Continuous Drift Coefficients Research Articles

Sharp lower error bounds for strong approximation of SDEs with piecewise Lipschitz continuous drift coefficient

We study pathwise approximation of strong solutions of scalar stochastic differential equations (SDEs) at a single time in the presence of discontinuities of the drift coefficient. Recently, it has been shown by Müller-Gronbach and Yaroslavtseva (2022) that for all p∈[1,∞) a transformed Milstein-type scheme reaches an Lp-error rate of at least 3/4 when the drift coefficient is a piecewise Lipschitz-continuous function with a piecewise Lipschitz-continuous derivative and the diffusion coefficient is constant. It has been proven by Müller-Gronbach and Yaroslavtseva (2023) that this rate 3/4 is optimal if one additionally assumes that the drift coefficient is bounded, increasing and has a point of discontinuity. While boundedness and monotonicity of the drift coefficient are crucial for the proof of the matching lower bound from Müller-Gronbach and Yaroslavtseva (2023), we show that both conditions can be dropped. For the proof we apply a transformation technique which was so far only used to obtain upper bounds.

Approximation of the invariant distribution for a class of ergodic SDEs with one-sided Lipschitz continuous drift coefficient using an explicit tamed Euler scheme

We study the behavior in a large time regime of an explicit tamed Euler-Maruyama scheme applied to a class of ergodic Ito stochastic differential equations with one-sided Lipschitz continuous drift coefficient and bounded globally Lipschitz diffusion coefficient. Our first main contribution is to prove moments for the numerical scheme, which, on the one hand, are uniform with respect to the time-step size, and which, on the other hand, may not be uniform but have at most polynomial growth with respect to time. Our second main contribution is to apply this result to obtain weak error estimates to quantify the error to approximate averages with respect to the invariant distribution of the continuous-time process, as a function of the time-step size and of the time horizon. The explicit tamed Euler scheme is shown to be computationally effective for the approximation of the invariant distribution: even if the moment bounds and error estimates are not proved to be uniform with respect to time, the obtained polynomial growth results in a marginal increase in the upper bound of the computational cost. To the best of our knowledge, this is the first result in the literature concerning the approximation of the invariant distribution for stochastic differential equations with non-globally Lipschitz coefficients using an explicit tamed Euler-Maruyama scheme.

Carathéodory approximate solutions for a class of stochastic differential equations involving the local time at point zero with one-sided Lipschitz continuous drift coefficients

Abstract In this paper, we study the Carathéodory approximate solution for a class of stochastic differential equations involving the local time at point zero. Based on the Carathéodory approximation procedure, we prove that stochastic differential equations involving the local time at point zero have a unique solution, and we show that the Carathéodory approximate solution converges to the solution of stochastic differential equations involving the local time at point zero with one-sided Lipschitz drift coefficient.

Comparison Theorems for Multi-Dimensional General Mean-Field BDSDES

In this paper we study multi-dimensional mean-field backward doubly stochastic differential equations (BDSDEs), that is, BDSDEs whose coefficients depend not only on the solution processes but also on their law. The first part of the paper is devoted to the comparison theorem for multi-dimensional mean-field BDSDEs with Lipschitz conditions. With the help of the comparison result for the Lipschitz case we prove the existence of a solution for multi-dimensional mean-field BDSDEs with an only continuous drift coefficient of linear growth, and we also extend the comparison theorem to such BDSDEs with a continuous coefficient.

Averaging principle for slow-fast stochastic partial differential equations with Hölder continuous coefficients

By using the technique of the Zvonkin's transformation and the classical Khasminskii's time discretization method, we prove the averaging principle for slow-fast stochastic partial differential equations with bounded and Hölder continuous drift coefficients. An example is also provided to explain our result.

Pathwise convergence of a numerical method for stochastic partial differential equations with correlated noise and local Lipschitz condition

In this paper we obtain a general statement concerning pathwise convergence of the full discretization of certain stochastic partial differential equations (SPDEs) with non-globally Lipschitz continuous drift coefficients. We focus on non-diagonal colored noise instead of the usual space–time white noise. By applying a spectral Galerkin method for spatial discretization and a numerical scheme in time introduced by Jentzen, Kloeden and Winkel we obtain the rate of path-wise convergence in the uniform topology. The main assumptions are either uniform bounds on the spectral Galerkin approximation or uniform bounds on the numerical data. Numerical examples illustrate the theoretically predicted convergence rate.

Well-posedness of semilinear stochastic wave equations with Hölder continuous coefficients

We prove that semilinear stochastic abstract wave equations, including wave and plate equations, are well-posed in the strong sense with an α-Hölder continuous drift coefficient, if α∈(2/3,1). The uniqueness may fail for the corresponding deterministic PDE and well-posedness is restored by adding an external random forcing of white noise type. This shows a kind of regularization by noise for the semilinear wave equation. To prove the result we introduce an approach based on backward stochastic differential equations. We also establish regularizing properties of the transition semigroup associated to the stochastic wave equation by using control theoretic results.

Strong rate of convergence for the Euler–Maruyama approximation of SDEs with Hölder continuous drift coefficient

In this paper, we consider a numerical approximation of the stochastic differential equation (SDE) Xt=x0+∫0tb(s,Xs)ds+Lt,x0∈Rd,t∈[0,T], where the drift coefficient b:[0,T]×Rd→Rd is Hölder continuous in both time and space variables and the noise L=(Lt)0≤t≤T is a d-dimensional Lévy process. We provide the rate of convergence for the Euler–Maruyama approximation when L is a Wiener process or a truncated symmetric α-stable process with α∈(1,2). Our technique is based on the regularity of the solution to the associated Kolmogorov equation.

Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations

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.

Multiple Solutions to Stochastic Differential Delay Equations and a Related Comparison Theorem

In this article, we discuss the existence of multiple solutions to a one-dimensional stochastic differential delay equation with continuous drift coefficients and derive a related comparison theorem.

The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients

For stochastic differential equations (SDEs) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient, the classical explicit Euler scheme fails to converge strongly to the exact solution. Recently, an explicit strongly convergent numerical scheme, called the tamed Euler method, has been proposed in [8] for such SDEs. Motivated by their work, we here introduce a tamed version of the Milstein scheme for SDEs with commutative noise. The proposed method is also explicit and easily implementable, but achieves higher strong convergence order than the tamed Euler method does. In recovering the strong convergence order one of the new method, new difficulties arise and kind of a bootstrap argument is developed to overcome them. Finally, an illustrative example confirms the computational efficiency of the tamed Milstein method compared to the tamed Euler method.

Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients

On the one hand, the explicit Euler scheme fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient. On the other hand, the implicit Euler scheme is known to converge strongly to the exact solution of such an SDE. Implementations of the implicit Euler scheme, however, require additional computational effort. In this article we therefore propose an explicit and easily implementable numerical method for such an SDE and show that this method converges strongly with the standard order one-half to the exact solution of the SDE. Simulations reveal that this explicit strongly convergent numerical scheme is considerably faster than the implicit Euler scheme.

B-convergence of split-step one-leg theta methods for stochastic differential equations

For stochastic differential equations (SDEs) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient, the explicit schemes fail to converge strongly to the exact solution (see, Hutzenthaler, Jentzen and Kloeden in Proc. R. Soc. A, rspa.2010.0348v1–rspa.2010.0348, 2010). In this article a class of implicit methods, called split-step one-leg theta methods (SSOLTM), are introduced and are shown to be mean-square convergent for such SDEs if the method parameter satisfies $\frac{1}{2}\leq\theta \leq1$ . This result gives an extension of B-convergence from the theta method for deterministic ordinary differential equations (ODEs) to SSOLTM for SDEs. Furthermore, the optimal rate of convergence can be recovered if the drift coefficient behaves like a polynomial. Finally, numerical experiments are included to support our assertions.