}}$ has finite variance.

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

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

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

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

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

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

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

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]