Lipschitz Continuous Coefficients Research Articles

Strong convergence of the tamed Euler-Maruyama method for stochastic singular initial value problems with non-globally Lipschitz continuous coefficients

In our previous works [1] and [2], we delved into numerical methods for solving stochastic singular initial value problems (SSIVPs) that involve coefficients satisfying the global Lipschitz condition. The paper addresses the limitations of our previous work by introducing an explicit method, called the tamed Euler-Maruyama method, for numerically solving SSIVPs with non-globally Lipschitz continuous coefficients, which is both easy-to-implement and exceptionally efficient. We prove the existence and uniqueness theorem and the boundedness of the moments of the solution to SSIVPs under the non-globally Lipschitz condition. Moreover, we provide a sharp analysis of the strong convergence of the proposed method, along with the uniform boundedness of numerical solutions. We also apply our results to the stochastic singular Ginzburg-Landau system and provide numerical simulations to illustrate our theoretical findings.

On the complexity of strong approximation of stochastic differential equations with a non-Lipschitz drift coefficient

We survey recent developments in the field of complexity of pathwise approximation in p-th mean of the solution of a stochastic differential equation at the final time based on finitely many evaluations of the driving Brownian motion. First, we briefly review the case of equations with globally Lipschitz continuous coefficients, for which an error rate of at least 1/2 in terms of the number of evaluations of the driving Brownian motion is always guaranteed by using the equidistant Euler-Maruyama scheme. Then we illustrate that giving up the global Lipschitz continuity of the coefficients may lead to a non-polynomial decay of the error for the Euler-Maruyama scheme or even to an arbitrary slow decay of the smallest possible error that can be achieved on the basis of finitely many evaluations of the driving Brownian motion. Finally, we turn to recent positive results for equations with a drift coefficient that is not globally Lipschitz continuous. Here we focus on scalar equations with a Lipschitz continuous diffusion coefficient and a drift coefficient that satisfies piecewise smoothness assumptions or has fractional Sobolev regularity and we present corresponding complexity results.

An exponential split-step double balanced $$\vartheta $$ Milstein scheme for SODEs with locally Lipschitz continuous coefficients

An exponential split-step double balanced $$\vartheta $$ Milstein scheme for SODEs with locally Lipschitz continuous coefficients

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.

Pathwise methods for the integration of a stochastic SVIR model

We propose an approach for the precise numerical integration of a stochastic SVIR model defined by a stochastic differential equation (SDE) with non‐globally Lipschitz continuous coefficients and multiplicative noise. This equation, based on a compartmental epidemic model, describes a continuous vaccination strategy with environmental noise effects. By means of an appropriate invertible continuous transformation, we link the solution to the stochastic SVIR model to the solution of an auxiliary random differential equation (RDE) that has an Ornstein–Uhlenbeck process as the only input parameter of the system. In this way, based on this explicit conjugacy between both equations, new pathwise numerical schemes are constructed for the SVIR model. In particular, we propose an exponential method that outperforms other integrators in the literature and is able to approximate, with high stability, meaningful probabilistic features of the continuous system, including its stationary distribution and ergodicity. A simulation study is presented to illustrate the practical performance of the introduced methods, and a comparative analysis with other integrators commonly used for the simulation of epidemiological models is performed.

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.

Exponential convergence of hp-time-stepping in space-time discretizations of parabolic PDES

For linear parabolic initial-boundary value problems with self-adjoint, time-homogeneous elliptic spatial operator in divergence form with Lipschitz-continuous coefficients, and for incompatible, time-analytic forcing term in polygonal/polyhedral domains D, we prove time-analyticity of solutions. Temporal analyticity is quantified in terms of weighted, analytic function classes, for data with finite, low spatial regularity and without boundary compatibility. Leveraging this result, we prove exponential convergence of a conforming, semi-discrete hp-time-stepping approach. We combine this semi-discretization in time with first-order, so-called “h-version’’ Lagrangian Finite Elements with corner-refinements in space into a tensor-product, conforming discretization of a space-time formulation. We prove that, under appropriate corner- and corner-edge mesh-refinement of D, error vs. number of degrees of freedom in space-time behaves essentially (up to logarithmic terms), to what standard FEM provide for one elliptic boundary value problem solve in D. We focus on two-dimensional spatial domains and comment on the one- and the three-dimensional case.

Higher regularity for minimizers of very degenerate integral functionals

In this article, we consider minimizers of integral functionals of the typeF(u)=∫Ωa(x)p(|Du|−1)+pdx with a bounded domain Ω⊂Rn(n≥2), a growth exponent p≥2 and Lipschitz continuous coefficients a:Ω→R. We consider the vectorial setting, i.e. u:Ω→RN with N≥1. We will prove that H(Du) is continuous for any continuous function H:RNn→R vanishing on {ξ∈RNn:|ξ|≤1}. This extends a recent result from [3] to the case of integrands that explicitly depend on the x-variable.

Stability of the analytic solution and the partially truncated Euler–Maruyama method for a class of stochastic Volterra integro-differential equations with non-globally Lipschitz continuous coefficients

In this paper, we develop the partially truncated Euler–Maruyama method of a class of stochastic Volterra integro-differential equations with non-globally Lipschitz continuous coefficients. The mean-square exponential stability and the mean-square stability of the analytic solution and the partially truncated Euler–Maruyama method are investigated. Moreover, the theoretical results are illustrated by some numerical examples.

Stability of mixed FEMs for non-selfadjoint indefinite second-order linear elliptic PDEs

For a well-posed non-selfadjoint indefinite second-order linear elliptic PDE with general coefficients {mathbf {A}}, {mathbf {b}},gamma in L^infty and symmetric and uniformly positive definite coefficient matrix {mathbf {A}}, this paper proves that mixed finite element problems are uniquely solvable and the discrete solutions are uniformly bounded, whenever the underlying shape-regular triangulation is sufficiently fine. This applies to the Raviart-Thomas and Brezzi-Douglas-Marini finite element families of any order and in any space dimension and leads to the best-approximation estimate in H({{,mathrm{div},}})times L^2 as well as in in L^2times L^2 up to oscillations. This generalises earlier contributions for piecewise Lipschitz continuous coefficients to L^infty coefficients. The compactness argument of Schatz and Wang for the displacement-oriented problem does not apply immediately to the mixed formulation in H({{,mathrm{div},}})times L^2. But it allows the uniform approximation of some L^2 contributions and can be combined with a recent L^2 best-approximation result from the medius analysis. This technique circumvents any regularity assumption and the application of a Fortin interpolation operator.

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.

On the exponential time-decay for the one-dimensional wave equation with variable coefficients

<p style='text-indent:20px;'>We consider the initial-value problem for the one-dimensional, time-dependent wave equation with positive, Lipschitz continuous coefficients, which are constant outside a bounded region. Under the assumption of compact support of the initial data, we prove that the local energy decays exponentially fast in time, and provide the explicit constant to which the solution converges for large times. We give explicit estimates of the rate of this exponential decay by two different techniques. The first one is based on the definition of a modified, weighted local energy, with suitably constructed weights. The second one is based on the integral formulation of the problem and, under a more restrictive assumption on the variation of the coefficients, allows us to obtain improved decay rates.</p>

Convergence and stability of the semi-tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients

A new explicit stochastic scheme of order 1 is proposed for solving commutative stochastic differential equations (SDEs) with non-globally Lipschitz continuous coefficients. The proposed method is a semi-tamed version of Milstein scheme to solve SDEs with the drift coefficient consisting of non-Lipschitz continuous term and globally Lipschitz continuous term. It is easily implementable and achieves higher strong convergence order. A stability criterion for this method is derived, which shows that the stability condition of the numerical methods and that of the solved equations keep uniform. Compared with some widely used numerical schemes, the proposed method has better performance in inheriting the mean square stability of the exact solution of SDEs. Numerical experiments are given to illustrate the obtained convergence and stability properties.

Split-step balanced [formula omitted]-method for SDEs with non-globally Lipschitz continuous coefficients

In this paper, a split-step balanced θ-method (SSBT) has been presented for solving stochastic differential equations (SDEs) under non-global Lipschitz conditions, where θ∈[0,1] is a parameter of the scheme. The moment boundedness and strong convergence of the numerical solution have been studied, and the convergence rate is 0.5. Moreover, under some conditions it is proved that the SSBT scheme can preserve the exponential mean-square stability of the exact solution when θ∈(1/2,1] for every step size h>0. Numerical examples verify the theoretical findings.

Strong Convergence of a Fully Discrete Scheme for Multiplicative Noise Driving SPDEs with Non-Globally Lipschitz Continuous Coefficients

Strong Convergence of a Fully Discrete Scheme for Multiplicative Noise Driving SPDEs with Non-Globally Lipschitz Continuous Coefficients

A well-posedness result of a characteristic hyperbolic mixed problem

We study the well-posedness of a hyperbolic characteristic initial boundary value problem with Lipschitz continuous coefficients. Assuming more general boundary assumptions than those of maximally dissipativeness, we deal with a Friedrichs symmetrizable system of first order satisfying a minimal structure boundary condition, the so-called Uniform Kreiss–Lopatinskii Condition. We show that a semi-group estimate holds, leading to the proof of the $$L^{2}$$ well-posedness of the initial boundary value problem, provided that the source data of the interior is only $$L^{1}([0,T],L^{2}(\varOmega )),$$ with the aid of the paradifferential calculus.

Interior decay of solutions to elliptic equations with respect to frequencies at the boundary

We prove decay estimates in the interior for solutions to elliptic equations in divergence form with Lipschitz continuous coefficients. The estimates explicitly depend on the distance from the boundary and on suitable notions of frequency of the Dirichlet boundary datum. We show that, as the frequency at the boundary grows, the square of a suitable norm of the solution in a compact subset of the domain decays in an inversely proportional manner with respect to the corresponding frequency. Under Lipschitz regularity assumptions, these estimates are essentially optimal and they have important consequences for the choice of optimal measurements for corresponding inverse boundary value problem.

Strong super convergence of the balanced Euler method for a class of stochastic Volterra integro-differential equations with non-globally Lipschitz continuous coefficients

In this paper, we propose the balanced Euler method of a class of stochastic Volterra integro-differential equations with non-globally Lipschitz continuous coefficients. The moment boundedness is studied and the strong convergence is shown to be 1. Moreover, the theoretical results are illustrated by a numerical example.

Convergence and asymptotic stability of an explicit numerical method for non-autonomous stochastic differential equations

This paper applies an explicit numerical method, i.e. the truncated Euler–Maruyama (EM) scheme, to investigate numerical approximations for nonlinear and non-autonomous stochastic differential equations (SDEs). The explicit method reproduces asymptotic stability of the equations. Under a weak condition and with locally Lipschitz continuous coefficients, strong convergence of the truncated EM scheme for the SDEs is proved. Combing the Hölder continuous condition for temporal variables, we obtain the convergence rate that is related to the Hölder continuity. Moreover, employing the discrete nonnegative semimartingale convergence theorem, we reformulate the explicit EM scheme to approximate asymptotic stability in an infinite horizon. Numerical simulations are provided to illustrate the validity of our theoretical results.

Strongly convergent error analysis for a spatially semidiscrete approximation of stochastic partial differential equations with non-globally Lipschitz continuous coefficients

We study a spatially semidiscrete approximation of nonlinear stochastic partial differential equations (SPDEs) driven by multiplicative noise under weak assumptions on the coefficients avoiding the standard global Lipschitz assumption in the literature. The discretization in space is done by a piecewise linear finite element method. Under some suitable regularity assumption for the exact solution, we present here a rigorous convergence error analysis for the proposed semidiscrete scheme—showing that the numerical scheme admits a convergent rate depending on the regularity of the exact solution.