The Euler scheme with irregular coefficients

Approximation of distribution-independent and distribution-dependent stochastic differential equations with singular drifts

On the MS-stability of predictor–corrector schemes for stochastic differential equations

In this article, we consider the so-called modified Euler scheme for stochastic differential equations (SDEs) driven by fractional Brownian motions (fBm) with Hurst parameter $\frac{1}{3} \frac{1}{2}$. The current contribution generalizes the modified Euler scheme to the rough case $\frac{1}{3}<H<\frac{1}{2}$. Namely, we show a convergence rate of order $n^{\frac{1}{2}-2H}$ for the scheme, and we argue that this rate is exact. We also derive a central limit theorem for the renormalized error of the scheme, thanks to some new techniques for asymptotics of weighted random sums. Our main idea is based on the following observation: the triple of processes obtained by considering the fBm, the scheme process and the normalized error process, can be lifted to a new rough path. In addition, the Holder norm of this new rough path has an estimate which is independent of the step-size of the scheme.

In this paper, we propose numerical schemes for stochastic differential equations driven by G-Lévy process under the G-expectation framework. By using G-Itô formula and G-expectation property, we propose Euler scheme and Milstein scheme which have order-1.0 convergence rate. And two numerical experiments including Ornstein-Uhlenbeck and Black-Scholes cases are given.

Wong-Zakai Corrections, Random Evolutions, and Simulation Schemes for SDE's

For solutions of Itô–Volterra equations and semilinear evolution-type equations we consider approximations via the Milstein scheme, approximations by finite-dimensional processes, and approximations by solutions of stochastic differential equations (SDEs) with bounded coefficients. We prove mean-square convergence for finite-dimensional approximations and establish results on the rate of mean-square convergence for two remaining types of approximation.

Monte Carlo simulation is becoming increasingly popular in areas with oceonographic and atmospheric applications. Examples are risk and uncertainty analysis. Usually what is done, is that a stochastic part is added to an already existing, deterministic model, providing the user with error bounds of the model outcome. The underlying deterministic model is often a discretization of a set of partial differential equations describing physical processes such as transport, turbulence, buoyancy effects, and continuity. A lot of effort is also put into numerically efficient schemes for the time integration. The resulting model is most of the time quite large and complex. In sharp contrast with this, the stochastic extension used for Monte Carlo experiments is usually achieved by adding white noise. Unfortunately the order of time integration for the complete stochastic model is reduced, because white noise is not smooth enough. Instead of completely replacing the old numerical scheme and implementing a higher order scheme for stochastic differential equations, we suggest a different approach that uses the existing scheme. The method uses a smooth colored noise process to obtain a higher order of convergence. The above idea was successfully tried on a stochastic version of the highly non-linear Lorenz model, where we performed an uncertainty analysis to gain insight in the accuracy of the model predictions. In the near future the method will be applied to real-life problems.

Projection scheme for stochastic differential equations with convex constraints

Some remarks on approximation of solutions of SDE's with reflecting boundary conditions

<title>Abstract</title> We introduce a lattice random walk discretisation scheme for stochastic differential equations (SDEs) that samples binary or ternary increments at each step, suppressing complex drift and diffusion computations to simple 1 or 2 bit random values. This approach is a significant departure from traditional floating point discretisations and offers several advantages; including compatibility with stochastic computing architectures that avoid floating-point arithmetic in place of directly manipulating the underlying probability distribution of a bitstream, elimination of Gaussian sampling requirements, robustness to quantisation errors, and handling of non-Lipschitz drifts. We prove weak convergence and demonstrate the advantages through experiments on various SDEs, including state-of-the-art diffusion models.

In this paper we present a numerical scheme for stochastic differential equations based upon the Wiener chaos expansion. The approximation of a square integrable stochastic differential equation is obtained by cutting off the infinite chaos expansion in chaos order and in number of basis elements. We derive an explicit upper bound for the $L^2$ approximation error associated with our method. The proofs are based upon an application of Malliavin calculus.

Previous article Next article A Simple Proof of the Existence of a Solution of Itô’s Equation with Monotone CoefficientsN. V. KrylovN. V. Krylovhttps://doi.org/10.1137/1135082PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] N. V. Krylov and , B. L. Rozovskii, Stochastic evolution equationsCurrent problems in mathematics, Vol. 14 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1979, 71–147, 256 81m:60116 Google Scholar[2] I. Gyongy and , N. V. Krylov, On stochastic equations with respect to semimartingales. I, Stochastics, 4 (1980/81), 1–21 82j:60104 CrossrefGoogle Scholar[3] N. V. Krylov, Extremal properties of solutions of stochastic equations, Theory Probab. Appl., 29 (1984), 205–217 10.1137/1129033 0557.60042 LinkGoogle Scholar[4] L. A. Alyushina, Euler polygonal lines for Itô equations with monotone coefficients, Theory Probab. Appl., 32 (1987), 340–346 10.1137/1132046 0663.60049 LinkGoogle Scholar[5] Gisirō Maruyama, Continuous Markov processes and stochastic equations, Rend. Circ. Mat. Palermo (2), 4 (1955), 48–90 17,166f 0053.40901 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails A splitting method for SDEs with locally Lipschitz drift: Illustration on the FitzHugh-Nagumo modelApplied Numerical Mathematics | 1 Apr 2022 Cross Ref Approximation of BSDEs with super-linearly growing generators by Euler’s polygonal line method: A simple proof of the existenceSystems & Control Letters, Vol. 153 | 1 Jul 2021 Cross Ref Fractional Stochastic Active Scalar Equations Generalizing the Multi-Dimensional Quasi-Geostrophic & 2D-Navier–Stokes Equations: The General CaseJournal of Mathematical Fluid Mechanics, Vol. 22, No. 4 | 4 September 2020 Cross Ref A note on strong convergence of implicit scheme for SDEs under local one-sided Lipschitz conditionsInternational Journal of Computer Mathematics, Vol. 5 | 9 March 2020 Cross Ref Strong Convergence Rates for Euler Approximations to a Class of Stochastic Path-Dependent Volatility ModelsAndrei Cozma and Christoph ReisingerSIAM Journal on Numerical Analysis, Vol. 56, No. 6 | 11 December 2018AbstractPDF (1095 KB)Well-Posedness of the Multidimensional Fractional Stochastic Navier–Stokes Equations on the Torus and on Bounded DomainsJournal of Mathematical Fluid Mechanics, Vol. 18, No. 1 | 22 October 2015 Cross Ref An Explicit Euler Scheme with Strong Rate of Convergence for Financial SDEs with Non-Lipschitz CoefficientsJean-François Chassagneux, Antoine Jacquier, and Ivo MihaylovSIAM Journal on Financial Mathematics, Vol. 7, No. 1 | 14 December 2016AbstractPDF (602 KB)A New Type of Stability Theorem for Stochastic Systems With Application to Stochastic StabilizationIEEE Transactions on Automatic Control, Vol. 61, No. 1 | 1 Jan 2016 Cross Ref Loss of regularity for Kolmogorov equationsThe Annals of Probability, Vol. 43, No. 2 | 1 Mar 2015 Cross Ref Strong Convergence of Euler Approximations of Stochastic Differential Equations with Delay Under Local Lipschitz ConditionStochastic Analysis and Applications, Vol. 32, No. 2 | 28 February 2014 Cross Ref A Note on Euler Approximations for Stochastic Differential Equations with DelayApplied Mathematics & Optimization, Vol. 68, No. 3 | 24 August 2013 Cross Ref Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equationsThe Annals of Applied Probability, Vol. 23, No. 5 | 1 Oct 2013 Cross Ref Rate of convergence of Euler’s approximations for SDEs with non-Lipschitz coefficientsActa Mathematica Sinica, English Series, Vol. 29, No. 8 | 15 July 2013 Cross Ref Stochastic Itô inclusion with upper separated multifunctionsJournal of Mathematical Analysis and Applications, Vol. 400, No. 2 | 1 Apr 2013 Cross Ref Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficientsThe Annals of Applied Probability, Vol. 22, No. 4 | 1 Aug 2012 Cross Ref Stochastic 3D tamed Navier–Stokes equations: Existence, uniqueness and small time large deviation principlesJournal of Differential Equations, Vol. 252, No. 1 | 1 Jan 2012 Cross Ref On the Strong Solution for the 3D Stochastic Leray-Alpha ModelBoundary Value Problems, Vol. 2010, No. 1 | 1 Jan 2010 Cross Ref ON STOCHASTIC EVOLUTION EQUATIONS WITH NON-LIPSCHITZ COEFFICIENTSStochastics and Dynamics, Vol. 09, No. 04 | 21 November 2011 Cross Ref Stochastic tamed 3D Navier–Stokes equations: existence, uniqueness and ergodicityProbability Theory and Related Fields, Vol. 145, No. 1-2 | 19 July 2008 Cross Ref Large deviations in the Langevin dynamics of a random field Ising modelStochastic Processes and their Applications, Vol. 105, No. 2 | 1 Jun 2003 Cross Ref Viable solutions of set-valued stochastic equationOptimization, Vol. 48, No. 2 | 1 Jan 2000 Cross Ref Volume 35, Issue 3| 1991Theory of Probability & Its Applications411-623 History Submitted:05 September 1988Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1135082Article page range:pp. 583-587ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

A class ofimplicit Runge-Kutta schemes for stochastic differential equations affected bymultiplicative Gaussian white noise is shown to be optimal with respect to global order of convergence in quadratic mean. A test equation is proposed in order to investigate the stability of discretization methods for systems of this kind. Herestability is intended in a truly probabilistic sense, as opposed to the recently introduced extension of A-stability to the stochastic context, given for systems with additive noise. Stability regions for the optimal class are also given.

Strong convergence of explicit numerical schemes for stochastic differential equations with piecewise continuous arguments

Higher-order weak extrapolation methods for the approximation of functionals of Ito diffusions are considered. Under appropriate regularity conditions it is shown that extrapolations allow a considerable increase in the weak order of convergence of a discrete-time one-step approximation method. Numerical experiments indicate the efficiency of extrapolations based on higher-order weak schemes for stochastic differential equations with additive noise.