• Corpus ID: 230437616

# A simple approach to proving the existence, uniqueness, and strong and weak convergence rates for a broad class of McKean-Vlasov equations

@article{HajiAli2021ASA,
title={A simple approach to proving the existence, uniqueness, and strong and weak convergence rates for a broad class of McKean-Vlasov equations},
author={Abdul-Lateef Haji-Ali and Hrakon Hoel and Ra{\'u}l Tempone},
journal={ArXiv},
year={2021},
volume={abs/2101.00886}
}
• Published 4 January 2021
• Mathematics
• ArXiv
By employing a system of interacting stochastic particles as an approximation of the McKean–Vlasov equation and utilizing classical stochastic analysis tools, namely Itô’s formula and Kolmogorov–Chentsov continuity theorem, we prove the existence and uniqueness of strong solutions for a broad class of McKean–Vlasov equations. Considering an increasing number of particles in the approximating stochastic particle system, we also prove the L strong convergence rate and derive the weak convergence…
3 Citations

## Figures from this paper

### Single Level Importance Sampling for McKean-Vlasov Stochastic Differential Equations

• Mathematics
• 2022
This paper investigates Monte Carlo methods to estimate probabilities of rare events associated with solutions to the d -dimensional McKean-Vlasov stochastic diﬀerential equation. The equation is

### Double Loop Monte Carlo Estimator with Importance Sampling for McKean-Vlasov Stochastic Differential Equation

• Mathematics
ArXiv
• 2022
This paper investigates Monte Carlo methods to estimate probabilities of rare events associated with solutions to the d -dimensional McKean-Vlasov stochastic diﬀerential equation. The equation is

### Multilevel Importance Sampling for McKean-Vlasov Stochastic Differential Equation

• Mathematics
ArXiv
• 2022
This work combines multilevel Monte Carlo methods with importance sampling (IS) to estimate rare event quantities that can be expressed as the expectation of a Lipschitz observable of the solution to

## References

SHOWING 1-10 OF 26 REFERENCES

### Existence and uniqueness theorems for solutions of McKean–Vlasov stochastic equations

• Mathematics
Theory of Probability and Mathematical Statistics
• 2021
New weak and strong existence and weak and strong uniqueness results for multi-dimensional stochastic McKean--Vlasov equations are established under relaxed regularity conditions. Weak existence is a

### A stochastic particle method for the McKean-Vlasov and the Burgers equation

• Mathematics
Math. Comput.
• 1997
A stochastic particle method for the McKean-Vlasov and the Burgers equation is introduced and numerical experiments are presented which confirm the theoretical estimates and illustrate the numerical efficiency of the method when the viscosity coefficient is very small.

### Convergence Rate for the Approximation of the Limit Law of Weakly Interacting Particles 2: Application to the Burgers Equation

• Mathematics
• 1996
In this paper, we construct a stochastic particles method for the Burgers equation with a monotonic initial condition; we prove that the convergence rate is \$\displaystyleO\left(\frac1\sqrtN

### Convergence Rates for Adaptive Weak Approximation of Stochastic Differential Equations

• Mathematics, Computer Science
• 2005
Convergence rates of adaptive algorithms for weak approximations of Itoˆ stochastic differential equations are proved for the Monte Carlo Euler method and both adaptive alogrithms are proven to stop with asymptotically optimal number of steps up to a problem independent factor defined in the algorithm.

### Numerical Solution of Stochastic Differential Equations

This paper provides an introduction to the main concepts and techniques necessary for someone who wishes to carryout numerical experiments involving Stochastic Differential Equation (SDEs). As SDEs

### The law of the Euler scheme for stochastic differential equations

• Mathematics
Monte Carlo Methods Appl.
• 1996
SummaryWe study the approximation problem ofEf(XT) byEf(XTn), where (Xt) is the solution of a stochastic differential equation, (XTn) is defined by the Euler discretization scheme with stepT/n, andf

### A new approach to quantitative propagation of chaos for drift, diffusion and jump processes

• Mathematics
• 2015
This paper is devoted the study of the mean field limit for many-particle systems undergoing jump, drift or diffusion processes, as well as combinations of them. The main results are quantitative

### Adaptive weak approximation of reflected and stopped diffusions

• Mathematics
Monte Carlo Methods Appl.
• 2010
An error representation for the projected Euler method of Costantini, Pacchiarotti and Sartoretto is derived and two new algorithms for stopped diffusion and stochastic refinement of the time grid are introduced.

### Nonlinear Markov Processes and Kinetic Equations

A nonlinear Markov evolution is a dynamical system generated by a measure-valued ordinary differential equation with the specific feature of preserving positivity. This feature distinguishes it from