scholarly journals Strong convergence rates of probabilistic integrators for ordinary differential equations

2019 ◽  
Vol 29 (6) ◽  
pp. 1265-1283 ◽  
Author(s):  
Han Cheng Lie ◽  
A. M. Stuart ◽  
T. J. Sullivan
Keyword(s):  
Differential Equations ◽  
Strong Convergence ◽  
Ordinary Differential Equations ◽  
Convergence Rates ◽  
Strong Convergence Rates

scholarly journals The Euler–Maruyama scheme for SDEs with irregular drift: convergence rates via reduction to a quadrature problem.

2020 ◽  
Author(s):  
Andreas Neuenkirch ◽  
Michaela Szölgyenyi
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Error Analysis ◽  
Strong Convergence ◽  
General Framework ◽  
Additive Noise ◽  
Convergence Rates ◽  
Convergence Order ◽  
Quadrature Problem ◽  
Weighted Quadrature

Abstract We study the strong convergence order of the Euler–Maruyama (EM) scheme for scalar stochastic differential equations with additive noise and irregular drift. We provide a general framework for the error analysis by reducing it to a weighted quadrature problem for irregular functions of Brownian motion. Assuming Sobolev–Slobodeckij-type regularity of order $\kappa \in (0,1)$ for the nonsmooth part of the drift, our analysis of the quadrature problem yields the convergence order $\min \{3/4,(1+\kappa )/2\}-\epsilon$ for the equidistant EM scheme (for arbitrarily small $\epsilon>0$). The cut-off of the convergence order at $3/4$ can be overcome by using a suitable nonequidistant discretization, which yields the strong convergence order of $(1+\kappa )/2-\epsilon$ for the corresponding EM scheme.

scholarly journals First-order convergence of Milstein schemes for McKean–Vlasov equations and interacting particle systems

2021 ◽  
Vol 477 (2245) ◽  
pp. 20200258
Author(s):  
Jianhai Bao ◽  
Christoph Reisinger ◽  
Panpan Ren ◽  
Wolfgang Stockinger
Keyword(s):  
Strong Convergence ◽  
Convergence Rates ◽  
Interacting Particle Systems ◽  
Particle System ◽  
First Order ◽  
Time Stepping ◽  
Vlasov Equations ◽  
Interacting Particle ◽  
Moment Stability ◽  
Strong Convergence Rates

In this paper, we derive fully implementable first-order time-stepping schemes for McKean–Vlasov stochastic differential equations, allowing for a drift term with super-linear growth in the state component. We propose Milstein schemes for a time-discretized interacting particle system associated with the McKean–Vlasov equation and prove strong convergence of order 1 and moment stability, taming the drift if only a one-sided Lipschitz condition holds. To derive our main results on strong convergence rates, we make use of calculus on the space of probability measures with finite second-order moments. In addition, numerical examples are presented which support our theoretical findings.

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