Schemes For Stochastic Differential Equations Research Articles

Mean-square convergence analysis of the semi-implicit scheme for stochastic differential equations driven by the Wiener processes

Mean-square convergence analysis of the semi-implicit scheme for stochastic differential equations driven by the Wiener processes

Tamed-adaptive Euler-Maruyama approximation for SDEs with locally Lipschitz continuous drift and locally Hölder continuous diffusion coefficients

We propose a tamed-adaptive Euler-Maruyama approximation scheme for stochastic differential equations with locally Lipschitz continuous, polynomial growth drift, and locally Hölder continuous, polynomial growth diffusion coefficients. We consider the strong convergence and the stability of the new scheme. In particular, we show that under some sufficient conditions for the stability of the exact solution, the tamed-adaptive scheme converges strongly in both finite and infinite time intervals.

Positivity preserving logarithmic Euler-Maruyama type scheme for stochastic differential equations

In this paper, we propose a class of explicit positivity preserving numerical methods for general stochastic differential equations which have positive solutions. Namely, all the numerical solutions are positive. Under some reasonable conditions, we obtain the convergence and the convergence rate results for these methods. The main difficulty is to obtain the strong convergence and the convergence rate for stochastic differential equations whose coefficients are of exponential growth. Some numerical experiments are provided to illustrate the theoretical results for our schemes.

Runge–Kutta Lawson schemes for stochastic differential equations

In this paper, we present a framework to construct general stochastic Runge-Kutta Lawson schemes. We prove that the schemes inherit the consistency and convergence properties of the underlying Runge-Kutta scheme, and confirm this in some numerical experiments. We also investigate the stability properties of the methods and show for some examples, that the new schemes have improved stability properties compared to the underlying schemes.

Uniform in time estimates for the weak error of the Euler method for SDEs and a pathwise approach to derivative estimates for diffusion semigroups

We present a criterion for uniform in time convergence of the weak error of the Euler scheme for Stochastic Differential equations (SDEs). The criterion requires (i) exponential decay in time of the space-derivatives of the semigroup associated with the SDE and (ii) bounds on (some) moments of the Euler approximation. We show by means of examples (and counterexamples) how both (i) and (ii) are needed to obtain the desired result. If the weak error converges to zero uniformly in time, then convergence of ergodic averages follows as well. We also show that Lyapunov-type conditions are neither sufficient nor necessary in order for the weak error of the Euler approximation to converge uniformly in time and clarify relations between the validity of Lyapunov conditions, (i) and (ii). Conditions for (ii) to hold are studied in the literature. Here we produce sufficient conditions for (i) to hold. The study of derivative estimates has attracted a lot of attention, however not many results are known in order to guarantee exponentially fast decay of the derivatives. Exponential decay of derivatives typically follows from coercive-type conditions involving the vector fields appearing in the equation and their commutators; here we focus on the case in which such coercive-type conditions are non-uniform in space. To the best of our knowledge, this situation is unexplored in the literature, at least on a systematic level. To obtain results under such space-inhomogeneous conditions we initiate a pathwise approach to the study of derivative estimates for diffusion semigroups and combine this pathwise method with the use of Large Deviation Principles.

Crank–Nicolson scheme for stochastic differential equations driven by fractional Brownian motions

We study the Crank–Nicolson scheme for stochastic differential equations (SDEs) driven by a multidimensional fractional Brownian motion with Hurst parameter H>1/2. It is well known that for ordinary differential equations with proper conditions on the regularity of the coefficients, the Crank–Nicolson scheme achieves a convergence rate of n−2, regardless of the dimension. In this paper we show that, due to the interactions between the driving processes, the corresponding Crank–Nicolson scheme for m-dimensional SDEs has a slower rate than for one-dimensional SDEs. Precisely, we shall prove that when the fBm is one-dimensional and when the drift term is zero, the Crank–Nicolson scheme achieves the convergence rate n−2H, and when the drift term is nonzero, the exact rate turns out to be n−12−H. In the general multidimensional case the exact rate equals n12−2H. In all these cases the asymptotic error is proved to satisfy some linear SDE. We also consider the degenerated cases when the asymptotic error equals zero.

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

Predictor–corrector schemes are designed to be a compromise to retain the stability properties of the implicit schemes and the computational efficiency of the explicit ones. In this paper a complete analytical study for the linear mean-square stability of the two-parameter family of Euler predictor–corrector schemes for scalar stochastic differential equations is given. The analyzed family is given in terms of two parameters that control the degree of implicitness of the method. For each selection of the parameters the stability region is obtained, letting its comparison. Particular cases of the counter-intuitive fact of losing numerical stability by reducing the step size, is confirmed and proved. Figures of the MS-stability regions and numerical examples that confirm the theoretical results are shown.

Optimal rate of convergence for two classes of schemes to stochastic differential equations driven by fractional Brownian motions

Abstract This paper investigates numerical schemes for stochastic differential equations driven by multi-dimensional fractional Brownian motions (fBms) with Hurst parameter $H\in (\frac 12,1)$. Based on the continuous dependence of numerical solutions on the driving noises, we propose the order conditions of Runge–Kutta methods for the strong convergence rate $2H-\frac 12$, which is the optimal strong convergence rate for approximating the Lévy area of fBms. We provide an alternative way to analyse the convergence rate of explicit schemes by adding ‘stage values’ such that the schemes are interpreted as Runge–Kutta methods. Taking advantage of this technique the strong convergence rate of simplified step-$N$ Euler schemes is obtained, which gives an answer to a conjecture in Deya et al. (2012) when $H\in (\frac 12,1)$. Numerical experiments verify the theoretical convergence rate.

Simulation of Stochastic Electromagnetic Transients in EMTP: A Bug Turned Into a Feature

This paper proposes a systematic approach to simulating stochastic electromagnetic transients in emtp, through the reliable numerical solution of stochastic differential equations. Actually, numerical integration schemes for stochastic differential equations (sdes) are not available in emtp. It is shown how to set up the Semi-Implicit Backward Euler scheme specific for stochastic differential equations. The implemented interaction strategy between the engine for the solution of electrical circuits and the engine for the Transient Analysis of Control Systems of emtp is exploited. This is a bolt from the blue feature of this simulator that makes it the only one on the shelf able to handle stochastic differential equations. Exploiting this unveiled capability, some basic and tutorial case studies are presented. They pave the way for the simulation of more complex circuits and systems that go beyond the scope of this work. Simulation files are provided as supplementary material and they are available on GitHub.

A note on strong convergence of implicit scheme for SDEs under local one-sided Lipschitz conditions

In this paper, we aim to study strong convergence of theta-EM scheme for stochastic differential equations under a local one-sided Lipschitz condition. Since the existence and uniqueness of theta-EM solution cannot be guaranteed under the local one-sided Lipschitz condition, a modified theta-EM scheme is proposed, and strong convergence of the numerical solution to the exact solution is given.

A machine learning solver for high-dimensional integrals: Solving Kolmogorov PDEs by stochastic weighted minimization and stochastic gradient descent through a high-order weak approximation scheme of SDEs with Malliavin weights

The paper introduces a very simple and fast computation method for high-dimensional integrals to solve high-dimensional Kolmogorov partial differential equations (PDEs). The new machine learning-based method is obtained by solving a stochastic weighted minimization with stochastic gradient descent which is inspired by a high-order weak approximation scheme for stochastic differential equations (SDEs) with Malliavin weights. Then solutions to high-dimensional Kolmogorov PDEs or expectations of functionals of solutions to high-dimensional SDEs are accurately approximated without suffering from the curse of dimensionality. Numerical examples for PDEs and SDEs up to 100 dimensions are shown by using second and third-order discretization schemes in order to demonstrate the effectiveness of our method.

Convergence Analysis of Parareal Algorithm Based on Milstein Scheme for Stochastic Differential Equations

Convergence Analysis of Parareal Algorithm Based on Milstein Scheme for Stochastic Differential Equations

Convergence, Non-negativity and Stability of a New Tamed Euler–Maruyama Scheme for Stochastic Differential Equations with Hölder Continuous Diffusion Coefficient

We propose and analyze a new tamed Euler–Maruyama approximation scheme for stochastic differential equations with Holder continuous diffusion. This new scheme preserves the stability and non-negativity of the exact solution.

The Euler scheme for stochastic differential equations with discontinuous drift coefficient: a numerical study of the convergence rate

The Euler scheme is one of the standard schemes to obtain numerical approximations of solutions of stochastic differential equations (SDEs). Its convergence properties are well known in the case of globally Lipschitz continuous coefficients. However, in many situations, relevant systems do not show a smooth behavior, which results in SDE models with discontinuous drift coefficient. In this work, we analyze the long time properties of the Euler scheme applied to SDEs with a piecewise constant drift and a constant diffusion coefficient and carry out intensive numerical tests for its convergence properties. We emphasize numerical convergence rates and analyze how they depend on the properties of the drift coefficient and the initial value. We also give theoretical interpretations of some of the arising phenomena. For application purposes, we study a rank-based stock market model describing the evolution of the capital distribution within the market and provide theoretical as well as numerical results on the long time ranking behavior.

First-Order Weak Balanced Schemes for Stochastic Differential Equations

We address the numerical solution of stochastic differential equations with multiplicative noise (SDEs) by means of balanced schemes. In particular, we study the design of balanced schemes that achieve the first order of weak convergence without involve the simulation of multiple stochastic integrals. We start by using the linear scalar SDE as a test problem to show that it is possible to construct almost sure stable first-order weak schemes based on the addition of stabilizing functions to the drift terms. Then, we consider multidimensional linear SDEs. In this case, we propose to find appropriate stabilizing weights through an optimization procedure. Finally, we illustrate the potential of the new methodology by solving the stochastic Duffing-Van Der Pol equation, which is a classical test non-linear SDE. Numerical experiments show a good performance of the numerical methods introduced in this paper.

Numerical Scheme for Stochastic Differential Equations Driven by Fractional Brownian Motion with $$ 1/4<H <1/2$$.

In this article, we study a numerical scheme for stochastic differential equations driven by fractional Brownian motion with Hurst parameter $$ H \in \left( 1/4, 1/2 \right) $$ . Toward this end, we apply Doss–Sussmann representation of the solution and an approximation of this representation using a first-order Taylor expansion. The obtained rate of convergence is $$n^{-2H +\rho }$$ , for $$\rho $$ small enough.

First-order Euler scheme for SDEs driven by fractional Brownian motions: The rough case

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.

The asymptotic error of chaos expansion approximations for stochastic differential equations

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.

An Adaptive Euler--Maruyama Scheme for Stochastic Differential Equations with Discontinuous Drift and its Convergence Analysis

We study the strong approximation of stochastic differential equations with discontinuous drift coefficients and (possibly) degenerate diffusion coefficients. To account for the discontinuity of the drift coefficient we construct an adaptive step sizing strategy for the explicit Euler-Maruyama scheme. As a result, we obtain a numerical method which has -- up to logarithmic terms -- strong convergence order $1/2$ with respect to the average computational cost. We support our theoretical findings with several numerical examples.

Convergence Rate of Euler\u2013Maruyama Scheme for SDEs with H\xf6lder\u2013Dini Continuous Drifts

In this paper, we are concerned with convergence rate of Euler–Maruyama scheme for stochastic differential equations with Hölder–Dini continuous drifts. The key contributions are as follows: (i) by means of regularity of non-degenerate Kolmogrov equation, we investigate convergence rate of Euler–Maruyama scheme for a class of stochastic differential equations which allow the drifts to be Dini continuous and unbounded; (ii) by the aid of regularization properties of degenerate Kolmogrov equation, we discuss convergence rate of Euler–Maruyama scheme for a range of degenerate stochastic differential equations where the drifts are Hölder–Dini continuous of order frac{2}{3} with respect to the first component and are merely Dini-continuous concerning the second component.