Computing the invariant distribution of McKean-Vlasov SDEs by ergodic simulation

Abstract

We design a fully implementable scheme to compute the invariant distribution of ergodic McKean-Vlasov SDE satisfying a uniform confluence property. Under natural conditions, we prove various convergence results notably we obtain rates for the Wasserstein distance in quadratic mean and almost sure sense.

Figures (7)

• Figure 1: Empirical estimation of the convergence rate for Model \ref{['eq de OU model']} as a function of $\mathfrak{b}$. Tests realised for three different specifications of the algorithm input step rate, $N=100000$ and $M=500$ in \ref{['eq de error']}. $\bar{\mathcal{X}}_0$ follows a uniform distribution on $[0,1].$
• Figure 2: Comparison of optimal rates for $\mathfrak{b}=0.9$ (left) and $\mathfrak{b}=0.5$ (right) for model \ref{['eq de OU model']}. $N=100000$ and $M=500$ in \ref{['eq de error']}. $\bar{\mathcal{X}}_0$ follows a uniform distribution on $[0,1].$
• Figure 3: Behavior of $(\bar{\mathcal{X}_n})$ and $(\bar{\nu}_{\Gamma_n})$ for model \ref{['eq de OU model']}. $\bar{\mathcal{X}}_0$ follows a uniform distribution on $[0,1].$
• Figure 4: Estimated convergence rate for the model \ref{['eq fancy drift new formulation']} for a fixed algorithm step rate $r=1/3$. Three different runs are plotted, $N=100000$ and $M=500$ in \ref{['eq de error']}. $\bar{\mathcal{X}}_0$ follows a uniform distribution on $[0,1].$
• Figure 5: Estimated second moment of the scheme for the model \ref{['eq fancy drift new formulation']} for a fixed algorithm step rate $r=1/3$ (M=20000). $\bar{\mathcal{X}}_0$ follows a uniform distribution on $[0,1].$

Theorems & Definitions (31)

• Theorem 1.1: Convergence of the empirical measure
• Theorem 1.2: Quadratic Mean $\mathcal{W}_2$-rates
• Theorem 1.3: Almost sure $\mathcal{W}_2$-rates
• Remark 1.1
• Lemma 1.1
• Lemma 1.2
• Lemma 2.1
• Proposition 2.1
• Corollary 2.1
• Proposition 2.2