No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Strong convergence of the stopped Euler–Maruyama method for nonlinear stochastic differential equations
Abstract
In this paper, numerical methods for the nonlinear stochastic differential equations (SDEs) with non-global Lipschitz drift coefficient are discussed. The existing known results have only so far shown that the classical (explicit) Euler-Maruyama (EM) approximate solutions converge to the true solution in probability [22,23]. More recently, the authors in [16] proved that the classical EM method will diverge in L-2 sense for the underlying SDEs in this paper (and those SDEs with superlinearly growing coefficients). These strongly indicate that the classical EM method is not good enough for the highly nonlinear SDEs. However, in this paper, we introduce a modified EM method using stopping time and show successfully that the discrete version of the modified EM approximate solution converges to the true solution in the strong sense (namely in L-2) with a order arbitrarily close to a half.
... In recent years, a few Euler-Maruyama (EM) discretization schemes have been developed for diffusion systems and SDSs including the implicit EM method [9,12], the tamed EM method [11,25,27,28], the tamed Milstein method [30], the stopped EM method [19] and the truncated EM method [18,22], to mention a few. In these EM methods, the approximation can potentially escape the domain of the exact solutions of systems. ...
... Consequently, in order to close the gap, a lot of effort has focused on deriving schemes staying in restricted domains for diffusion systems with non-Lipschitz continuous coefficients [1,2,6,8,24]. Several modified EM methods have been developed such as the implicit schemes [8,24] and the explicit EM schemes [6,19], in the context of mathematical finance, a thorough overview of these can be found in [13]. A now classical trick is to apply a suitable Lamperti transform in order to obtain diffusion systems with constant diffusion coefficient, thereby translating all the non-smoothness to the drift. ...
... One observes that the schemes proposed in [18,19,22,31] are not preserve positivity and therefore are not well defined when directly applied to SDS (1.1), which don't work for the above stochastic logistic models. However, the performance of Scheme (3.2) is very nice for this case, see Figs. 9 and 11. ...
The stochastic logistic model with regime switching is an important model in the ecosystem. While analytic solution to this model is positive, current numerical methods are unable to preserve such boundaries in the approximation. So, proposing appropriate numerical method for solving this model which preserves positivity and dynamical behaviors of the model's solution is very important.
In this paper, we present a positivity preserving truncated Euler-Maruyama scheme for this model, which taking advantages of being explicit and easily implementable. Without additional restriction conditions, strong convergence of the numerical algorithm is studied, and 1/2 order convergence rate is obtained.
In the particular case of this model without switching the first order strong convergence rate is obtained. Furthermore, the approximation of long-time dynamical properties is realized, including the stochastic permanence, extinctive and stability in distribution. Some simulations and examples are provided to confirm the theoretical results and demonstrate the validity of the approach.
... In recent years, a few Euler-Maruyama (EM) discretization schemes have been developed for diffusion systems and SDSs including the implicit EM method [9,12], the tamed EM method [11,25,27,28], the tamed Milstein method [30], the stopped EM method [19] and the truncated EM method [18,22], to mention a few. In these EM methods, the approximation can potentially escape the domain of the exact solutions of systems. ...
... Consequently, in order to close the gap, a lot of effort has focused on deriving schemes staying in restricted domains for diffusion systems with non-Lipschitz continuous coefficients [1,2,6,8,24]. Several modified EM methods have been developed such as the implicit schemes [8,24] and the explicit EM schemes [6,19], in the context of mathematical finance, a thorough overview of these can be found in [13]. A now classical trick is to apply a suitable Lamperti transform in order to obtain diffusion systems with constant diffusion coefficient, thereby translating all the non-smoothness to the drift. ...
... One observes that the schemes proposed in [18,19,22,31] are not preserve positivity and therefore are not well defined when directly applied to SDS (1.1), which don't work for the above stochastic logistic models. However, the performance of Scheme (3.2) is very nice for this case, see Figs. 9 and 11. ...
The stochastic logistic model with regime switching is an important model in the ecosystem. While analytic solution to this model is positive, current numerical methods are unable to preserve such boundaries in the approximation. So, proposing appropriate numerical method for solving this model which preserves positivity and dynamical behaviors of the model's solution is very important. In this paper, we present a positivity preserving truncated Euler-Maruyama scheme for this model, which taking advantages of being explicit and easily implementable. Without additional restriction conditions, strong convergence of the numerical algorithm is studied, and 1/2 order convergence rate is obtained. In the particular case of this model without switching the first order strong convergence rate is obtained. Furthermore, the approximation of long-time dynamical properties is realized, including the stochastic permanence, extinctive and stability in distribution. Some simulations and examples are provided to confirm the theoretical results and demonstrate the validity of the approach.
... To relax the strong convexity condition of h, [4,5,23,24,27,28,29] considered replacing it with certain (local) dissipativity or convexity at infinity condition, and obtain convergence results using techniques developed in [10,11]. To relax the global Lipschitz condition (in θ) and replace it with a local Lipschitz (or Hölder) condition, certain techniques need to be applied to modify the algorithms (see, e.g., [13,19,21,25] and references therein). This is due to the fact that the absolute moments of the aforementioned algorithms could diverge to infinity at finite time point [12]. ...
... Proof of Lemma D.3-(i). For any 0 < λ ≤ λ max ≤ 1 with λ max given in (19), t ∈ (n, n + 1], n ∈ N 0 , we define ∆ λ n,t := Θ λ n + λϕ λ ...
In this paper, we propose two new algorithms, namely aHOLA and aHOLLA, to sample from high-dimensional target distributions with possibly super-linearly growing potentials. We establish non-asymptotic convergence bounds for aHOLA in Wasserstein-1 and Wasserstein-2 distances with rates of convergence equal to 1 + q/2 and 1/2 + q/4, respectively, under a local Hölder condition with exponent q ∈ (0, 1] and a convexity at infinity condition on the potential of the target distribution. Similar results are obtained for aHOLLA under certain global continuity conditions and a dissipativity condition. Crucially, we achieve state-of-the-art rates of convergence of the proposed algorithms in the non-convex setting which are higher than those of the existing algorithms. Numerical experiments are conducted to sample from several distributions and the results support our main findings.
... Therefore, much attention has been devoted to modifying the classical EM method such that the modified method is effective to deal with the super-linear SDEs. For example, Hutzenthaler et al. established the tamed EM method [7], Liu et al. proved the stopping EM method [8], Mao proposed the truncated EM method [9,10], Guo modified the truncated EM method to obtain the partially truncated EM method [11,12] and Beyn discussed the projected EM method [13,14]. ...
... lytical solution X (t) of SDE Eq. 2.1, which has been investigated in many works, such as [6][7][8][9][10][15][16][17] and references therein. Moreover, we can know from these works that the Ait-Sahalia, Cox-Ingersoll-Ross, 3/2 and SIS models all belong to this class. ...
This paper is concerned with the numerical approximations for stochastic differential equations with non-Lipschitz drift or diffusion coefficients. A modified truncated Euler-Maruyama discretization scheme is developed. Moreover, by establishing the criteria on stochastic C-stability and B-consistency of the truncated Euler-Maruyama method, we obtain the strong convergence and the convergence rate of the numerical method. Finally, numerical examples are given to illustrate our theoretical results.
... To overcome this issue, tamed Euler approximations were introduced in [26,23]. Subsequently further tamed Euler approximations were introduced and analyzed; see, e.g., [10,31,40,41,43,44,47] for stochastic ordinary differential equtaions and, e.g., [2,3,21,29,38,39,42] for SEEs. Strong convergence rates for explicit time discrete and explicit space-time discrete numerical methods for SEEs with a non-globally Lipschitz continuous but globally monotone nonlinearity have been derived in, e.g., [2,3,7,38,45]. ...
... Combining (39), (40), (41) (42), and (43) proves for all t ∈ (0, h] that ...
In this article we establish strong convergence rates on the whole probability space for explicit full-discrete approximations of stochastic Burgers equations with multiplicative trace-class noise. The key step in our proof is to establish uniform exponential moment estimates for the numerical approximations.
... Nevertheless, additional computational efforts are required for its implementation since the solution of an algebraic equation has to be found before each iteration. Due to the advantages of explicit schemes (e.g., simple structure and cheap computational cost), a few modified EM methods have been developed for diffusion systems with nonlinear coefficients including the tamed EM method [12,9,13,14], the tamed Milstein method [15], the stopped EM method [16] and the truncated EM method [17]. These modified EM methods have shown their abilities to approximate the solutions of nonlinear diffusion systems. ...
... Combining (3.14)-(3. 16) and (3.18), using (2.5) and (2.7), we obtain that ...
Focusing on hybrid diffusion dynamics involving continuous dynamics as well as discrete events, this article investigates the explicit approximations for nonlinear switching diffusion systems modulated by a Markov chain. Different kinds of easily implementable explicit schemes have been proposed to approximate the dynamical behaviors of switching diffusion systems with local Lipschitz continuous drift and diffusion coefficients in both finite and infinite intervals. Without additional restriction conditions except those which guarantee the exact solutions posses their dynamical properties, the numerical solutions converge strongly to the exact solutions in finite horizon, moreover, realize the approximation of long-time dynamical properties including the moment boundedness, stability and ergodicity. Some simulations and examples are provided to support the theoretical results and demonstrate the validity of the approach.
... q 3 2 in the notation of Theorem 2.4) yields for all q 1 , q 2 ∈ (0, ∞] with 1 ds (39) and yields for all q 1 , q 2 , q 3 ∈ (0, ∞] with q 3 < p and 1 . (40) This completes the proof of Corollary 2.5. ...
... The classical (exponential) Euler approximations diverge in the strong and weak sense for most one-dimensional SDEs with super-linearly growing coefficients (see [31,33]) and also for some SPDEs (see Beccari et al. [3]). It was shown in [29,32] that minor modifications of the Euler method -so called tamed Euler methodsavoid this divergence problem; see also the Euler-type methods, e.g., in [5][6][7]9,10,12,16,20,23,25,35,36,39,40,42,48,49,51,52]. Now, analogously to Hutzenthaler & Jentzen [30], Corollary 3.10 is a powerful tool to establish uniform strong convergence rates (in combination with exponential moment estimates for suitably tamed Euler approximations, e.g., Hutzenthaler et al. [34]). ...
... The most popular numerical scheme is the Euler-Maruyama (EM) method [5], which, as already shown by [6], produces divergent approximations when used to solve SDEs with super-linear growing coefficients such as (1.2). In order to numerically approximate such SDEs with super-linear growing coefficients, various convergent schemes have been proposed and analyzed in the past few years, including implicit methods [7-13] and modified explicit methods [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. In these works, a global monotonicity condition was usually imposed on the drift and diffusion coefficients, which was essentially used in the analysis when one attempts to prove strong convergence. ...
... As implied by the numerical comparison (see Table 1), the proposed explicit scheme in this paper is computationally much more efficient than the LBE method, which needs to implicitly solve a transcendental equation for every time step. We also note that some authors have studied convergence rates of explicit methods for SDEs with the polynomially growing coefficients, see [12][13][14][15][16][17][18]20,22,23,[25][26][27]. In [24], a boundary truncated method combined with the Lamperti-type transformation was introduced for strong approximations of financial SDEs with non-Lipschitz coefficients. ...
A novel explicit time-stepping scheme, called Lamperti smoothing truncation scheme, is devised in this paper to strongly approximate a stochastic SIS epidemic model, whose solution process takes values in a bounded domain and whose coefficients violate the global monotonicity condition. The proposed scheme is based on combining a Lamperti-type transformation with an explicit truncation method. The new scheme results in numerical approximations preserving the domain of the original SDEs and is proved to retain a mean-square convergence rate of order one. Numerical examples are finally reported to confirm our theoretical findings.
... This is not desirable for practical problems such as predicting economic crises and epidemic outbreak, for which modelling results are expected to meet immediate needs. In order to meet the practical needs, improved EM methods have been proposed to solve nonlinear SDEs, such as the tamed EM scheme [5,21], the tamed Milstein scheme [25], the stopped EM scheme [10], and the truncated EM scheme [16]. Moreover, Li et al. [8] proposed a new and easy to implement truncated EM method to obtain numerical approximations of the autonomous SDEs. ...
... This lemma is similar to Lemma 3.11 in [27], and its proof is thus not shown. (9), (10) and (23), respectively. Using Young's inequality, we have ...
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.
... The explicit EM approximate solution to nonlinear SDEs may diverge to infinity in finite time [4] . Therefore some modified EM methods have been proposed to numerically solve nonlinear SDEs, such as the stopped EM method [5] , the tamed EM method [6] and the tamed Milstein method [7] . Especially, Mao [8,9] invented the truncated Euler-Maruyama scheme (TEM for short) with strong convergence theory, which stimulates many researchers interest. ...
In this paper, our main aim is to investigate the strong convergence rate of the truncated Euler-Maruyama approximations for stochastic differential equations with superlinearly growing drift coefficients. When the diffusion coefficient is polynomially growing or linearly growing, the strong convergence rate of arbitrarily close to one half is established at a single time T or over a time interval [0,T], respectively. In both situations, the common one-sided Lipschitz and polynomial growth conditions for the drift coefficients are not required. Two examples are provided to illustrate the theory.
... The modified SEIR model is the SEIR model with demographics, where Λ represents the influence rate, that is, the average number of new susceptible populations per unit of time [10]. The emigration rate is denoted as μ, and γ is the recovered rate. ...
This paper studies the updated estimation method for estimating the transmission rate changes over time. The models for the population dynamics under SEIR epidemic models with stochastic perturbations are analysed the dynamics of the COVID-19 pandemic in Bogotá, Colombia. We performed computational experiments to interpret COVID-19 dynamics using actual data for the proposed models. We estimate the model parameters and updated their estimates for reported infected and recovered data.
... Fortunately, great achievements have been made in the research of explicit numerical methods for super-linear SDEs, for examples, the tamed EM scheme [20,22,38,39], the tamed Milstein scheme [45], the stopped EM scheme [31], the truncated EM scheme [27,28,34] and therein. ...
This work focuses on solving super-linear stochastic differential equations (SDEs) involving different time scales numerically. Taking advantages of being explicit and easily implementable, a multiscale truncated Euler-Maruyama scheme is proposed for slow-fast SDEs with local Lipschitz coefficients. By virtue of the averaging principle, the strong convergence of its numerical solutions to the exact ones in pth moment is obtained. Furthermore, under mild conditions on the coefficients, the corresponding strong error estimate is also provided. Finally, two examples and some numerical simulations are given to verify the theoretical results.
... But Hutzenthaler-Jentzen-Kloeden [10, Theorem 2.1] have showed that, for SDEs whose coefficients are growing superlinearly, EM approximations would inevitably involve the pth moments diverging to infinity. In recent years, several modified EM schemes, including implicit schemes and explicit schemes, have been proposed for approximating the exact solutions of nonlinear SDEs (see, e.g., [9,11,16,18,19,21,22,25,26,28,31,32] and the references therein). Even so, these methods mentioned above requiring a global monotonicity condition cannot be directly used for (1.2). ...
This paper aims to establish a novel explicit method for the stochastic SIS epidemic model, which can preserve the bounded positive domain and asymptotic properties. The proposed new method is based on combining a logarithmic transformation with a truncated Euler-Maruyama method, and it has the first-order rate of convergence for the pth-moment with p>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>0$$\end{document}. Moreover, without additional restriction conditions except those necessary to guarantee the extinction of the exact solution, the approximation of the extinction is achieved for the stochastic SIS model whose coefficients violate the global monotonicity condition. Some numerical experiments are given to illustrate the theoretical results and testify to the efficiency of our algorithm.
... Because explicit EM methods own simple algebraic structure, cheap computational costs and acceptable convergence rate, they have attracted more and more attention. Researchers have developed various versions of these methods for nonlinear SDEs: the stopped EM method [6], the tamed EM method [7][8][9][10][11], and the tamed Milstein method [12][13][14], to mention a few. Recently, Mao [15,16] introduced the truncated EM method and established the strong convergence theory under the local Lipschitz and Khasminskii-type conditions: ...
The main aim of this paper is to investigate the strong convergence order for the truncated Euler-Maruyama (TEM) method to solve stochastic differential delay equations (SDDEs) with multiple time delays and super-linearly growing coefficients. The strong Lp (1 ≤ p < 2) convergence rate of the TEM method under the one-sided polynomial growth condition is first established. Imposing additional conditions on the diffusion coefficient, the p-th moment uniform boundedness of both the exact and approximate solutions is then proved. Next, we show that the strong order of Lq-convergence can be arbitrarily close to 1/2 for 2 ≤ q ≤ p. Several examples and a numerical simulation are provided to illustrate the main results at the end.
... For example, [12] proposed the tamed Euler-Maruyama schemes to approximate SDEs with the global Lipschitz diffusion coefficient and one-sided Lipschitz drift coefficient, whose numerical solutions converge strongly to the exact solution with order 1 2 . Moreover, the tamed Milstein [19] and the stopped Euler-Maruyama method [14] as well as their variants have also been proposed to deal with the strong convergence problem for nonlinear SDEs. Recently, Mao [15,16] proposed a new explicit method, called the truncated Euler-Maruyama method for nonlinear stochastic differential equations, and established the strong convergence and obtained the convergence rate under the local Lipschitz condition and the Khasminskii-type condition. ...
Recently, Mao developed a new explicit method, called the truncated Euler–Maruyama method for nonlinear SDEs, and established the strong convergence theory under the local Lipschitz condition plus the Khasminskii-type condition. The key aim of this paper is to establish the rate of strong convergence of the truncated Euler–Maruyama method for one-dimensional stochastic differential equations involving that the local time at point zero under the drift coefficient satisfies a one-sided Lipschitz condition and plus some additional conditions.
... Unfortunately, the divergence of EM scheme for SDEs with super-linear coefficients was proved in [18]. Whereafter, different kinds of modified EM methods have been established to approximate nonlinear SDEs, such as truncated EM method [24,29], tamed EM method [19,38], stopped EM method [26], multilevel EM method [3], projected EM method [5] and others. Furthermore, the implicit methods have also been studied and developed on account of their better convergence rates in the last decades [2,4,20,40,44]. ...
Over the last few decades, the numerical methods for stochastic differential delay equations (SDDEs) have been investigated and developed by many scholars. Nevertheless, there is still little work to be completed. By virtue of the novel technique, this paper focuses on the mean-square convergence and stability of the backward Euler method (BEM) for SDDEs whose drift and diffusion coefficients can both grow polynomially. The upper mean-square error bounds of BEM are obtained. Then the convergence rate, which is one-half, is revealed without using the moment boundedness of numerical solutions. Furthermore, under fairly general conditions, the novel technique is applied to prove that the BEM can inherit the exponential mean-square stability with a simple proof. At last, two numerical experiments are implemented to illustrate the reliability of the theories.
... As an explicit numerical method, the Euler-Maruyama (EM) scheme is easily implementable and hence is widely used in practice. Since the EM method is not strongly convergent without global Lipchitz continuous coefficients, several modified EM methods have been proposed to address this issue, including the tamed EM method [9], the stopped EM method [14], the truncated EM method [16] and the multilevel EM method [1], etc. ...
This work concerns about the numerical solution to the stochastic epidemic model proposed by Cai et al. [2]. The typical features of the model including the positivity and boundedness of the solution and the presence of the square-root diffusion term make this an interesting and challenging work. By modifying the classical Euler-Maruyama (EM) scheme, we generate a positivity and boundedness preserving numerical scheme, which is proved to have a strong convergence to the true solution over finite time intervals. We also demonstrate that the principle of this method is applicable to a bunch of popular stochastic differential equation (SDE) models, e.g. the mean-reverting square-root process, an important financial model, and the multi-dimensional SDE SIR epidemic model.
... The increment-tamed Euler method in [7,Theorem 3.15] leaves the coefficients unchanged on with N growing subsets of the state space and rescales them outside of these large sets. The stopped Euler method in [15] (generalized in [12]) also leaves the coefficients unchanged on with N growing subsets of the state space, and sets the coefficients to zero outside of these large sets. This latter method is appealing from a traditional point of view since it can be formulated in terms of the classical Euler-Maruyama method. ...
In this article we propose a new explicit Euler-type approximation method for stochastic differential equations (SDEs). In this method, Brownian increments in the recursion of the Euler method are replaced by suitable bounded functions of the Brownian increments. We prove strong convergence rate one-half for a large class of SDEs with polynomial coefficient functions whose local monotonicity constant grows at most like the logarithm of a Lyapunov-type function.
... In the past several years, a number of numerical methods have been developed with comprehensive numerical analysis. [21][22][23][24] For example, Li et al 21 studied the strong convergence properties of partially truncated EM method for the stochastic age-structured Susceptible-Infected-Removed epidemic model with a local Lipschtiz condition. Liu et al 23 constructed a stopped EM method for nonlinear stochastic differential equations (SDEs) to discuss the strong convergence properties. ...
In this paper, we present a periodic averaging (PA) method for impulsive stochastic age‐structured population model (ISASPM) in a polluted environment. By using Young's inequality and Gronwall's lemma and under the stochastic Lipschitz condition, we reveal that both the standard ISASPM and the averaged ISASPM have a unique solution. We also study the mean‐square convergence criteria of the numerical solutions produced by the PA method. Finally, asimulation example is given to demonstrate that the PA method is efficient for our results.
... Milstein's numerical scheme is a first-order method which can be weakly or strongly convergent [18]. Due to the Lipschitzian characteristics of the deterministic and stochastic parts of our model, we have here a strong convergence of the Milstein scheme [27]. The convergence of this numerical method has been validated in for many models having an explicit expression of their exact solution, [15,18,34]. ...
In this paper, we propose a mathematical
model on the oncolytic virotherapy incorporating
virus-specific cytotoxic T lymphocyte (CTL) response,
which contribute to killing infected tumor cells. In
order to improve the understanding of the dynamic
interactions between tumor cells and virus-specific
CTLs, stochastic differential equation models are constructed.
We obtain sufficient conditions for existence,
persistence and extinction of the stochastic system. In
relation to the therapy control, we also analyze the
stochasticity role of equilibrium point stabilities. The
Monte Carlo algorithm is used to estimate the mean
extinction time and the extinction probability of cancer
cells or viruses-specific CTLs. Our simulations highlighted
the switch of the system leaving the attractor
basin of the three species co-existence equilibrium
toward that of cancer cell extinction or that of virusspecific
CTLs depletion. This allowed us to characterize
the spaces of cancer control parameters. Finally, we determine the model solution robustness by analyzing
the sensitivity of the model characteristic parameters.
Our results demonstrate the high dependence of the
virotherapy success or failure on the combination of
stochastic diffusion parameters with the maximum per
capita growth rate of uninfected tumor cells, the transmission
rate, the viral cytotoxicity and the strength of
the CTL response.
... The stochastic numerical methods such as Euler, Milstein, Wagner-Platen have order of convergence 1/2, 1, 3/2, respectively, which is exactly the case with analytic approximations described in this paper, obtained by applying Taylor expansions of zero, first and second degrees, respectively. The authors in [9,10,18], for example, under non-Lipschitz and polynomial conditions for the coefficients of the stochastic differential equation, proved that Euler-Maruyama solution converges strongly at the rate one half. Milstein-type [7,11,30] schemes, under the same conditions, may achieve a strong convergence order greater than that of Euler-type schemes and additional computational effort is required to approximate the iterated Ito integrals for every time step. ...
The subject of this paper is an analytic approximate method for a class of stochastic differential equations with coefficients that do not necessarily satisfy the Lipschitz and linear growth conditions but behave like a polynomials. More precisely, equations from the observed class have unique solutions with bounded moments and their coefficients satisfy polynomial condition. Approximate equations are defined on partitions of a time interval, and their coefficients are Taylor approximations of the coefficients of the initial equation. The rate of Lp convergence increases when degrees in Taylor approximations of coefficients increase. At the end of the paper, an example is provided to support the main theoretical result.
... The following lemma confirms f ∆ and g ∆ nicely reproduce (19). [21]. ...
While the original Ait-Sahalia interest rate model has been found considerable use as a model for describing time series evolution of interest rates, it may not possess adequate specifications to explain responses of interest rates to empirical phenomena such as volatility 'skews' and 'smiles', jump behaviour, market regulatory lapses, economic crisis, financial clashes, political instability, among others collectively. The aim of this paper is to propose a modified version of this model by incorporating additional features to collectively describe these empirical phenomena adequately. Moreover, due to lack of a closed-form solution to the proposed model, we employ several new truncated EM techniques to examine this model and justify the scheme within Monte Carlo framework to compute expected payoffs of some financial quantities such as a bond and a path-dependent barrier option.
... For this scheme they then proved moment bounds and strong convergence for SDEs with one sided globally Lipschitz drift coefficient µ. There follows a number of different taming type schemes such as Sabanis [2013Sabanis [ , 2016, Liu and Mao [2013], İzgi and Çetin [2018], Zong et al. [2014]. We mention in particular Wang and Gan Wang and Gan [2013] who devised a tamed version of Milstein for SDEs with commutative noise that we consider later Section 5.3 and the generalization considered in the work of Kumar and Sabanis [2019]. ...
We introduce a tamed exponential time integrator which exploits linear terms in both the drift and diffusion for Stochastic Differential Equations (SDEs) with a one sided globally Lipschitz drift term. Strong convergence of the proposed scheme is proved, exploiting the boundedness of the geometric Brownian motion (GBM) and we establish order 1 convergence for linear diffusion terms. In our implementation we illustrate the efficiency of the proposed scheme compared to existing fixed step methods and utilize it in an adaptive time stepping scheme. Furthermore we extend the method to nonlinear diffusion terms and show it remains competitive. The efficiency of these GBM based approaches are illustrated by considering some well-known SDE models.
... We now observe through simulations the smallest values of R SIS 0,E and R SEIR 0,E for which the asymptotic stability holds. We now apply the Euler-Maruyama method for simulating the SIS and SEIR models with random perturbations [21]. The approximation equations of the models are given by ...
In this paper, we discuss the basic reproduction number of stochastic epidemic models with random perturbations. We define the basic reproduction number in epidemic models by using the integral of a function or survival function. We study the systems of stochastic differential equations for SIR, SIS, and SEIR models and their stability analysis. Some results on deterministic epidemic models are also obtained. We give the numerical conditions for which the disease-free equilibrium point is asymptotically stable.
... erefore, many implicit methods have been proposed to approximate the solutions of the stochastic differential equations with nonlinear growing coefficients [11][12][13]. In addition, considering that the amount of calculations of the explicit schemes is less, some modified EM methods have been used to approximate the nonlinear stochastic differential equations [14][15][16]. In particular, the truncated EM method was initialized by Mao in [17] with both the drift and diffusion coefficients growing superlinearly. ...
In this paper, we study a class of nonlinear and nonautonomous hybrid stochastic differential delay equations with Poisson jumps (HSDDEwPJs). The convergence rate of the truncated theta-EM numerical solutions to HSDDEwPJs is investigated under given conditions. An example is shown to support our theory.
1. Introduction
Stochastic differential equations have been widely used in many fields and have attracted many scholars [1–3]. Sometimes, an emergency may occur in the system, and it is necessary to consider the influence of the emergency. For example, the surprising outbreak of COVID-19 has a huge impact on the world economy, especially on the stock market. Therefore, stochastic differential equations with jumps considering continuous and discontinuous random effects have been investigated to analyze these situations [4–7]. In practical applications, the parameters and forms in the stochastic systems will change when certain emergencies occur. In this case, we could use stochastic differential equations with Markovian switching to describe them [8]. In this paper, we will take the Markovian switching and jumps into consideration; i.e., we shall study hybrid stochastic differential delay equations with Poisson jumps (HSDDEwPJs).
Numerical methods have been extensively studied, due to the fact that many true solutions of plenty of stochastic differential equations could not be obtained. For example, the explicit Euler–Maruyama (EM) schemes are well known for approximating the true solutions [9]. However, when the coefficients grow superlinearly, Hutzenthaler et al. in [10] proved that, for all , the th moment of the EM approximations diverges to infinity. Therefore, many implicit methods have been proposed to approximate the solutions of the stochastic differential equations with nonlinear growing coefficients [11–13]. In addition, considering that the amount of calculations of the explicit schemes is less, some modified EM methods have been used to approximate the nonlinear stochastic differential equations [14–16]. In particular, the truncated EM method was initialized by Mao in [17] with both the drift and diffusion coefficients growing superlinearly. The convergence rate of the truncated EM method was given in [18]. Subsequently, there have been many papers discussing the truncated EM method for stochastic differential equations with superlinear coefficients [19–25]. In addition, there are many papers which consider the stability of the systems [26–30]. The truncated EM scheme for time-changed nonautonomous stochastic differential equations was shown in [31]. In [32], it was extended to the truncated theta-EM scheme on the basis of truncated EM scheme, and the strong convergence rate of the truncated theta-EM scheme for stochastic delay differential equations under local Lipschitz condition was investigated. The truncated theta-EM method will become the EM method when and degenerate to the backward EM method when . Additionally, there are a few results on the numerical solutions for HSDDEwPJs. The convergence of EM approximation solution to the true solution in probability under some weaker conditions was proved in [33]. The EM approximate solutions converge to the true solutions for stochastic differential delay equations with Poisson jumps and Markovian switching under local Lipschitz condition [34]. The convergence of EM method for stochastic differential delay equations with Poisson jumps and Markovian switching in the sense of -norm under one non-Lipschitz condition was discussed in [35]. The strong convergence between the true solutions and the numerical solutions to stochastic differential delay equations with Poisson jumps and Markovian switching was studied when the drift and diffusion coefficients are Taylor approximations [36]. To the best of our knowledge, there are few papers concerning the numerical solutions of the nonlinear and nonautonomous HSDDEwPJs. Thus, in this paper, we will give the strong convergence rate of the truncated theta-EM method for nonlinear and nonautonomous HSDDEwPJs.
This paper is organized as follows. We will introduce some necessary notations in Section 2. The rate of convergence in sense will be discussed in Section 3. Finally, in Section 4, we will give an example to illustrate that our main result could cover a large class of nonlinear and nonautonomous HSDDEwPJs.
2. Mathematical Preliminaries
Throughout this paper, unless otherwise specified, we will use the following notations. If is a vector or matrix, its transpose is denoted by . , let denote its Euclidean norm. If is a matrix, its trace norm is denoted by . and mean that is nonpositive and negative definite, respectively. If are real numbers, then and . Let be the largest integer which does not exceed . Let and . Let be the family of continuous functions from to with the norm . If is a set, let denote its indicator function which means if and if . Let be a generic positive real constant which could be different in different cases.
Let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is increasing and right continuous while contains all -null sets). In addition, let denote the probability expectation with respect to . For , let denote the family of all -measurable and -value random variables such that . Let be an -dimensional Brownian motion defined on the probability space. Let denote a scalar Poisson process with the compensated Poisson process , where the parameter is the jump intensity. Moreover, we assume that and are independent in this paper.
Let () be a right-continuous Markov chain on the probability space taking values in a finite state space with the generator given bywhere and is the transition rate from to with if , while . We suppose that the Markov chain is independent of and . As we know in [37], almost every sample path of is a right-continuous step function with a finite number of simple jumps in any finite subinterval of . Thus, there exists a sequence of stopping times , almost surely such that
Hence, is constant on each interval ,
In this paper, we consider the nonlinear and nonautonomous hybrid stochastic differential delay equations with Poisson jumps of the formwith the initial data
Here, , , . They are all Borel-measurable functions.
To estimate the truncated theta-EM method for (4), we need the following lemma ([8], Theorem 1.44).
Lemma 1. Given , let for . Then, {} is a discrete Markov chain with the one-step transition probability matrix
Then, we impose the standard hypothesis on the initial data.
Assumption 1. There exist constants and such that
Since is independent of , the paths of could be generated before approximating . The discrete Markovian chain could be generated as follows: Compute the one-step transition probability matrix . Let , and generate a random number which is uniformly distributed in . Definewhere we set as usual. Then, we generate a new random number independently which is uniformly distributed in as well. Define
Repeating this procedure, we could obtain a trajectory of . The procedure could be applied independently to get more trajectories. After generating the discrete Markov chain , we can now define the truncated theta-EM approximate solution for HSDDEwPJs (4) with the initial data (5).
In order to define the truncated theta-EM scheme, we first choose a strictly decreasing function : such thatwhere is a function that depends on . For example, we could choosefor some .
For a given step size , we give the definition of the truncated mappingwhere we let when . The truncated functions are defined as
Now we give the definition of the discrete-time truncated theta-EM scheme to approximate the true solution of (4). Assume that there exist two positive integers and such that . Hence, will become sufficiently small when we choose sufficiently large. Define for . Set for and then formfor , where , . To form the continuous-time scheme, define
It is well known that there exist two kinds of the continuous-time step approximations. The first one is that
The other one is that
Then, we could observe that . Namely, they coincide at . For simplicity, we write
3. Convergence Rate
To obtain the rate of convergence for the truncated theta-EM method for (4) in sense, we need to impose the following assumptions on the coefficients.
Assumption 2. For any , there exists a constant such thatfor all , and any with .
Assumption 3. There exists a constant such thatfor all , any , and .
From Assumption 3, we could derive that there exists a constant such thatwhere .
Before presenting the next assumption, we need more notations. Let be the family of continuous functions such that, for each , there exists a positive constant satisfyingfor any with .
Assumption 4. There exist constants , and , such thatfor all , any , and .
Assumption 5. There exist constants , such thatfor all , any , and .
Assumption 6. There exist constants , and such thatfor all , , and any with .
Assumption 7. There exist constants , such thatfor all , any , and .
The boundedness of the -moment of the true solution is shown in the following lemma which could be proved by using the standard method.
Lemma 2. Let Assumptions 2, 3, and 5 hold. Then, for any given initial data (5), there exists a unique solution to (4) on . Moreover,
Furthermore, we could obtain the next two lemmas in an analogous way to the proof of Lemmas 2.2 and 2.3 in [32].
Lemma 3. Let Assumption 2 holds. For any with , we havefor all , , and any .
From Lemma 3, we derive that
In addition, by the monotone operator theory in [38], needs to be satisfied to ensure the existence and uniqueness of for given . We get from (10) that ; thus, we need . Moreover, to get the main result, should be satisfied in the proof of Lemma 6. Denote . Let and in the rest of this paper.
Lemma 4. Let Assumption 5 hold. Then, for any and , we derive thatwhere .
Before the next lemma, define
Lemma 5. Let Assumptions 2 and 3 hold. For any and , we obtain that
Proof. Fix any . By (17), for any and , we get thatwhere .
For any , there always exists an integer such that . By Hölder’s inequality and BDG’s inequality, we obtain thatBy the characteristic function’s argument in [39], for any , we havewhere is a constant independent of . Then, we get from Assumption 3 thatThus,For , the use of Jensen’s inequality yields thatThe proof is completed.
Lemma 6. Let Assumptions 2, 3, and 5 hold; then, we have
Proof. By (17), we get thatwhere .
Using Itô’s formula, we derive thatWith the help of (10), Lemma 4, and Young’s inequality, we could obtainMoreover,By (10), Young’s inequality, and Lemmas 3 and 5, we could show thatIn the same way, with (10) and Lemmas 3 and 5, we haveBy inserting (44) and (45) into (43), it follows thatFrom Assumption 3, one can see thatCombining (41), (42), (46) , and (47) together, we obtainBy the inequality and (29), we getNote that ; hence,Applying Gronwall’s inequality gives the desired result.
Lemma 7. Let Assumptions 2, 3, and 5 hold. For any and , we derive thatHence,
Proof. By Lemmas 5 and 6, we could obtain (51) and (52). Then, the use of a similar technique in Lemma 6 gives the following when :We could get (54) by applying Jensen’s inequality.
By Lemmas 2 and 6 and Chebyshev’s inequality, we could get the following lemma right away.
Lemma 8. Let Assumptions 2, 3, and 5 hold. For any number , define the stopping timeThen, we obtain that
Lemma 9. Let Assumptions 1–7 hold. Let be sufficiently small such that . Then, we getwhere .
Proof. Let for and . For simplicity, we rewrite . Recalling the definition of and , we havefor any . By Itô’s formula, we could show thatLet , soThen, we haveBy Assumptions 1 and 4 and Lemma 7, it follows thatWe derive from Assumption 6 thatIn addition, let denote the integer part of . Thus,where we have utilized the fact that and are conditionally independent of with reference to the -algebra generated by . The application of the Markov property givesFrom Lemma 6, it follows thatInserting (68) into (65), we getSimilar to , we obtainCombining (63), (64), (69), and (70) together givesBy Lemmas 3 and 7, we could show thatwhere the same techniques used in the proofs of and have been applied. Similarly,By Assumptions 3 and 7, we obtain thatwhere the skills of estimating have been used. Inserting (71)–(74) into (61), one can see thatThus,The use of Gronwall’s inequality yields thatThe proof is completed.
Let us now give the rate of convergence in sense.
Theorem 1. Let Assumptions 1–7 hold. For any sufficiently small , we assume there exists a positive constant such thatThen, for any , we have
Proof. Let be arbitrary. By Young’s inequality, we obtainTherefore,By Lemmas 2 and 6, we getFrom Lemma 8, it follows thatWe can choose and such thatWith condition (78), we obtainFrom Lemma 9 and (85), we derive thatThen, combining Lemma 7 and (79) gives (80). We complete the proof.
4. Example
An example is presented to illustrate our theory in this section.
Example 1. Consider the following nonlinear and nonautonomous scalar hybrid stochastic differential delay equation with Poisson jumps:with the initial value which satisfies Assumption 1. is a scalar Brownian motion, and is a scalar Poisson process with intensity . Furthermore, is a Markovian chain defined on the state space with generatorMoreover, for any and , letObviously, Assumptions 2 and 3 hold with . Moreover, when ,Let . In a similar way, we haveThus,Similarly, for , we haveTherefore,Hence, Assumption 4 is satisfied for any . For Assumption 5, by letting , we obtain thatThus, Assumption 5 is satisfied. Then, it is easy to see that Assumptions 6 and 7 hold with .
For any , define . Therefore, when , we have and . Choose ; then, we derive thatBy Theorem 1, we could get thatwhere is defined in Assumption 1. Thus, the convergence rate of the truncated theta-EM method for (88) is . This example shows that our main result could cover a large class of nonlinear and nonautonomous HSDDEwPJs.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant nos. 11901584, 61876192, 61976228, and 62076106) and the Fundamental Research Funds for the Central Universities of South-Central University for Nationalities (Grant nos. CZY20013, CZY20014, CZT20020, CZT20022, KTZ20051, and YZZ19004).
... Thus, many implicit methods have been put forward to approximate the solutions of the equations with nonlinear growing coefficients [2,11,24,27]. In addition, since the explicit schemes require less computation, some modified EM methods have also been established for nonlinear stochastic equations [14,18,25,26]. In particular, the truncated EM method was originally proposed by Mao [21] with drift and diffusion coefficients growing super-linearly. ...
We study a class of super-linear stochastic differential delay equations with Poisson jumps (SDDEwPJs). The convergence and rate of the convergence of the truncated Euler-Maruyama numerical solutions to SDDEwPJs are investigated under the generalized Khasminskii-type condition.
... Sabanis (2013Sabanis ( , 2016 developed tamed EM schemes for SDEs with nonlinear growth coefficients. Moreover, stopped EM schemes (Liu & Mao, 2013), truncated EM schemes (Mao, 2015), multilevel EM schemes (Anderson et al., 2016) and their variants have also been developed to deal with the strong convergence problem for nonlinear SDEs. However, to the best of our knowledge, these modified EM methods still cannot handle the convergence of a large class of SDEs with nonlinear drift and diffusion coefficients, for example, the constant elasticity of volatility model (CEV model) arising in finance for an asset price of the form (Lewis, 2000) dr(t) = β 0 − β 1 r(t) dt + σ r(t) 3/2 dB(t), (1 .3) ...
Solving stochastic differential equations (SDEs) numerically, explicit Euler–Maruyama (EM) schemes are used most frequently under global Lipschitz conditions for both drift and diffusion coefficients. In contrast, without imposing the global Lipschitz conditions, implicit schemes are often used for SDEs but require additional computational effort; along another line, tamed EM schemes and truncated EM schemes have been developed recently. Taking advantages of being explicit and easily implementable, truncated EM schemes are proposed in this paper. Convergence of the numerical algorithms is studied, and pth moment boundedness is obtained. Furthermore, asymptotic properties of the numerical solutions such as the exponential stability in pth moment and stability in distribution are examined. Several examples are given to illustrate our findings.
... Sabanis [28,29] developed tamed EM schemes for nonlinear stochastic differential equations. More detail on the other explicit numerical methods can be found in [1,16,18]. In addition, Mao initialized the truncated EM method in [21] and obtained the convergence rate in [22]. ...
In this paper, we establish a partially truncated Euler–Maruyama scheme for highly nonlinear and nonautonomous neutral stochastic differential delay equations with Markovian switching. We investigate the strong convergence rate and almost sure exponential stability of the numerical solutions under the generalized Khasminskii-type condition.
... Thus, many implicit methods have been put forward to approximate the solutions of the equations with nonlinear growing coefficients [2,10,24,27]. In addition, since the explicit schemes require less computation, some modified EM methods have also been established for nonlinear stochastic equations [14,18,25,26]. In particularly, the truncated EM method was originally proposed by Mao in [22] with drift and diffusion coefficients growing super-linearly. ...
In this paper, we study a class of super-linear stochastic differential delay equations with Poisson jumps (SDDEwPJs). The convergence and rate of the convergence of the truncated Euler-Maruyama numerical solutions to SDDEwPJs are investigated under the generalized Khasminskii-type condition.
... More recently results concerning explicit schemes for equations with drift operator not satisfying the linear growth condition have been obtained for finite dimensional stochastic differential equations (SDEs) driven by Wiener noise in [13], [14], [58], [59], [60], [64], [65], [66], [69], [90], [95], [101], [102], [106], [117], [118], [122], [123], [126], [129], [133], [135] and for SDEs with jump-diffusive noise in [32], [89], [124]. ...
Stochastic evolution equations with compensated Poisson noise are considered in the variational approach with monotone and coercive coefficients. Here the Poisson noise is assumed to be time homogeneous with $\sigma$-finite intensity measure on a metric space. By using finite element methods and Galerkin approximations, some explicit and implicit discretizations for this equation are presented and their convergence is proved. Polynomial growth condition and linear growth condition are assumed on the drift operator, respectively for the implicit and explicit schemes.
... For example, [4] proposed the tamed EM schemes to approximate SDEs with the global Lipschitz diffusion coefficient and one-sided Lipschitz drift coefficient, whose numerical solutions converge strongly to the exact solution with order 1/2. Moreover, the tamed Milstein [5] and the stopped EM method [6] as well as their variants have also been proposed to deal with the strong converge problem for nonlinear SDEs. Recently, Mao [7,8] proposed a new explicit method called the truncated EM method for nonlinear SDEs and established the strong convergence and obtained the convergence rate under the local Lipschitz condition and the Khasminskii-type condition. ...
In stochastic financial and biological models, the diffusion coefficients often involve the term x, or more general |x|r for r∈(0,1), which is non-Lipschitz. In this paper, we study the strong convergence of the truncated Euler–Maruyama (EM) approximation first proposed by Mao (2015) for one-dimensional stochastic differential equations (SDEs) with superlinearly growing drifts and the Hölder continuous diffusion coefficients.
... The classical (exponential) Euler approximations diverge in the strong and weak sense for most one-dimensional SDEs with super-linearly growing coefficients (see [25,27]) and also for some SPDEs (see Beccari et al. [2]). It was shown in [26,24] that minor modifications of the Euler method -so called tamed Euler methods -avoid this divergence problem; see also the Euler-type methods, e.g., in [4,5,6,8,9,12,16,19,21,29,30,33,35,36,39,44,45,47,48]. Now, analogously to Hutzenthaler & Jentzen [23], Corollary 3.11 is a powerful tool to establish uniform strong convergence rates (in combination with exponential moment estimates for suitably tamed Euler approximations, e.g., Hutzenthaler et al. [28]). ...
There are numerous applications of the classical (deterministic) Gronwall inequality. Recently, Michael Scheutzow has discovered a stochastic Gronwall inequality which provides upper bounds for the $p$-th moments, $p\in(0,1)$, of the supremum of nonnegative scalar continuous processes which satisfy a linear integral inequality. In this article we complement this with upper bounds for $p$-th moments, $p\in[2,\infty)$, of the supremum of multi-dimensional It\^o processes which satisfy a suitable one-sided affine-linear growth condition. As example applications, we improve known estimates on strong local Lipschitz continuity of solutions of stochastic differential equations (SDEs) in the starting point, on (exponential) moment estimates for SDEs, on perturbation estimates for SDES, and we improve known results on strong completeness of SDES.
The main objective of this paper is to develop and analyze a numerical method for the strong approximation of solutions to highly nonlinear stiff It\^{o} stochastic differential equations (SDEs). In this method, we first divide the coefficients of the SDEs into linear and nonlinear parts. Then, in two fully explicit recursive phases, we apply the appropriate truncation transformation only to the nonlinear parts of the coefficients of the SDEs at each step. Theoretical aspects are introduced to establish the $L^q$-convergence theory of the new method in the finite time interval $[0,T]$. The stability and boundedness properties of the solutions generated by the new method are also investigated. An important contribution is that the region of MS-stability of the new method includes the region of MS-stability of the partially truncated method proposed by Yang et al. (2022). Moreover, we show that the proposed method preserves the exponential MS-stability and the asymptotic boundedness of the SDEs. Finally, numerical experiments are performed to compare the efficiency of this method with an existing explicit method developed for nonlinear SDEs.
MSC Classification: 65C30 , 65L07 , 65L04
To obtain strong convergence rates of numerical schemes, an overwhelming majority of existing works impose a global monotonicity condition on coefficients of SDEs. On the contrary , a majority of SDEs from applications do not have globally monotone coefficients. As a recent breakthrough, the authors of [Hutzenthaler, Jentzen, Ann. Probab., 2020] originally presented a perturbation theory for stochastic differential equations (SDEs), which is crucial to recovering strong convergence rates of numerical schemes in a non-globally monotone setting. However, only a convergence rate of order 1/2 was obtained there for time-stepping schemes such as a stopped increment-tamed Euler-Maruyama (SITEM) method. As an open problem, a natural question was raised by the aforementioned work as to whether higher convergence rate than 1/2 can be obtained when higher order schemes are used. The present work attempts to solve the tough problem. To this end, we develop some new perturbation estimates that are able to reveal the order-one strong convergence of numerical methods. As the first application of the newly developed estimates, we identify the expected order-one pathwise uniformly strong convergence of the SITEM method for additive noise driven SDEs and multiplicative noise driven second order SDEs with non-globally monotone coefficients. As the other application, we propose and analyze a positivity preserving explicit Milstein-type method for Lotka-Volterra competition model driven by multi-dimensional noise, with a pathwise uniformly strong convergence rate of order one recovered under mild assumptions. These obtained results are completely new and significantly improve the existing theory. Numerical experiments are also provided to confirm the theoretical findings.
In this paper, we construct a two-stage truncated Runge–Kutta (TSRK2) method for highly nonlinear stochastic differential equations (SDEs) with non-global Lipschitz coefficients. TSRK2 is an explicit method and includes some free parameters that can extend the accuracy of the results and stability regions. We show that this method can achieve a strong convergence rate arbitrarily close to half under local Lipschitz and Khasiminskii conditions. We study the mean square stability properties (MS-stability) of the method based on a scalar linear test equation with multiplicative noise, and the advantages of our results are highlighted by comparing them with those of the truncated Euler–Maruyama method. We also analyze the asymptotic stability properties of the method. We show that the proposed method can preserve the asymptotic stability of the original system under mild conditions. Finally, we report some numerical experiments to illustrate the effectiveness of the proposed method. As a result, we show that the new method has good properties not only in terms of practical errors but also in terms of stability.
The dynamics of viral infection within-plant hosts are of critical importance for characterizing the prevalence and impact of plant diseases. However, few mathematical modeling efforts have been made to characterize viral dynamics within plants. In this study, the dynamics of the vector-borne plant epidemic (VBPE) model are modeled with two nonlinear mathematical models, the deterministic vector-borne plant epidemic (DVBPE) model and the stochastic vector-borne plant epidemic (SVBPE) model to portray and forecast the virus dynamical behavior. The VBPE model segregated the population into three classes: susceptible plants (S), infectious plants (I), and infectious vectors (Y). For classes S, I, and Y, the approximate solution is established by creating a sufficient number of scenarios by varying the ratio of infection, infected vector biting rate, infection rates of plant model, host’s capacity of infection, disease-induced death rate of infectious hosts, insect vectors’ natural mortality rate, the stochastic term for susceptible plants, the stochastic term for infected plants and stochastic term for infected vectors The Adams method is utilized to determine the approximate solution for the DVBPE model, while the Euler-Maruyama and Kloeden-Platen-Schurz methods are used to evaluate the SVBPE model. Finally, a comparison between the DVBPE and SVBPE models is presented.
Influenced by Mao (J Comput Appl Math 296:362–375, 2016), we present a truncated split-step forward Euler–Maruyama (TSSFEM) method for stochastic differential equations with non-global Lipschitz coefficients. We study the strong convergence of the new method under local Lipschitz and Khasiminskii conditions. We analyze the mean square stability and asymptotic stability properties of the new method. And finally, we test the advantages of our new findings using various examples. The newly proposed method achieves a strong convergence rate arbitrarily close to half under some additional conditions. We investigate the mean square stability properties of the method based on a scalar linear test equation with multiplicative noise, and the advantages of our results are highlighted by comparing with some well-known methods. We show that the proposed method can preserve the asymptotic stability of the original system under mild conditions. The illustrative examples could successfully demonstrate the improved stability and accuracy of the TSSFEM method.
Classical approximation results for stochastic differential equations analyze the Lp-distance between the exact solution and its Euler–Maruyama approximations. In this article we measure the error with temporal-spatial Hölder-norms. Our motivation for this are multigrid approximations of the exact solution viewed as a function of the starting point. We establish the classical strong convergence rate 0.5 with respect to temporal-spatial Hölder-norms if the coefficient functions have bounded derivatives of first and second order.
Mechanical equipment is always influenced by environmental factors or random excitations, such as random collision, rock fall, sea wave and wind force. It is significant to analyze the statistical information of the system performance for the operation security and kinematic accuracy. Considering time-varying stiffness and nonlinearity, a stochastic iteration method is proposed for dynamic responses in the gear drive subjected to deterministic and stochastic excitations. Also, the convergence analysis is investigated. Considering the parametric correlation, the improved iterative schemes are analyzed for the mathematical expectation and standard deviation of any multi-stage gear drive with the backlash and mesh stiffness, respectively. Also, covariance and correlation coefficient are investigated to represent the dynamic relationship between each gear pair. The crude Monte Carlo simulation is performed to provide benchmark results, and the data of simulation have good coincidence with the iteration results. It is shown that auxiliary hole of gear drive can reduce the fluctuation of stochastic response in operation. The reasonable modification gear can decrease the dispersion of system response at the transient and stable stage.
In this paper, a stochastic age-structured HIV/AIDS model with nonlinear incidence rates is proposed. It is of great importance to develop efficient numerical approximation methods to solve this HIV/AIDS model since most stochastic partial differential equations (SPDEs) cannot be solved analytically. From the perspective of biological significance, the exact solution of the HIV/AIDS model must be nonnegative and bounded. Then a modified explicit Euler–Maruyama (EM) scheme is constructed based on a projection operator. The EM scheme could preserves the nonnegativity of the numerical solutions and also make the numerical solutions not outside the domain of the exact solutions. The convergence results between the numerical solutions and the exact solutions are analyzed, and some numerical examples are given to verify our theoretical results.
The well-known stochastic SIS model characterized by highly nonlinear in epidemiology has a unique positive solution taking values in a bounded domain with a series of dynamical behaviors. However, the approximation methods to maintain the positivity and long-time behaviors for the stochastic SIS model, while very important, are also lacking. In this paper, based on a logarithmic transformation, we propose a novel explicit numerical method for a stochastic SIS epidemic model whose coefficients violate the global monotonicity condition, which can preserve the positivity of the original stochastic SIS model. And we show the strong convergence of the numerical method and derive that the rate of convergence is of order one. Moreover, the extinction of the exact solution of stochastic SIS model is reproduced. Some numerical experiments are given to illustrate the theoretical results and testify the efficiency of our algorithm.
Recently, Yang et al. (2020) established the strong convergence of the truncated Euler–Maruyama (EM) approximation, that was first proposed by Mao (2015), for one-dimensional stochastic differential equations with superlinearly growing drift and the Hölder continuous diffusion coefficients. However, there are some restrictions on the truncation functions and these restrictions sometimes might force the step size to be so small that the truncated EM method would be inapplicable. The key aim of this paper is to construct several new techniques of the partially truncated EM method to establish the optimal convergence rate in theory without these restrictions. The other aim is to study the stability of the partially truncated EM method. Finally, some simulations and examples are provided to support the theoretical results and demonstrate the validity of the approach.
We introduce a tamed exponential time integrator which exploits linear terms in both the drift and diffusion for Stochastic Differential Equations (SDEs) with a one sided globally Lipschitz drift term. Strong convergence of the proposed scheme is proved, exploiting the boundedness of the geometric Brownian motion (GBM) and we establish order 1 convergence for linear diffusion terms. In our implementation we illustrate the efficiency of the proposed scheme compared to existing fixed step methods and utilize it in an adaptive time stepping scheme. Furthermore we extend the method to nonlinear diffusion terms and show it remains competitive. The efficiency of these GBM based approaches is illustrated by considering some well-known SDE models.
In this article, we first construct a positivity-preserving numerical method for the stochastic Susceptible-Infected-Susceptible (SIS) epidemic model by combining the logarithmic transformation and Euler–Maruyama (EM) method. Then, we show that the algorithm not only preserves the domain of the original SDE, but also has the first-order rate of the pth-moment convergence over a finite time interval for all p>0. Finally, some numerical experiments are provided to illustrate the theoretical results.
The well-known stochastic Lotka–Volterra model for interacting multi-species in ecology has some typical features: highly nonlinear, positive solution and multi-dimensional. The known numerical methods including the tamed/truncated Euler–Maruyama (EM) applied to it do not preserve its positivity. The aim of this paper is to modify the truncated EM to establish a new positive preserving truncated EM (PPTEM). To simplify the proof as well as to make our theory more understandable, we will first develop a nonnegative preserving truncated EM (NPTEM) and then establish the PPTEM. Of course, we should point out that the NPTEM has its own right as many SDE models in applications have their nonnegative solutions.
Traditional explicit schemes such as the Euler-Maruyama, Milstein and stochastic Runge-Kutta methods, in general, result in strong and weak divergence when solving stochastic differential equations (SDEs) with super-linearly growing coefficients. Motivated by this, various modified versions of explicit Euler and Milstein methods were constructed and analyzed in the literature. In the present paper, we aim to introduce a family of explicit tamed stochastic Runge-Kutta (TSRK) methods for commutative SDEs with super-linearly growing drift and diffusion coefficients. Strong convergence rates of order 1.0 are successfully identified for the proposed methods under certain non-globally Lipschitz conditions. Compared to the Milstein-type methods involved with derivatives of coefficients, the newly proposed derivative-free TSRK methods can be computationally more efficient. Numerical experiments are reported to confirm the expected strong convergence rate of the TSRK methods.
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.
In this paper, a simple class of stochastic partitioned Runge–Kutta (SPRK) methods is proposed for solving stochastic Hamiltonian systems with additive noise. Firstly, the order conditions and symplectic condictions are analysised by using colored rooted tree theory. Then a family of mean-square order 1.5 diagonally implicit stochastic symplectic partitioned Runge–Kutta (SSPRK) methods is presented. Moreover, several explicit SSPRK methods are constructed for systems with a separable Hamiltonian H0=V(p)+U(t,q). Furthermore, these methods are proved to converge with mean-square order 2.0 to the solution when they are applied to second-order stochastic Hamiltonian systems with a separable Hamiltonian and additive noise. Finally, several numerical examples are performed to demonstrate efficiency of those SSPRK methods.
A strategy for controlling the stepsize in the numerical integration of stochastic differential equations (SDEs) is presented. It is based on estimating the pth mean of local errors. The strategy leads to stepsize sequences that are identical for all computed paths. For the family of Euler schemes for SDEs with small noise, we derive computable estimates for the dominating term of the pth mean of local errors and show that the strategy becomes efficient for reasonable stepsizes. Numerical experience is reported for test examples including scalar SDEs and a stochastic circuit model.
A two-stage, Runge–Kutta algorithm for vector Ito (and, by transform, also Stratonovich) stochastic differential equations with multiplicative noise has been developed. The method is second order accurate; but, for vanishing drift the algorithm yields a martingale independent of step size. Several examples are included to illustrate our method. A discussion of errors shows that sample size can be as important as truncation.
Stochastic dierential,equations provide a useful means of intro- ducing stochasticity into models across a broad range of systems from chem- istry to population biology. However, in many applications the resulting equa- tions have so far proved intractable to direct analytical solution. Numerical ap- proximations, such as the Euler scheme, are therefore a vital tool in exploring model behaviour. Unfortunately, current results concerning the convergence of such schemes impose conditions on the drift and diusion,coecients,of the stochastic dierential,equation, namely the linear growth and global Lipschitz conditions, which are often not met by systems of interest. In this paper we relax these conditions and prove that numerical solutions based on the Euler scheme will converge to the true solution of a broad class of stochastic dif- ferential equations. The results are illustrated by application to a stochastic Lotka-Volterra model and a model of chemical auto-catalysis, neither of which satisfy either the linear growth nor the global Lipschitz conditions. AMS: 60H10 60H05 60H35 65C30 91B70
Ion channels are membrane proteins that open and close at random and play a vital role in the electrical dynamics of excitable cells. The stochastic nature of the conformational changes these proteins undergo can be significant, however current stochastic modeling methodologies limit the ability to study such systems. Discrete-state Markov chain models are seen as the “gold standard,” but are computationally intensive, restricting investigation of stochastic effects to the single-cell level. Continuous stochastic methods that use stochastic differential equations (SDEs) to model the system are more efficient but can lead to simulations that have no biological meaning. In this paper we show that modeling the behavior of ion channel dynamics by a reflected SDE ensures biologically realistic simulations, and we argue that this model follows from the continuous approximation of the discrete-state Markov chain model. Open channel and action potential statistics from simulations of ion channel dynamics using the reflected SDE are compared with those of a discrete-state Markov chain method. Results show that the reflected SDE simulations are in good agreement with the discrete-state approach. The reflected SDE model therefore provides a computationally efficient method to simulate ion channel dynamics while preserving the distributional properties of the discrete-state Markov chain model and also ensuring biologically realistic solutions. This framework could easily be extended to other biochemical reaction networks.
For a stopped diffusion process in a multidimensional time-dependent domain D, we propose and analyse a new procedure consisting in simulating the process with an Euler scheme with step size Δ and stopping it at discrete times in a modified domain, whose boundary has been appropriately shifted. The shift is locally in the direction of the inward normal n(t,x) at any point (t,x) on the parabolic boundary of D, and its amplitude is equal to where σ stands for the diffusion coefficient of the process. The procedure is thus extremely easy to use. In addition, we prove that the rate of convergence w.r.t. Δ for the associated weak error is higher than without shifting, generalizing the previous results by Broadie et al. (1997) [6] obtained for the one-dimensional Brownian motion. For this, we establish in full generality the asymptotics of the triplet exit time/exit position/overshoot for the discretely stopped Euler scheme. Here, the overshoot means the distance to the boundary of the process when it exits the domain. Numerical experiments support these results.
We propose a new concept which allows us to apply any numerical method of weak approximation to a very broad class of stochastic differential equations (SDEs) with nonglobally Lipschitz coefficients. Following this concept, we discard the approximate trajectories which leave a sufficiently large sphere. We prove that accuracy of any method of weak order p is estimated by $\varepsilon +O(h^{p}),$ where $\varepsilon $ can be made arbitrarily small with increasing radius of the sphere. The results obtained are supported by numerical experiments.
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.
In the present interdisciplinary review, we focus on the applications of the symmetry principles to quantum and statistical physics in connection with some other branches of science. The profound and innovative idea of quasiaverages formulated by N. N. Bogoliubov, gives the so-called macro-objectivation of the degeneracy in the domain of quantum statistical mechanics, quantum field theory and quantum physics in general. We discuss the complementary unifying ideas of modern physics, namely: spontaneous symmetry breaking, quantum protectorate and emergence. The interrelation of the concepts of symmetry breaking, quasiaverages and quantum protectorate was analyzed in the context of quantum theory and statistical physics. The chief purposes of this paper were to demonstrate the connection and interrelation of these conceptual advances of the many-body physics and to try to show explicitly that those concepts, though different in details, have certain common features. Several problems in the field of statistical physics of complex materials and systems (e.g., the chirality of molecules) and the foundations of the microscopic theory of magnetism and superconductivity were discussed in relation to these ideas.
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, is proposed in [Hutzenthaler, Jentzen, & Kloeden (2010); Strong
convergence of an explicit numerical method for SDEs with non-globally
Lipschitz continuous coefficients, arXiv:1010.3756v1] 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.
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.
We present a numerical method for the approximation of solutions for the class of stochastic differential equations driven by Brownian motions which induce stochastic variation in fixed directions. This class of equations arises naturally in the study of population processes and chemical reaction kinetics. We show that the method constructs paths that are second order accurate in the weak sense. The method is simpler than many second order methods in that it neither requires the construction of iterated Itˆo integrals nor the evaluation of any derivatives. The method consists of two steps. In the first an explicit Euler step is used to take a fractional step. The resulting fractional point is then combined with the initial point to obtain a higher order, trapezoidal like, approximation. The higher order of accuracy stems from the fact that both the drift and the quadratic variation of the underlying SDE are approximated to second order.
The stochastic Euler scheme is known to converge to the exact solution of a
stochastic differential equation with globally Lipschitz continuous drift and
diffusion coefficient. Recent results extend this convergence to coefficients
which grow at most linearly. For superlinearly growing coefficients finite-time
convergence in the strong mean square sense remained an open question according
to [Higham, Mao & Stuart (2002); Strong convergence of Euler-type methods for
nonlinear stochastic differential equations, SIAM J. Numer. Anal. 40, no. 3,
1041-1063]. In this article we answer this question to the negative and prove
for a large class of stochastic differential equations with non-globally
Lipschitz continuous coefficients that Euler's approximation converges neither
in the strong mean square sense nor in the numerically weak sense to the exact
solution at a finite time point. Even worse, the difference of the exact
solution and of the numerical approximation at a finite time point diverges to
infinity in the strong mean square sense and in the numerically weak sense.
Stochastic differential equations (SDEs) arise from physical systems where the parameters describing the system can only be estimated or are subject to noise. Much work has been done recently on developing higher order Runge-Kutta methods for solving SDEs numerically. Fixed stepsize implementations of numerical methods have limitations when, for example, the SDE being solved is stiff as this forces the stepsize to be very small. This paper presents a completely general variable stepsize implementation of an embedded Runge Kutta pair for solving SDEs numerically; in this implementation, there is no restriction on the value used for the stepsize, and it is demonstrated that the integration remains on the correct Brownian path.
We consider one-dimensional stochastic differential equations
in the particular case of diffusion coefficient functions of the form
|x|α, α ∈ [1/2,1). In that case, we study the rate of convergence of a
symmetrized version of the Euler scheme. This symmetrized version is
easy to simulate on a computer.
We prove its strong convergence and obtain the same rate of
convergence as when the coefficients are Lipschitz.
We analyze numerical methods for the pathwise approximation of a system of stochastic differential equations. As a measure of performance we consider the $q$th mean of the maximum distance between the solution and its approximation on the whole unit interval. We introduce an adaptive discretization that takes into account the local smoothness of every trajectory of the solution. The resulting adaptive Euler approximation performs asymptotically optimal in the class of all numerical methods that are based on a finite number of observations of the driving Brownian motion.
. This paper introduces some implicitness in stochastic terms of numerical methods for solving sti# stochastic di#erential equations and especially a class of fully implicit methods, the balanced methods. Their order of strong convergence is proved. Numerical experiments compare the stability properties of these schemes with explicit ones. Key words. stochastic di#erential equations, implicit numerical methods, sti# equations, simulation experiments AMS subject classifications. 65C20, 60H10 PII. S0036142994273525 1. Introduction. During the last few years several authors have proposed implicit numerical methods for stochastic di#erential equations with respect to strong and weak convergence criterions. We refer here to the papers of Talay ([9], 1982), Klauder and Petersen ([3], 1985), Milstein ([6], 1988), Hernandez and Spigler ([2], 1990), Saito and Mitsui ([8], 1993), Drummond and Mortimer ([1], 1991), Kloeden and Platen ([4], [5], 1992) just to mention a few of them. As in the de...
Several applications of the strong schemes that were derived in the preceding chapters will be indicated in this chapter. These are the direct simulation of trajectories of stochastic dynamical systems, including stochastic flows, the testing of parametric estimators and Markov chain filters. In addition, some results on asymptotically efficient schemes will be presented.
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.
Using concrete examples, we discuss the current and potential use of stochastic ordinary differential equations (SDEs) from
the perspective of applied and computational mathematics. Assuming only a minimal background knowledge in probability and
stochastic processes, we focus on aspects that distinguish SDEs from their deterministic counterparts. To illustrate a multiscale
modelling framework, we explain how SDEs arise naturally as diffusion limits in the type of discrete-valued stochastic models
used in chemical kinetics, population dynamics and, most topically, systems biology. We outline some key issues in existence,
uniqueness and stability that arise when SDEs are used as physical models and point out possible pitfalls. We also discuss
the use of numerical methods to simulate trajectories of an SDE and explain how both weak and strong convergence properties
are relevant for highly efficient multilevel Monte Carlo simulations. We flag up what we believe to be key topics for future
research, focussing especially on non-linear models, parameter estimation, uncertainty quantification, model comparison and
multiscale simulation.
We discuss a semi-implicit time discretization scheme to approximate the solution of a kind of stiff stochastic differential equations. Roughly speaking, by stiffness for an SDE we mean that the drift coefficient of the considered equtaions satisfies the so-called one-side Lipschitz condition and the diffusion coefficients have the bounded derivatives. We prove that the semi-implicit scheme has the convergence rate 0.5 (as for the explicit Euler scheme for nonstiff problems). The proof of this result needs some results on moment estimation of the solution of the original stiff stochastic differential equations, which is also done in this paper.
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.
Stochastic differential equations (SDEs) represent physical phenomena dominated by stochastic processes. As for deterministic ordinary differential equations (ODEs), various numerical schemes are proposed for SDEs. In this paper we study the stability of numerical schemes for scalar SDEs with respect to the mean-square norm, which we call $MS$-stability. We will show some figures of the $MS$-stability domain or regions for some numerical schemes and present numerical results which confirm it. This notion is an extension of absolute stability in numerical methods for ODEs.
1. Probability and Statistics.- 2. Probability and Stochastic Processes.- 3. Ito Stochastic Calculus.- 4. Stochastic Differential Equations.- 5. Stochastic Taylor Expansions.- 6. Modelling with Stochastic Differential Equations.- 7. Applications of Stochastic Differential Equations.- 8. Time Discrete Approximation of Deterministic Differential Equations.- 9. Introduction to Stochastic Time Discrete Approximation.- 10. Strong Taylor Approximations.- 11. Explicit Strong Approximations.- 12. Implicit Strong Approximations.- 13. Selected Applications of Strong Approximations.- 14. Weak Taylor Approximations.- 15. Explicit and Implicit Weak Approximations.- 16. Variance Reduction Methods.- 17. Selected Applications of Weak Approximations.- Solutions of Exercises.- Bibliographical Notes.
1 Mean-square approximation for stochastic differential equations.- 2 Weak approximation for stochastic differential equations.- 3 Numerical methods for SDEs with small noise.- 4 Stochastic Hamiltonian systems and Langevin-type equations.- 5 Simulation of space and space-time bounded diffusions.- 6 Random walks for linear boundary value problems.- 7 Probabilistic approach to numerical solution of the Cauchy problem for nonlinear parabolic equations.- 8 Numerical solution of the nonlinear Dirichlet and Neumann problems based on the probabilistic approach.- 9 Application of stochastic numerics to models with stochastic resonance and to Brownian ratchets.- A Appendix: Practical guidance to implementation of the stochastic numerical methods.- A.1 Mean-square methods.- A.2 Weak methods and the Monte Carlo technique.- A.3 Algorithms for bounded diffusions.- A.4 Random walks for linear boundary value problems.- A.5 Nonlinear PDEs.- A.6 Miscellaneous.- References.
For applications in finance, we study the stochastic differential equationdXs = (2βXs + δs) ds + g(Xs) dBswith β a negative real number, g a continuous function vanishing at zero which satisfies a Hölder condition and δ a measurable and adapted stochastic process such that∫t0 δu du < ∞ a.e. for all t ∈ ℝ+and which may have a random correlation with the process X itself. In this paper, we concentrate on the Euler discretization scheme for such processes and we study the convergence in L1-supnorm and in ℋ1-norm towards the solution of the stochastic differential equation with stochastic drift term. We also check the order of strong convergence. © 1998 John Wiley & Sons, Ltd.
Empirical studies show that the most successful continuous-time models of the short-term rate in capturing the dynamics are those that allow the volatility of interest changes to be highly sensitive to the level of the rate. However, from the mathematics, the high sensitivity to the level implies that the coefficients do not satisfy the linear growth condition, so we can not examine its properties by traditional techniques. This paper overcomes the mathematical difficulties due to the nonlinear growth and examines its analytical properties and the convergence of numerical solutions in probability. The convergence result can be used to justify the method within Monte Carlo simulations that compute the expected payoff of financial products. For illustration, we apply our results compute the value of a bond with interest rate given by the highly sensitive mean-reverting process as well as the value of a single barrier call option with the asset price governed by this process.
A numerical scheme for stochastic differential equations with convex constraints is considered. The solutions to the SDEs are constrained to the domain of convex lower semicontinuous function through a multivalued monotone drift component and a variational inequality. The projection scheme is a time discrete version of the constrained SDE. In the particular case when the constraining function is an indicator of a closed convex domain, the SDE is reflected. Previous convergence results for the projection scheme applied to reflected SDEs are recovered.
We are interested in approximating a multidimensional hypoelliptic diffusion process (Xt)t[greater-or-equal, slanted]0 killed when it leaves a smooth domain D. When a discrete Euler scheme with time step h is used, we prove under a noncharacteristic boundary condition that the weak error is upper bounded by , generalizing the result obtained by Gobet in (Stoch. Proc. Appl. 87 (2000) 167) for the uniformly elliptic case. We also obtain a lower bound with the same rate , thus proving that the order of convergence is exactly 1/2. This provides a theoretical explanation of the well-known bias that we can numerically observe in that kind of procedure.
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, we propose two local error estimates based on drift and diffusion terms of the stochastic differential equations
in order to determine the optimal step-size for the next stage in an adaptive variable step-size algorithm. These local error
estimates are based on the weak approximation solution of stochastic differential equations with one-dimensional and multi-dimensional
Wiener processes. Numerical experiments are presented to illustrate the effectiveness of this approach in the weak approximation
of several standard test problems including SDEs with small noise and scalar and multi-dimensional Wiener processes.
KeywordsStochastic differential equations-Scalar noise-Multi-dimensional Wiener process-Adaptive variable step-size algorithm-Weak approximation-Local error estimate
We consider the Euler discretisation of a scalar linear test equation with positive solutions and show for both strong and weak approximations that the probability of positivity over any finite interval of simulation tends to unity as the step size approaches zero. Although a.s. positivity in an approximation is impossible to achieve, we develop for the strong (Maruyama) approximation an asymptotic estimate of the number of mesh points required for positivity as our tolerance of non-positive trajectories tends to zero, and examine the effectiveness of this estimate in the context of practical numerical simulation. We show how this analysis generalises to equations with a drift coefficient that may display a high level of nonlinearity, but which must be linearly bounded from below (i.e. when acting towards zero), and a linearly bounded diffusion coefficient. Finally, in the linear case we develop a refined asymptotic estimate that is more useful as an a priori guide to the number of mesh points required to produce positive approximations with a given probability.
The classical existence-and-uniqueness theorem of the solution to a stochastic differential delay equation (SDDE) requires the local Lipschitz condition and the linear growth condition (see e.g. [11,12,20]). The numerical solutions under these conditions have also been discussed intensively (see e.g. [4,10,13,16–18,21,22,24]). Recently, Mao and Rassias [14,15] established the generalized Khasminskii-type existence-and-uniqueness theorems for SDDEs, where the linear growth condition is no longer imposed. These generalized Khasminskii-type theorems cover a wide class of highly nonlinear SDDEs but these nonlinear SDDEs do not have explicit solutions, whence numerical solutions are required in practice. However, there is so far little numerical theory on SDDEs under these generalized Khasminskii-type conditions. The key aim of this paper is to close this gap.
In this paper we construct predictor-corrector (PC) methods based on the trivial predictor and stochastic implicit Runge--Kutta (RK) correctors for solving stochastic differential equations. Using the colored rooted tree theory and stochastic B-series, the order condition theorem is derived for constructing stochastic RK methods based on PC implementations. We also present detailed order conditions of the PC methods using stochastic implicit RK correctors with strong global order 1.0 and 1.5. A two-stage implicit RK method with strong global order 1.0 and a four-stage implicit RK method with strong global order 1.5 used as the correctors are constructed in this paper. The mean-square stability properties and numerical results of the PC methods based on these two implicit RK correctors are reported.
- V L Ginzurg
- L D Landau
- J Exptl
Ginzurg V.L., Landau, L.D., J. Exptl. Theoret. Phys. 20(1950) (USSR)