Strong convergence of semi-implicit split-step methods for SDE with locally Lipschitz coefficients

İzgi, Burhaneddin; Çetin, Coşkun

doi:10.1016/j.cnsns.2020.105574

Cited by 3 publications

(4 citation statements)

References 16 publications

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“…The bounds for the error norm and the estimated error norms (with a standard error of roughly 0.001) are given in Table 3 which includes improved approximations for y (4) 2 (•) and y (5) 2 (•), too. We applied repeated (Matlab) simulations of discretized solutions to (4.1) using its pathwise solution (3.9) as well as partially drift-implicit and semiimplicit split-step (SISS, in short) methods which are known to be convergent for such nonlinear SDE ( [12], [28]). We observe that the (estimated) actual error terms are much smaller than the error bounds (both in Table 2 and Table 3).…”

Section: Appendix Some Approximate Moment Functions For Stochastic Ve...mentioning

confidence: 99%

“…Simulation-based methods may be useful for estimating expectations at a fixed time T . One can check [12] for details about about the SISS methods and their comparison with other convergent numerical schemes for SDE with locally Lipschitz coefficients (including tamed, implicit and truncated Euler methods). The advantage of our method in this paper is finding explicit and simple approximate moment functions over an entire interval [0, T ].…”

Section: Appendix Some Approximate Moment Functions For Stochastic Ve...mentioning

confidence: 99%

“…We refer the reader to [22] for failure of such explicit numerical schemes to inherit ergodicity of the actual solution, and to [8] for a proof that E-M procedure does not converge in neither strong nor weak sense for a large class of nonlinear SDE. There are several alternatives to explicit Euler schemes including implicit, semi-implicit, balanced, tamed and truncated numerical procedures with desirable convergence and stability properties ( [9], [10], [11], [12], [19], [21], and [28]). The focus of these papers has been on the "strong convergence" of numerical solutions at a fixed terminal time T .…”

Section: Introductionmentioning

confidence: 99%

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Approximate Moment Functions for Logistic Stochastic Differential Equations

Çetin

Đorđević

2023

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In this paper, we introduce an iterative procedure for approximate moment functions of logistic stochastic differential equations. We first reduce the solutions of the moment functions of such an equation to an infinite system of linear ordinary differential equations. Then, we determine certain upper and lower bounds on the moment functions, and utilize these bounds to solve the corresponding systems approximately via some truncations, iterations and a local improvement step. By focusing on the stochastic Verhulst systems, we compare these moment approximations with numerical solutions via simulation-based methods that include discretizations of the pathwise solutions as well as convergent numerical procedures like partially implicit Euler methods. In particular, we discuss performance of these approximations for certain parameter values.

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Section: Appendix Some Approximate Moment Functions For Stochastic Ve...mentioning

confidence: 99%

Section: Appendix Some Approximate Moment Functions For Stochastic Ve...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Approximate Moment Functions for Logistic Stochastic Differential Equations

Çetin

Đorđević

2023

Preprint

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“…Numerical methods, which can be broadly categorized as explicit and implicit, are important tools for approximating solutions to these equations [1,6]. While implicit methods are an accepted approach for numerically solving highly nonlinear SDEs [7][8][9][10][11][12][13][14], there are some explicit methods that are well suited for solving problems with non-global Lipschitz coefficients. For example, Hutzenthaler et al [15] consider SDEs with one-sided Lipschitz drift coefficient and the linear growth diffusion coefficient and propose the tamed Euler-Maruyama (EM) method.…”

Section: Introductionmentioning

confidence: 99%

A split-step Euler-Maruyama based method with partial truncation coefficients for nonlinear SDEs

Haghighi

2024

Preprint

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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

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