An Euler scheme for McKean SDEs with Besov drift: convergence rate and implementation

TL;DR

This work proposes the first implementable numerical scheme for a one-dimensional McKean-Vlasov stochastic differential equation with a drift equal to a product of a distribution depending on the state of the process and a non-linear function depending pointwise on the law density of the solution.

Abstract

We study a one-dimensional McKean-Vlasov stochastic differential equation (SDE) with a drift equal to a product of a distribution depending on the state of the process and a non-linear function depending pointwise on the law density of the solution. Building on recent well-posedness results, we propose the first implementable numerical scheme for this class of SDEs. Our approach combines mollification of the distributional drift with the Euler-Maruyama scheme and a PDE-based approximation of the law via the associated Fokker-Planck equation. We prove strong convergence of the scheme and derive an explicit rate, showing how to balance the smoothing parameter with the time discretisation. Numerical experiments confirm the applicability of our scheme and demonstrate the significant influence of the McKean interaction term on the law of the solution.

Figures (3)

• Figure 1: Left panel: $F(x) = \sin(x)$. Right panel: $F(x) = 1/(1 + \exp(-100(x - 0.2)))$. Top: the plot of $b^N$ for two different values of $N$ (green line for a 'small' $N$, red line for a 'large' $N$) for $b\in C^{-\beta}$ with $\beta=0.49$. Middle: the plot of $F(\rho^N)$ for two different values of $N$. The approximation of $\rho^N$ is obtained solving the PDE on a grid with $2^{11}$ time steps and $4\times10^3$ points in $x$. Bottom: the plot of the product $F(\rho^N) b^N$ for two different values of $N$.
• Figure 2: Plot of theoretical (broken line) and empirical (dotted-broken line) convergence rate (vertical axis) against parameter $\beta$ (horizontal axis) obtained with $F(\hat{\rho}^N)b^N$ with $b\in \mathcal{C}^{-\beta}$, $X_0 \sim \mathcal{N}(0, 1)$ and $F(x) = \sin(x)$. Left panel: we used a singular fixed drift, and 10,000 sample paths of a Brownian motion, run 40 times to obtain the empirical rate and its 95% confidence inteval (shaded area around empirical rate). Right panel: same as before but we used multiple drifts, in particular 40 drifts (a different drift for evey run).
• Figure 3: Representation of $\hat{\rho}^N$ for $b^N \in C^{-\beta}$ with different values of $F$ and of $\beta$. Left panels $\beta= 0.49$, right panels $\beta= 0.01$. From top to bottom $F(x) = \sin(x)$, $F(x) = 5\sin(x)$, $F(x) = 1/(1 + \exp(-100(x - 0.2)))$, $F(x) = \cos(x)$. SDE solved with $2^{11}$ time steps and $10^4$ sample paths. PDE solved on a grid with $2^{11}$ time steps and $4\times10^3$ points in $x \in [-10, 10]$.

Theorems & Definitions (36)

• Remark 3.1
• Proposition 3.2: issoglioMcKeanSDEsSingular2023
• Theorem 3.3: issoglioMcKeanSDEsSingular2023
• Theorem 3.4
• proof
• Theorem
• Remark 3.5
• Lemma 4.1: Schauder estimates
• Remark 4.2
• Lemma 4.3: Bernstein inequality