## No full-text available

To read the full-text of this research,

you can request a copy directly from the authors.

In this note, we establish under mild smoothness assumptions the pathwise convergence rate of an Euler-type method with projection for delay stochastic differential equations on unbounded domains.

To read the full-text of this research,

you can request a copy directly from the authors.

... If the delay function depends only on time, then it is called time-dependent, see, e.g. [1,6,8,22]. But if, in addition to the time, it depends on the solution process, then it is named state-dependent. ...

... have different signs. Without any loss of generality, we assume that h1 (a, b, c, d) < h 2 (a, b, c, d), then (4.7) is satisfied for all t k ∈ (h 1 (a, b, c, d), h 2 (a, b, c, d)). Hence, if the stepsize is taken in the interval (0, h 2 (a, b, c, d)), then the stability of the scheme will be guaranteed.If A(a, b,c, d) = 0, then B(a, b, c, d) = b 2a 2 and by Theorem 4.5, b 2 < a 2 , and so B(a, b, c, d) < 0. In this case, for all stepsizes, relation (4.7) is satisfied, and so unconditional stability arises. ...

... The rate of strong convergence for Example 5.1 with T = 1. The logarithm of T is denoted by the green line with asterisk which is parallel with the dashed red line by slope1 The rate of strong convergence for Example 5.2 with T = 1. The logarithm of T is denoted by the green line with asterisk which is parallel with the dashed red line by slope 1 2 ...

Abstract Numerical analysis of stochastic delay differential equations has been widely developed but frequently for the cases where the delay term has a simple feature. In this paper, we aim to study a more general case of delay term which has not been much discussed so far. We mean the case where the delay term takes random values. For this purpose, a new continuous split-step scheme is introduced to approximate the solution and then convergence in the mean-square sense is investigated. Moreover, given a test equation, the mean-square asymptotic stability of the scheme is presented. Numerical examples are provided to further illustrate the obtained theoretical results.

... In this section, we present two examples that demonstrate our main results. To perform a numerical simulation of the specific example, we utilize the Euler-Maruyama scheme for system (2.1) in the following manner (see [32]). First, we approximate the infinite delay terms by truncating them to a finite number of delayed values. ...

In this study, we explore the realm of practical stability within stochastic functional differential equations with infinite delay (SFDEID). Specifically, we investigate the notions of pth moment and almost sure stability and develop a fresh set of criteria to effectively assess and quantify these properties. To showcase the practical significance of our findings, we apply our newly established criteria to a compelling numerical example, thereby validating the robustness and applicability of our main result.

... Several researchers such as [2,3,4] used Euler-Maruyama scheme to formulate continuous split-step schemes of SDDE on a continuous interval 0 a t tt for the numerical solutions and encountered some setbacks in the use of interpolation techniques in evaluating the delay term and noise term. [5] adopted Malliavin calculus and a refined Lindeberg principle in solving some examples of SDDE and encountered setbacks in obtaining week approximations of Page 2 uniform error bounds. ...

The main objective of this study is to tackle the setbacks encountered by many researchers in the evaluations of the delay term and noise term of Stochastic Delay Differential Equation (SDDE) in order to obtain its numerical solution. These setbacks affect the preservation of the accuracy and efficiency of the numerical solution of Stochastic Delay Differential Equation (SDDE) which need to be addressed. Three new mathematical expressions are developed for the evaluations of the delay term and noise term different from the interpolation techniques been used by other researchers which causes setbacks in obtaining and preserving the accuracy of the numerical solutions. These mathematical expressions developed for the evaluations of the delay term and the noise term together with the discrete schemes of Hybrid Extended Block Backward Differentiation Formulae Method (HEBBDFM) are incorporated into some examples of Stochastic Delay Differential Equation (SDDE) for its numerical solutions. The discrete schemes of the method (HEBBDFM) are obtained by continuous formulations of multistep collocation method by matrix inversion approach. The basic properties for the convergence and stability analysis of the method were analyzed and proved satisfactory. The analysis of the numerical results with its comparison and graphical presentations proved that the method performs better by producing the Least Minimum Absolute Random Errors (LMAREs) at Lesser Computational Processing Unit Time (LCPUT) than other existing methods in terms of accuracy and efficiency using the developed mathematical expressions. Thus, this study recommends the three new mathematical expressions developed for the evaluations of the delay term and noise term different from the interpolation techniques suitable for obtaining better approximate solutions of SDDE.

... The applications of SDDEs can be seen in applied sciences, economics and engineering. Several authorssuch as Evelyn (2000), Zhang et al. (2009), Bahar (2019), Wang et al. (2011), Kazmerchuk (2005), Kazmerchuk et al. (2004), Akhtari et al. (2015) used Euler-Maruyama scheme to formulate continuous split-step schemes of SDDE on a continuous interval for the numerical solutions and encountered some challenges in the use of interpolation techniques in evaluating their delay terms as studied by Majid et al. (2013). In order to circumvent these challenges, we applied Hybrid Block Extended Adams Moulton Methods (HBEAMM) as a linear multistep collocation method to discretize ASTDDEs on a discrete interval [ ) 0 , a t t . ...

This study examined the implementation of Hybrid Block Extended Adams Moulton Methods (HBEAMM) for the computational solution of Advanced Stochastic Time-Delay Differential Equations (ASTDDEs) through the use of various electronic payment systems such as ATM, POS and internet banking offered by the Nigerian banks. This work attempts to proffer solution to customer challenges in using the ATM of most banks in Nigeria. The challenges are considered in this work as capable of resulting to advanced time-delay and volatility noise. It is therefore modeled as advanced stochastic time-delay differential equation (ASTDDE) and solved using Hybrid Extended Block Adams Moulton Methods (HEBAMMs). The discrete schemes of the proposed method (HBEAMM) were obtained by continuous formulations of multistep collocation method by matrix inversion approach. The necessary and sufficient conditions for convergence and stability of the method were analyzed and proved satisfactory. The discrete schemes obtained were applied in solving some advanced stochastic time-delay differential equations and the results obtained revealed the degree of customers’ satisfaction in the use of e-payment systems. The accuracy of the method is compared with other existing methods and presented graphically to prove its superiority.

... These discrete schemes obtained shall be applied in solving some SDDEs with accurate and efficient formulae in evaluating the delay terms which gives more accurate results, lesser time to compute and can also solve different classifications of DDEs. From, [17] stochastic delay differential equation (SDDE) can be express as ...

This paper presents the application of Block Backward Differentiation Formulae Methods for the numerical solution of Stochastic Delay Differential Equations with two new formulated expressions for evaluation of the delay terms. This was carried-out by formulating a continuous representation of the proposed method through multistep collocation method by matrix inversion technique. After the evaluation and simplification of the continuous representations of each step number, the discrete schemes were obtained. The convergence and stability analysis of the method were investigated. The performances of the method were demonstrated by solving some stochastic delay differential equations to show the accuracy and efficiency advantages over other existing methods. It was observed that the scheme for step number k = 4 performed slightly better and faster in terms of accuracy than the schemes for step number k = 2 and 3 respectively when compared with the exact solutions.

... To reveal the applications of SDDEs in applied sciences, economics and engineering, authors such as [6,19,2,17,9,4,1] applied Euler-Maruyama scheme and interpolation techniques such as Hermite, Nordsieck, Newton divided difference and Neville's interpolation to construct continuous split-step schemes of SDDE on a continuous interval 0 a t tt and evaluation of the delay terms for the numerical solutions of SDDEs but encountered some obstacles as studied by [12]. In order to overcome the advanced random/stochastic time-delays experienced by the policyholders in settling their claims by the insurance companies and the obstacles encountered in the use of interpolation techniques for the evaluation of the delay terms, we applied Block Adams Moulton Methods (BAMM) as a linear multistep collocation method to discretize ASTDDEs on a discrete interval ) 0 , a tt in order to obtain its discrete schemes for its numerical solution. ...

In this paper, Block Adams Moulton Methods (BAMM) is implemented for the approximate solution of Advanced Stochastic Time-Delay Differential Equations (ASTDDEs) to determine the level of policyholders' satisfaction in settling their claims in Nigerian Insurance Industry. With the help of continuous development of multistep collocation method by matrix inversion techniques, the discrete schemes of the proposed method (BAMM) were derived. The analysis of basic properties of the method such as order and error constant, consistency, zero stability, convergence and region of absolute stability were carried-out and proved satisfactory. The numerical solutions of the method were obtained, computed and presented graphically by solving some advanced stochastic time-delay differential equations using the derived discrete schemes which revealed the effect of advanced stochastic time-delay on the policyholders' claims settlement. The accuracy of the method was examined by comparing the results obtained with other existing methods and was found to give better approximation.

... From Akhtari et al. [14], advanced stochastic delay differential equation (ASTDDE) can be express as ...

This paper examined the stochastic time effects on the degree of customers' satisfaction through the use of e-commerce for business transactions. These days, customers' satisfaction in the use of e-commerce face serious challenges due to unforeseen, unpredicted and unexpected circumstances between the time orders/purchases of products are made and the delivery time. To improve customers' satisfaction in the use of e-commerce and drastically reduce these uncertainties which result to advanced stochastic time-delay or random change in network failures during the verification of credit alerts by the sellers when orders are made, road accidents, bad road and long distance, we applied of Block Backward Differentiation Formulae Methods to solve some Advanced Stochastic Time-Delay Differential Equations (ASTDDEs). The convergence and stability analysis of the method were investigated. Based on the findings, it was recommended that companies and suppliers should ensure steady networks, provide alternative means for effective e-commerce between buyers and sellers to improve customer and supplier relationships and orders/purchases should be made from close-by companies/suppliers to avoid advanced stochastic time-delay. The results obtained were presented graphically and compared with other existing methods to prove the computational efficiency and accuracy of our method.

... These discrete schemes obtained shall be applied in solving some SDDEs with accurate and efficient formulae in evaluating the delay terms which gives more accurate results, lesser time to compute and can also solve different classifications of DDEs. From, [17] stochastic delay differential equation (SDDE) can be express as ...

The formulation of extended third derivative block backward differentiation formulae was presented for the solution of first order delay differential equations (DDEs) without the application of interpolation techniques in computing the delay term. The delay term was computed by a valid expression of sequence. By matrix transposition procedure, the discrete schemes of the proposed method were carried-out through its continuous derivations with the help of linear multistep collocation approach. The convergence and stability analysis of the method were satisfied. The performances of the proposed method on numerical experiments of some first order DDEs revealed that the scheme for step number k = 4 performed better and faster in terms of efficiency, accuracy, consistency, convergence, region of absolute stability and Central Processing Unit Time (CPUT) at fixed step size than the schemes for step numbers k = 3 and 2 when compared with their exact solutions and other existing methods. Keywords: First order delay differential equations; extended third derivative backward differentiation formulae; block method Publication details

... In this research work, we are concerned with using Simpson's Mechanism for pace number k = 2, 3 and 4 to obtain the numerical solutions of an autonomous d-dimensional Ito stochastic time-dependent delay differential equation (STDDDE) of the form as studied by [1] Impact Factor (JCC): 6 Most researchers applied Euler-Maruyama scheme to develop a continuous split-step scheme of SDDE on a continuous interval 0 a t t t in order to obtain its numerical solutions with the application of intercalation techniques in appraising the lag argument but in this work, we used Block Simpson's Method as a linear multistep collocation method to discretize STDDDE in (1) on a discrete interval 0 , a t t in order to derive its discrete schemes from the continuous formulations of each step number through matrix inversion method. These discrete schemes obtained were applied in solving some STDDDEs without the application of the intercalation method in evaluating the lag argument containing random values. ...

This paper deals with numerical solution of stochastic time-dependent delay differential equations (STDDDEs) using block Simpson's mechanism for pace number k = 2, 3 and 4 without intercalation approach in calculating the delay term containing random values. The multistep collocation method was used to derive the discrete schemes of the proposed method through matrix inversion technique of each step number. The convergence and region of absolute stability of these discrete schemes were examined. The implementation of these discrete schemes was worked-out in block forms to solve some stochastic time-dependent first order delay differential equations. It was observed that the scheme for step number k = 4 performed better and faster in terms of accuracy than the schemes for step number k = 3 and 2 respectively after the comparisons with their exact solutions and other existing methods.

In this note, we will obtain the first quasi-surely convergence rate of approximation of stochastic differential equations driven by G-Brownian motion. We carry it out by making a relation between Lp and quasi-surely convergences by considering this point that in general Lp convergence does not imply quasi-surely one and neither does quasi-surely convergence. This result will show that the rate of quasi-surely convergence can not exceed that of p-th mean.

The explicit Euler scheme and similar explicit approximation schemes (such as the Milstein scheme) are known to diverge strongly and numerically weakly in the case of one-dimensional stochastic ordinary differential equations with superlinearly growing nonlinearities. It remained an open question whether such a divergence phenomenon also holds in the case of stochastic partial differential equations with superlinearly growing nonlinearities such as stochastic Allen-Cahn equations. In this work we solve this problem by proving that full-discrete exponential Euler and full-discrete linear-implicit Euler approximations diverge strongly and numerically weakly in the case of stochastic Allen-Cahn equations. This article also contains a short literature overview on existing numerical approximation results for stochastic differential equations with superlinearly growing nonlinearities.

The paper studies the almost sure asymptotic sta-bility of a class of scalar nonlinear Itô stochastic delay-differential equation with polynomial nonlinearity in the drift, and determinis-tic and fading diffusion coefficient. We show, under conditions that guarantee the stability of the unperturbed deterministic equation, that the condition σ 2 (t) log t → 0 as t → ∞ is sufficient for the almost sure asymptotic stability of solutions. If σ is decreasing, this rate of decay is also necessary. It is also possible to show that all solutions approach zero at a polynomial rate. If σ decays sufficiently rapidly, we obtain the same upper bound on the rate of decay of the deterministic prob-lem. Under some positivity assumptions, we can show that the result we obtain is optimal. When σ decays more slowly, a weaker almost sure polynomial upper bound on the decay rate of solutions is established.

We study approximation methods for stochastic differential equations and point out a simple relation between their order of convergence in the pth mean and their order of convergence in the pathwise sense: Convergence in the pth mean of order α for all p 1 implies pathwise convergence of order α − ε for arbitrary ε > 0. We apply this result to several one-step and multi-step approximation schemes for stochastic differen-tial equations and stochastic delay differential equations. In addition, we give some numerical examples.

We consider the pathwise numerical approximation of nonlinear parabolic stochastic partial differential equations (SPDEs)
driven by additive white noise under local assumptions on the coefficients only. We avoid the standard global Lipschitz assumption
in the literature on the coefficients by first showing convergence under global Lipschitz coefficients but with a strong error
criteria and then by applying a localization technique for one sample path on a bounded set.

Many stochastic differential equations (SDEs) in the literature have a
superlinearly growing nonlinearity in their drift or diffusion coefficient.
Unfortunately, moments of the computationally efficient Euler-Maruyama
approximation method diverge for these SDEs in finite time. This article
develops a general theory based on rare events for studying integrability
properties such as moment bounds for discrete-time stochastic processes. Using
this approach, we establish moment bounds for fully and partially
drift-implicit Euler methods and for a class of new explicit approximation
methods which require only a few more arithmetical operations than the
Euler-Maruyama method. These moment bounds are then used to prove strong
convergence of the proposed schemes. Finally, we illustrate our results for
several SDEs from finance, physics, biology and chemistry.

Traditional finite-time convergence theory for numerical methods applied to stochastic differential equations (SDEs) requires a global Lipschitz assumption on the drift and diffusion coefficients. In practice, many important SDE models satisfy only a local Lipschitz property and, since Brownian paths can make arbitrarily large excursions, the global Lipschitz-based theory is not directly relevant. In this work we prove strong convergence results under less restrictive conditions. First, we give a convergence result for Euler--Maruyama requiring only that the SDE is locally Lipschitz and that the pth moments of the exact and numerical solution are bounded for some p >2. As an application of this general theory we show that an implicit variant of Euler--Maruyama converges if the diffusion coefficient is globally Lipschitz, but the drift coefficient satisfies only a one-sided Lipschitz condition; this is achieved by showing that the implicit method has bounded moments and may be viewed as an Euler--Maruyama approximation to a perturbed SDE of the same form. Second, we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial.

Measure theory is a classical area of mathematics that continues intensive development and has fruitful connections with most other fields of mathematics as well as important applications in physics. This book gives a systematic presentation of modern measure theory as it has developed over the past century and offers three levels of presentation: a standard university graduate course, an advanced study containing some complements to the basic course (the material of this level corresponds to a variety of special courses), and, finally, more specialized topics partly covered by more than 850 exercises. Bibliographical and historical comments and an extensive bibliography with 2000 works covering more than a century are provided. Volume 1 is devoted to the classical theory of measure and integral. Whereas the first volume presents the ideas that go back mainly to Lebesgue, the second volume is to a large extent the result of the later development up to the recent years. The central subjects of Volume 2 are: transformations of measures, conditional measures, and weak convergence of measures. These topics are closely interwoven and form the heart of modern measure theory. The target readership includes graduate students interested in deeper knowledge of measure theory, instructors of courses in measure and integration theory, and researchers in all fields of mathematics. The book may serve as a source for many advanced courses or as a reference. © Springer-Verlag Berlin Heidelberg 2007. All rights are reserved.

We prove that Euler's approximations for stochastic differential equations on domains of R-d converge almost surely if the drift satisfies the monotonicity condition and the diffusion coefficient is Lipschitz continuous.

This paper establishes the existence-and-uniqueness theorems of the global solution to a class of stochastic delay differential equations only with the local Lipschitz condition but does not consider the linear growth condition and studies stochastic ultimate boundedness and the moment average in time. Here we allow the coefficients of these equations to be polynomial or their growths are controlled by polynomial speed. To illustrate our idea clearly, we consider a scalar equation and a special n-dimensional equation carefully.

This advanced undergraduate and graduate text has now been revised and updated to cover the basic principles and applications of various types of stochastic systems, with much on theory and applications not previously available in book form. The text is also useful as a reference source for pure and applied mathematicians, statisticians and probabilists, engineers in control and communications, and information scientists, physicists and economists.

An existence and uniqueness theorem for a class of stochastic delay
differential equations is presented, and the convergence of Euler
approximations for these equations is proved under general conditions.
Moreover, the rate of almost sure convergence is obtained under local Lipschitz
and also under monotonicity conditions.

We study the approximation of stochastic differential equations on domains. For this, we introduce modified Itô–Taylor schemes,
which preserve approximately the boundary domain of the equation under consideration. Assuming the existence of a unique non-exploding
solution, we show that the modified Itô–Taylor scheme of order γ has pathwise convergence order γ − ε for arbitrary ε >0 as long as the coefficients of the equation are sufficiently differentiable. In particular, no global Lipschitz conditions
for the coefficients and their derivatives are required. This applies for example to the so called square root diffusions.

We prove that Euler''s approximations for stochastic differential equations on domains of
d
converge almost surely if the drift satisfies the monotonicity condition and the diffusion coefficient is Lipschitz continuous.

In this paper a variant of the Euler–Maruyama method is used to define the numerical solutions for stochastic differential delay equations (SDDEs) with variable delay. The key aim is to show that the numerical solutions will converge to the true solutions of the SDDEs under the local Lipschitz condition.

In this paper we stochastically perturb the delay Lotka–Volterra model into the stochastic delay differential equation (SDDE) The main aim is to reveal the effects of environmental noise on the delay Lotka–Volterra model. Our results can essentially be divided into two categories: (i)If the delay Lotka–Volterra model already has some nice properties, e.g., nonexplosion, persistence, and asymptotic stability, then the SDDE will preserve these nice properties provided the noise is sufficiently small.(ii)When the delay Lotka–Volterra model does not have some desired properties, e.g., nonexplosion and boundedness, the noise might make the SDDE achieve these desired properties.

We present an error analysis for the pathwise approximation of a general semilinear
stochastic evolution equation in d dimensions. We discretise in space by a
Galerkin method and in time by a stochastic exponential integrator. We show that
for spatially regular (smooth) noise the number of nodes needed for the noise can
be reduced and that the rate of convergence degrades as the regularity of the noise
reduces (and the noise is rougher)