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 σ : R d × P ( R d ) → M d ( R ) {\displaystyle \sigma :\mathbb {R} ^{d}\times {\mathcal {P}}(\mathbb {R} ^{d})\to {\mathcal {M}}_{d}(\mathbb {R} )} where P ( R d ) {\displaystyle {\mathcal {P}}(\mathbb {R} ^{d})} is the space of probability distributions on R d {\displaystyle \mathbb {R} ^{d}} equipped with the Wasserstein metric W 2 {\displaystyle W_{2}} and M d ( R ) {\displaystyle {\mathcal {M}}_{d}(\mathbb {R} )} is the space of square matrices of dimension d {\displaystyle d} . Consider a measurable function b : R d × P ( R d ) → R d {\displaystyle b:\mathbb {R} ^{d}\times {\mathcal {P}}(\mathbb {R} ^{d})\to \mathbb {R} ^{d}} . Define a ( x , μ ) := σ ( x , μ ) σ ( x , μ ) T {\displaystyle a(x,\mu ):=\sigma (x,\mu )\sigma (x,\mu )^{T}} . A stochastic process ( X t ) t ≥ 0 {\displaystyle (X_{t})_{t\geq 0}} is a McKean–Vlasov process if it solves the following system:
X 0 {\displaystyle X_{0}} has law f 0 {\displaystyle f_{0}}
d X t = σ ( X t , μ t ) d B t + b ( X t , μ t ) d t {\displaystyle dX_{t}=\sigma (X_{t},\mu _{t})dB_{t}+b(X_{t},\mu _{t})dt}
where μ t = L ( X t ) {\displaystyle \mu _{t}={\mathcal {L}}(X_{t})} describes the law of X {\displaystyle X} and B t {\displaystyle B_{t}} denotes a d {\displaystyle d} -dimensional Wiener process. This process is non-linear, in the sense that the associated Fokker–Planck equation for μ t {\displaystyle \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 ≤ i ≤ N {\displaystyle (X_{t}^{i})_{1\leq i\leq N}} . Formally, define ( X i ) 1 ≤ i ≤ N {\displaystyle (X^{i})_{1\leq i\leq N}} to be the d {\displaystyle d} -dimensional solutions to:
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