Strong approximations of stochastic differential equations with jumps

Bruti‐Liberati, Nicola; Platen, Eckhard

doi:10.1016/j.cam.2006.03.040

Cited by 67 publications

(39 citation statements)

References 20 publications

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“…Despite their wide interest, however, few analytical solutions have been proposed so far, thus, it is necessary to develop numerical methods and study the properties of these methods. Higham and Kloeden [8,9] studied the convergence of numerical solutions to SDEs with jumps and the stability of implicit method for jump-diffusion systems, Bruti-Liberati and Platen [2,4] developed strong and weak approximations of SDEs with jumps, Chalmers and Higham [7] considered the convergence and stability for the implicit simulation of SDEs with random jump magnitudes. There are extensive literatures on the numerical simulation of stochastic differential delay equations (SDDEs) [5,6,[10][11][12]15,16], and efforts are now being made to bring SDDEs with jumps up to a similar level.…”

Section: Introductionmentioning

confidence: 99%

Mean-square stability of the Euler–Maruyama method for stochastic differential delay equations with jumps

Tan

Wang

2010

International Journal of Computer Mathematics

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This paper deals with the mean-square (MS) stability of the Euler-Maruyama method for stochastic differential delay equations (SDDEs) with jumps. First, the definition of the MS-stability of numerical methods for SDDEs with jumps is established, and then the sufficient condition of the MS-stability of the EulerMaruyama method for SDDEs with jumps is derived, finally a class scalar test equation is simulated and the numerical experiments verify the results obtained from theory.

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Section: Introductionmentioning

confidence: 99%

Mean-square stability of the Euler–Maruyama method for stochastic differential delay equations with jumps

Tan

Wang

2010

International Journal of Computer Mathematics

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“…x → q x is continuous for all (6) Furthermore X t is conservative by theorem 5.2. in [19] using (A1)-(A3). Now the proof will be divided into three parts.…”

Section: Proof Of the Theoremmentioning

confidence: 89%

The Euler Scheme for Feller Processes

Böttcher

Schnurr

2011

Stochastic Analysis and Applications

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We consider the Euler scheme for stochastic differential equations with jumps, whose intensity might be infinite and the jump structure may depend on the position. This general type of SDE is explicitly given for Feller processes and a general convergence condition is presented.In particular, the characteristic functions of the increments of the Euler scheme are calculated in terms of the symbol of the Feller process in a closed form. These increments are increments of Lévy processes and, thus, the Euler scheme can be used for simulation by applying standard techniques from Lévy processes.

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“…Therefore, it is natural to use stochastic differential equations with jumps for modeling these characteristics. The rate of the strong convergence of numerical schemes for stochastic differential equations of this type with Lipschitz coefficients was studied in [6]. However, of special interest is the combination of non-Lipschitz diffusion, which is inherent in most stochastic financial models, and jumps, which are modeled, e.g., by using the Poisson measure.…”

Section: Introductionmentioning

confidence: 99%