McKean–Vlasov process

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.

|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 T0 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. 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

References

References

1. Des Combes, Rémi Tachet. (2011). "Non-parametric model calibration in finance: Calibration non paramétrique de modèles en finance".
2. (1984). "A certain class of diffusion processes associated with nonlinear parabolic equations". Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete.
3. McKean, H. P.. (1966). "A Class of Markov Processes Associated with Nonlinear Parabolic Equations". [[Proceedings of the National Academy of Sciences of the United States of America.
4. (2022). "Propagation of chaos: A review of models, methods and applications. I. Models and methods". Kinetic and Related Models.
5. "Control of McKean-Vlasov Dynamics versus Mean Field Games".
6. Chan, Terence. (January 1994). "Dynamics of the McKean-Vlasov Equation". The Annals of Probability.