Lipschitz Coefficients Research Articles

Strong and weak divergence of the backward Euler method for neutral stochastic differential equations with time-dependent delay

. In this article, we consider the backward Euler method for a class of neutral stochastic differential equations with time-dependent delay and reveal conditions of the divergence of the p-th absolute moments of the corresponding approximate solutions when p ∈ ( 0 , ∞ ) . This implies the strong L p -divergence of the method at finite time, for p ∈ [ 1 , + ∞ ) , and shows that its numerically weak convergence fails to hold. This article is motivated by the paper Hutzenthaler, M., Jentzen, A., Kloeden, P. E.: Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proc. R. Soc. A 467 (2011), no. 2130, 1563–1576, where a class of ordinary stochastic differential equations with superlinearly growing coefficients is studied.

A note on reflected BSDEs in infinite horizon with stochastic Lipschitz coefficients

We consider an infinite horizon, obliquely reflected backward stochastic differential equation (RBSDE). The main contribution of the present work is that we generalize previous results on infinite horizon RBSDEs to the setting where the driver has a stochastic Lipschitz coefficient. As an application, we consider robust optimal stopping problems for functional stochastic differential equations (FSDEs) where the driver has linear growth.

Stochastic fractional integrodifferential equations with jumps: application to an averaging principle

In this work we deal with a stochastic fractional integrodifferential equation associated to a Poisson random measure. We first prove existence and uniqueness of solution in the case of Lipschitz coefficients but also in the non Lipschitz case. In the second part, we establish an averaging principle in the sense of Khasminskii approach for a class of this equation with non lipschitz coefficients and weak averaging conditions.

Antithetic multilevel Monte Carlo method for approximations of SDEs with non-globally Lipschitz continuous coefficients

In the field of computational finance, one is commonly interested in the expected value of a financial derivative whose payoff depends on the solution of stochastic differential equations (SDEs). For multi-dimensional SDEs with non-commutative diffusion coefficients in the globally Lipschitz setting, a kind of one-half order truncated Milstein-type scheme without Lévy areas was recently introduced by Giles and Szpruch (2014), which combined with the antithetic multilevel Monte Carlo (MLMC) gives the optimal overall computational cost O(ϵ−2) for the required target accuracy ϵ. Nevertheless, many nonlinear SDEs in applications have non-globally Lipschitz continuous coefficients and the corresponding theoretical guarantees for antithetic MLMC are absent in the literature. In the present work, we aim to fill the gap and analyze antithetic MLMC in a non-globally Lipschitz setting. First, we propose a family of modified Milstein-type schemes without Lévy areas to approximate SDEs with non-globally Lipschitz continuous coefficients. The expected one-half order of strong convergence is recovered in a non-globally Lipschitz setting, where even the diffusion coefficients are allowed to grow superlinearly. This then helps us to analyze the relevant variance of the multilevel estimator and the optimal computational cost is finally achieved for the antithetic MLMC. Since getting rid of the Lévy areas destroys the martingale properties of the scheme, the analysis of both the convergence rate and the desired variance becomes highly non-trivial in the non-globally Lipschitz setting. By introducing an auxiliary approximation process, we develop non-standard arguments to overcome the essential difficulties. Numerical experiments are provided to confirm the theoretical findings.

Weak convergence of the split-step backward Euler method for stochastic delay integro-differential equations

In this paper, our primary objective is to discuss the weak convergence of the split-step backward Euler (SSBE) method, renowned for its exceptional stability when used to solve a class of stochastic delay integro-differential equations (SDIDEs) characterized by global Lipschitz coefficients. Traditional weak convergence analysis techniques are not directly applicable to SDIDEs due to the absence of a Kolmogorov equation. To bridge this gap, we employ modified equations to establish an equivalence between the SSBE method used for solving the original SDIDEs and the Euler–Maruyama method applied to modified equations. By demonstrating first-order strong convergence between the solutions of SDIDEs and the modified equations, we establish the first-order weak convergence of the SSBE method for SDIDEs. Finally, we present numerical simulations to validate our theoretical findings.

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.

A linearly implicit finite element full-discretization scheme for SPDEs with nonglobally Lipschitz coefficients

Abstract The present article deals with strong approximations of additive noise driven stochastic partial differential equations (SPDEs) with nonglobally Lipschitz nonlinearity in a bounded domain $ \mathcal{D} \in{\mathbb{R}}^{d}$, $ d \leq 3$. As the first contribution, we establish the well-posedness and regularity of the considered SPDEs in space dimension $d \le 3$, under more relaxed assumptions on the stochastic convolution. This improves relevant results in the literature and covers both the space-time white noise ($d=1$) and the trace-class noises ($\text{Tr} (Q) < \infty $) in multiple dimensions $d=2,3$. Such an improvement is achieved based on a key perturbation estimate for a perturbed PDE, with the aid of which we prove the convergence and uniform regularity of a spectral approximation of the SPDEs and thus get the improved regularity results. The second contribution of the paper is to propose and analyze a spatio-temporal discretization of the SPDEs, by incorporating a standard finite element method in space and a linearly implicit nonlinearity-tamed Euler method for the temporal discretization. The proposed time-stepping scheme is linearly implicit and does not suffer from solving nonlinear algebra equations as the backward Euler scheme does. Based on the improved regularity results, we recover the expected strong convergence rates of the fully discrete scheme and reveal how the convergence rates rely on the regularity of the noise process. In particular, a classical convergence rate of order $O(h^{2} +\tau )$ can be obtained even in high dimension $d=3$, as the driven noise is of trace class and satisfies certain regularity assumptions. The optimal error estimates turn out to be challenging and face some essential difficulties when the tamed time-stepping scheme meets the finite element spatial discretization, particularly in the context of low regularity and multiple dimensions $d \le 3$. Some highly nontrivial arguments are introduced to overcome the difficulties. Finally, numerical examples corroborate the claimed strong orders of convergence.

Tamed Stochastic Runge-Kutta-Chebyshev Methods for Stochastic Differential Equations with Non-Globally Lipschitz Coefficients

Tamed Stochastic Runge-Kutta-Chebyshev Methods for Stochastic Differential Equations with Non-Globally Lipschitz Coefficients

Averaging Principle for BSDEs and one Barrier Reflected BSDEs With Non-Liphschitz Coefficients

In this paper, the averaging principle for BSDEs and one barrier RBSDEs, with non Lipschitz coefficients, is investigated. An averaged BSDEs for the original BSDEs is proposed, as well as the one barrier RBSDEs, and their solutions are quantitatively compared. Under some appropriate assumptions, the solutions to original systems can be approximated by the solutions to averaged stochastic systems in the sense of mean square.

The concentration of zero-noise limits of invariant measures for stochastic dynamical systems

We study the zero-noise concentration phenomena of invariant measures for SDE. The models are with locally Lipschitz coefficients and have more than one ergodic state. By the large deviations method and an expression of invariant measures, we estimate the invariant measures in neighborhoods of stable sets, unstable sets and their complements. Our results illustrate that the zero-noise limiting invariant measures will concentrate on the stable sets, where a cost functional W(Ki) is minimized. This implies that the long time behaviors of ODE and SDE must have different judging criteria. Furthermore, we prove the large deviations principle of invariant measures.

Linear implicit approximations of invariant measures of semi-linear SDEs with non-globally Lipschitz coefficients

This article investigates the weak approximation towards the invariant measure of semi-linear stochastic differential equations (SDEs) under non-globally Lipschitz coefficients. For this purpose, we propose a linear-theta-projected Euler (LTPE) scheme, which also admits an invariant measure, to handle the potential influence of the linear stiffness. Under certain assumptions, both the SDE and the corresponding LTPE method are shown to converge exponentially to the underlying invariant measures, respectively. Moreover, with time-independent regularity estimates for the corresponding Kolmogorov equation, the weak error between the numerical invariant measure and the original one can be guaranteed with convergence of order one. In terms of computational complexity, the proposed ergodicity preserving scheme with the nonlinearity explicitly treated has a significant advantage over the ergodicity preserving implicit Euler method in the literature. Numerical experiments are provided to verify our theoretical findings.

Reflected BSDEs driven by G-Brownian motion with time-varying Lipschitz coefficients

In this paper, we consider the reflected backward stochastic differential equations driven by [Formula: see text]-Brownian motion (reflected [Formula: see text]-BSDEs) with time-varying Lipschitz coefficients. We obtain the uniqueness result by establishing a priori estimates. For the existence, the solution can be approximated by a family of reflected [Formula: see text]-BSDEs with Lipschitz conditions and by penalized [Formula: see text]-BSDEs with time-varying coefficients. The latter approximation is useful to get the comparison theorem. Finally, we study the reflected [Formula: see text]-BSDEs with infinite time horizon.

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

Backward doubly stochastic differential equations driven by fractional Brownian motion with stochastic integral-Lipschitz coefficients

Abstract This paper deals with a class of backward doubly stochastic differential equations driven by fractional Brownian motion with Hurst parameter H greater than 1 2 {\frac{1}{2}} . We essentially establish the existence and uniqueness of a solution in the case of stochastic Lipschitz coefficients and stochastic integral-Lipschitz coefficients. The stochastic integral used throughout the paper is the divergence-type integral.

Averaging Principle for BSDEs driven by fractional Brownian motion with non Lipschitz coefficients

Averaging Principle for BSDEs driven by fractional Brownian motion with non Lipschitz coefficients

Existence and uniqueness of solutions for stochastic differential equations with locally one-sided Lipschitz condition

<p>This paper investigated stochastic differential equations (SDEs) with locally one-sided Lipschitz coefficients. Apart from the local one-sided Lipschitz condition, a more general condition was introduced to replace the monotone condition. Then, in terms of Euler's polygonal line method, the existence and uniqueness of solutions for SDEs was established. In the meanwhile, the $ p $th moment boundedness of solutions was also provided.</p>

Subgradient Extra-Gradient Algorithm for Pseudomonotone Equilibrium Problems and Fixed-Point Problems of Bregman Relatively Nonexpansive Mappings

In this article, we introduce a new subgradient extra-gradient algorithm to find the common element of a set of fixed points of a Bregman relatively nonexpansive mapping and the solution set of an equilibrium problem involving a Pseudomonotone and Bregman–Lipschitz-type bifunction in reflexive Banach spaces. The advantage of the algorithm is that it is run without prior knowledge of the Bregman–Lipschitz coefficients. Finally, two numerical experiments are reported to illustrate the efficiency of the proposed algorithm.

Approximation of Stochastic Volterra Equations with kernels of completely monotone type

In this work, we develop a multifactor approximation for d d -dimensional Stochastic Volterra Equations (SVE) with Lipschitz coefficients and kernels of completely monotone type that may be singular. First, we prove an L 2 L^2 -estimation between two SVEs with different kernels, which provides a quantification of the error between the SVE and any multifactor Stochastic Differential Equation (SDE) approximation. For the particular rough kernel case with Hurst parameter lying in ( 0 , 1 / 2 ) (0,1/2) , we propose various approximating multifactor kernels, state their rates of convergence and illustrate their efficiency for the rough Bergomi model. Second, we study a Euler discretization of the multifactor SDE and establish a convergence result towards the SVE that is uniform with respect to the approximating multifactor kernels. These obtained results lead us to build a new multifactor Euler scheme that reduces significantly the computational cost in an asymptotic way compared to the Euler scheme for SVEs. Finally, we show that our multifactor Euler scheme outperforms the Euler scheme for SVEs for option pricing in the rough Heston model.

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.