Approximation of the invariant measure for stable stochastic differential equations by the Euler–Maruyama scheme with decreasing step sizes

Approximation of the invariant measure of stable SDEs by an Euler–Maruyama scheme

This paper deals with numerical analysis of solutions to stochastic differential equations with jumps (SDEJs) with measurable drifts that may have quadratic growth. The main tool used is the Zvonkin space transformation to eliminate the singular part of the drift. More precisely, the idea is to transform the original SDEJs to standard SDEJs without singularity by using a deterministic real-valued function that satisfies a second-order differential equation. The Euler–Maruyama scheme is used to approximate the solution to the equations. It is shown that the rate of convergence is 12. Numerically, two different methods are used to approximate solutions for this class of SDEJs. The first method is the direct approximation of the original equation using the Euler–Maruyama scheme with specific tests for the evaluation of the singular part at simulated values of the solution. The second method consists of taking the inverse of the Euler–Maruyama approximation for Zvonkin’s transformed SDEJ, which is free of singular terms. Comparative analysis of the two numerical methods is carried out. Theoretical results are illustrated and proved by means of an example.

For a family of stochastic differential equations driven by additive Gaussian noise, we study the asymptotic behaviors of its corresponding Euler-Maruyama scheme by deriving its convergence rate in terms of relative entropy. Our results for the convergence rate in terms of relative entropy complement the conventional ones in the strong and weak sense and induce some other properties of the Euler-Maruyama scheme. For example, the convergence in terms of the total variation distance can be implied by Pinsker's inequality directly. Moreover, when the drift is β(0<β<1)-Hölder continuous in the spatial variable, the convergence rate in terms of the weighted variation distance is also established. Both of these convergence results do not seem to be directly obtained from any other convergence results of the Euler-Maruyama scheme. The main tool this paper relies on is the Girsanov transform.

Strongly asymptotically optimal schemes for the strong approximation of stochastic differential equations with respect to the supremum error

For a stochastic differential equation(SDE) driven by a fractional Brownian motion(fBm) with Hurst parameter H&gt;12, it is known that the existing (naive) Euler scheme has the rate of convergence n1−2H. Since the limit H→12 of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler scheme is the extension of the classical Euler scheme for Itô SDEs for H=12, the convergence rate of the naive Euler scheme deteriorates for H→12. In this paper we introduce a new (modified Euler) approximation scheme which is closer to the classical Euler scheme for Stratonovich SDEs for H=12, and it has the rate of convergence γ−1n, where γn=n2H−1/2 when H&lt;34, γn=n/logn−−−−√ when H=34 and γn=n if H&gt;34. Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if {Xt,0≤t≤T} is the solution of a SDE driven by a fBm and if {Xnt,0≤t≤T} is its approximation obtained by the new modified Euler scheme, then we prove that γn(Xn−X) converges stably to the solution of a linear SDE driven by a matrix-valued Brownian motion, when H∈(12,34]. In the case H&gt;34, we show the Lp convergence of n(Xnt−Xt), and the limiting process is identified as the solution of a linear SDE driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this scheme. We also apply our approach to the naive Euler scheme.

On Stochastic Processes Defined by Differential Equations with a Small Parameter

This note addresses the following question. Suppose we have a sequence of distributions of sums of independent identically distributed (i.i.d.) random summands converging to a given stable law. What properties of the summands have the most influence on the rate of convergence under consideration? At once it is necessary to note that a general answer is well known at present (a good reference for the accuracy of approximation with stable laws is the recent monograph [3], see also [6]-[11]), and it can be roughly stated that the existence of pseudomoments of higher order ensures better rates of convergence. In turn, the existence of pseudomoments depends on how close are the distribution of the summand and the stable law. But contrary to the case of the Gaussian limit law, where the existence of moments (or more precisely, tail behavior of the distribution) defines the rate of convergence and necessary and sufficient conditions for a given rate of convergence can be formulated, there is no such natural measure of closeness of the distributions in the case of the stable limit law. Taking the particular form of the density of the summand, namely, the Paretian distribution, which can be regarded in some sense as the "main" distribution in the domain of normal attraction (DNA) of a given stable distribution, we show how the rate of convergence depends on small perturbations of this density. Another goal of this note was to give an answer to the question raised in [1]. In that paper it was noted that the convergence to a stable distribution may be extremely slow (this was shown by means of simulation). It turns out that in the situation described in that paper summands belong to the domain of attraction (DA) but do not belong to the DNA. Our example shows that in this case the rate of convergence is only of logarithmic rate. This means that although theoretically both DNA and DA are equally important, in practice, limit theorems in the case of summands in DA but not in DNA are less useful due to the extremely slow rate of convergence and the fact that even asymptotic expansions cannot help very much (see Propositions 2 and 3, where new types of asymptotic expansions are given).

Tamed EM scheme of neutral stochastic differential delay equations

It is shown that the Euler–Maruyama scheme applied to a stochastic differential equation with a discontinuous monotone drift coefficient, such as a Heaviside function, and additive noise converges strongly to a solution of the stochastic differential equation with the same initial condition. The proof uses upper and lower solutions of the stochastic differential equations and the Euler–Maruyama scheme.

The aim of this work is to show the numerical approximation of random periodic solutions for path‐dependent stochastic differential equations with a finite and an infinite delay, in which the drifts are only dissipative on average with respect to the time parameter. The key findings are as follows: (i) by using a combination of synchronous coupling approaches and continuous‐time Euler–Maruyama schemes, we study the existence and uniqueness of numerical random periodic solutions to path‐dependent stochastic differential equations on an infinite time horizon; (ii) by introducing a new technical method, we investigate the strong convergence in the mean‐square sense of a numerical approximation to path‐dependent stochastic differential equations in the case of an infinite time horizon; (iii) to the best of our knowledge, our result is the first one upon numerical approximations of random periodic solutions for path‐dependent stochastic differential equations (SDEs) with infinite delay. We expect that our approximated method can be extended to a more general structure.

The Euler–Maruyama scheme is known to diverge strongly and numerically weakly when applied to nonlinear stochastic differential equations (SDEs) with superlinearly growing and globally one-sided Lipschitz continuous drift coefficients. Classical Monte Carlo simulations do, however, not suffer from this divergence behavior of Euler’s method because this divergence behavior happens on rare events. Indeed, for such nonlinear SDEs the classical Monte Carlo Euler method has been shown to converge by exploiting that the Euler approximations diverge only on events whose probabilities decay to zero very rapidly. Significantly more efficient than the classical Monte Carlo Euler method is the recently introduced multilevel Monte Carlo Euler method. The main observation of this article is that this multilevel Monte Carlo Euler method does—in contrast to classical Monte Carlo methods—not converge in general in the case of such nonlinear SDEs. More precisely, we establish divergence of the multilevel Monte Carlo Euler method for a family of SDEs with superlinearly growing and globally one-sided Lipschitz continuous drift coefficients. In particular, the multilevel Monte Carlo Euler method diverges for these nonlinear SDEs on an event that is not at all rare but has probability one. As a consequence for applications, we recommend not to use the multilevel Monte Carlo Euler method for SDEs with superlinearly growing nonlinearities. Instead we propose to combine the multilevel Monte Carlo method with a slightly modified Euler method. More precisely, we show that the multilevel Monte Carlo method combined with a tamed Euler method converges for nonlinear SDEs with globally one-sided Lipschitz continuous drift coefficients and preserves its strikingly higher order convergence rate from the Lipschitz case.

We propose three methods of parameter estimation based on discrete observation data for stochastic differential equations (SDEs). The first method is designed for linear stochastic differential equations (SDEs). For these equations we deduce distribution of certain operation of the exact solution and assume that the relevant operation of the observed data obey this distribution, from which we estimate the unknown parameters in the drift and diffusion coefficients. In the second method, we suppose that certain operation of the observation data and that of the numerical solution arising from the Euler-Maruyama scheme for the SDEs of Ito sense obey the same distribution, from which the unknown parameters can be estimated. We use the third method for SDEs of Stratonovich sense. For these equations we derive the distribution of relevant operation of the numerical solution produced by the midpoint scheme and let the same operation of the data obey this distribution to get estimation of the unknown parameters. Numerical results show validity of the proposed methods, and illustrate that the estimation error produced by the Euler-Maruyama scheme is about of order O(h0.5) while that by the midpoint scheme is about of order O(h), with h being the time step size of the numerical methods. Furthermore, the numerical results show that our methods are more accurate than the existing EM-MLE estimator.

A new explicit stabilized scheme of weak order one for stiff and ergodic stochastic differential equations (SDEs) is introduced. In the absence of noise, the new method coincides with the classical deterministic stabilized scheme (or Chebyshev method) for diffusion dominated advection-diffusion problems and it inherits its optimal stability domain size that grow quadratically with the number of internal stages of the method. For mean-square stable stiff stochastic problems, the scheme has an optimal extended mean-square stability domain that grows at the same quadratic rate as the deterministic stability domain size in contrast to known existing methods for stiff SDEs [A. Abdulle and T. Li. Commun. Math. Sci., 6(4), 2008, A. Abdulle, G. Vilmart, and K. C. Zygalakis, SIAM J. Sci. Comput., 35(4), 2013]. Combined with postprocessing techniques, the new methods achieve a convergence rate of order two for sampling the invariant measure of a class of ergodic SDEs, achieving a stabilized version of the non-Markovian scheme introduced in [B. Leimkuhler, C. Matthews, and M. V. Tretyakov, Proc. R. Soc. A, 470, 2014].

Growth curve data consist of repeated measurements of a continuous growth process over time in a population of individuals. These data are classically analyzed by nonlinear mixed models. However, the standard growth functions used in this context prescribe monotone increasing growth and can fail to model unexpected changes in growth rates. We propose to model these variations using stochastic differential equations (SDEs) that are deduced from the standard deterministic growth function by adding random variations to the growth dynamics. A Bayesian inference of the parameters of these SDE mixed models is developed. In the case when the SDE has an explicit solution, we describe an easily implemented Gibbs algorithm. When the conditional distribution of the diffusion process has no explicit form, we propose to approximate it using the Euler-Maruyama scheme. Finally, we suggest validating the SDE approach via criteria based on the predictive posterior distribution. We illustrate the efficiency of our method using the Gompertz function to model data on chicken growth, the modeling being improved by the SDE approach.

A numerical method for some stochastic differential equations with multiplicative noise