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Efficient Numerical Approximation of Invariant Probability Measures for McKean-Vlasov Stochastic Differential Equations


Core Concepts
This paper establishes the convergence rate of the weighted empirical measure of a self-interacting process to the invariant probability measure of McKean-Vlasov stochastic differential equations (MV-SDEs). It then designs an Euler-Maruyama (EM) scheme for the self-interacting process and derives the convergence rate between the weighted empirical measure of the EM numerical solution and the invariant measure of MV-SDEs.
Abstract
The paper focuses on the numerical approximation of the invariant probability measure of McKean-Vlasov stochastic differential equations (MV-SDEs). Key highlights: Establishes the convergence rate of the weighted empirical measure of a self-interacting process to the invariant probability measure of MV-SDEs. Designs an Euler-Maruyama (EM) scheme for the self-interacting process and derives the convergence rate between the weighted empirical measure of the EM numerical solution and the invariant measure of MV-SDEs. Compares the computational cost of the weighted empirical approximation and the averaged weighted empirical approximation, showing the former has a lower cost. Validates the theoretical results through numerical experiments. The paper first introduces the MV-SDE model and the assumptions required for the well-posedness of the solution and the existence of a unique invariant probability measure. It then constructs a self-interacting process whose coefficients depend only on the current and historical information of the solution. The key results are: The convergence rate of the weighted empirical measure of the self-interacting process and the invariant measure of MV-SDEs is obtained in the W2-Wasserstein metric. An EM scheme is constructed for the self-interacting process, and the convergence rate between the weighted empirical measure of the EM numerical solution and the invariant measure of MV-SDEs is derived. The convergence rate between the averaged weighted empirical measure of the EM numerical solution of the corresponding multi-particle system and the invariant measure of MV-SDEs is also provided. The computational cost analysis shows the averaged weighted empirical approximation of the particle system has a lower cost than the weighted empirical approximation.
Stats
The paper does not contain any explicit numerical data or statistics. The focus is on theoretical analysis and convergence rate derivations.
Quotes
None.

Deeper Inquiries

How can the proposed numerical approximation methods be extended to handle more general classes of MV-SDEs, such as those with super-linear growth coefficients

The proposed numerical approximation methods can be extended to handle more general classes of MV-SDEs, including those with super-linear growth coefficients, by adapting the existing framework to accommodate the specific characteristics of these equations. One approach could involve modifying the self-interacting process to incorporate the nonlinearity introduced by the super-linear growth coefficients. This may require adjusting the convergence analysis to account for the increased complexity in the dynamics of the system. Additionally, the numerical schemes, such as the EM scheme, may need to be tailored to handle the specific form of the coefficients in the SDEs. By carefully considering the properties of the super-linear growth terms and their impact on the invariant measure, the numerical approximation methods can be extended to effectively capture the behavior of these more general MV-SDEs.

What are the potential limitations or challenges in applying the self-interacting process approach to high-dimensional MV-SDEs

Applying the self-interacting process approach to high-dimensional MV-SDEs may pose several limitations and challenges. One significant challenge is the computational complexity associated with simulating and analyzing high-dimensional systems. As the dimensionality of the problem increases, the number of particles or interactions in the self-interacting process grows exponentially, leading to a substantial increase in computational cost. This can make it challenging to obtain accurate and efficient approximations of the invariant measure for high-dimensional MV-SDEs. Additionally, the curse of dimensionality may impact the convergence rates of the numerical methods, requiring sophisticated techniques to mitigate these effects. Furthermore, the interpretation and visualization of results in high-dimensional spaces can be challenging, making it harder to gain insights from the approximation methods in practice.

Are there any connections between the invariant measure approximation of MV-SDEs and the mean-field game theory, and how can these connections be further explored

There are indeed connections between the invariant measure approximation of MV-SDEs and mean-field game theory that offer avenues for further exploration. Mean-field game theory deals with the study of strategic interactions among a large number of agents, where each agent's behavior depends on the distribution of the entire population. In the context of MV-SDEs, the invariant measure represents the long-term distribution of the system, capturing the collective behavior of all particles. By drawing parallels between the invariant measure approximation and mean-field game theory, researchers can potentially leverage insights from one field to enhance understanding in the other. Exploring these connections further could lead to the development of novel methodologies for analyzing and solving both MV-SDEs and mean-field games, offering new perspectives on complex systems with interacting components.
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