Stochastic C-Stability and B-Consistency of Explicit and Implicit Milstein-Type Schemes

Beyn W-J, Isaak E, Kruse R (2017)
Journal of Scientific Computing 70(3): 1042-1077.

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Zeitschriftenaufsatz | Veröffentlicht | Englisch
Abstract / Bemerkung
This paper focuses on two variants of the Milstein scheme, namely the split-step backward Milstein method and a newly proposed projected Milstein scheme, applied to stochastic differential equations which satisfy a global monotonicity condition. In particular, our assumptions include equations with super-linearly growing drift and diffusion coefficient functions and we show that both schemes are mean-square convergent of order 1. Our analysis of the error of convergence with respect to the mean-square norm relies on the notion of stochastic C-stability and B-consistency, which was set up and applied to Euler-type schemes in Beyn et al. (J Sci Comput 67(3):955-987, 2016. doi:10.1007/s10915-015-0114-4). As a direct consequence we also obtain strong order 1 convergence results for the split-step backward Euler method and the projected Euler-Maruyama scheme in the case of stochastic differential equations with additive noise. Our theoretical results are illustrated in a series of numerical experiments.
Erscheinungsjahr
Zeitschriftentitel
Journal of Scientific Computing
Band
70
Ausgabe
3
Seite(n)
1042-1077
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eISSN
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Beyn W-J, Isaak E, Kruse R. Stochastic C-Stability and B-Consistency of Explicit and Implicit Milstein-Type Schemes. Journal of Scientific Computing. 2017;70(3):1042-1077.
Beyn, W. - J., Isaak, E., & Kruse, R. (2017). Stochastic C-Stability and B-Consistency of Explicit and Implicit Milstein-Type Schemes. Journal of Scientific Computing, 70(3), 1042-1077. doi:10.1007/s10915-016-0290-x
Beyn, W. - J., Isaak, E., and Kruse, R. (2017). Stochastic C-Stability and B-Consistency of Explicit and Implicit Milstein-Type Schemes. Journal of Scientific Computing 70, 1042-1077.
Beyn, W.-J., Isaak, E., & Kruse, R., 2017. Stochastic C-Stability and B-Consistency of Explicit and Implicit Milstein-Type Schemes. Journal of Scientific Computing, 70(3), p 1042-1077.
W.-J. Beyn, E. Isaak, and R. Kruse, “Stochastic C-Stability and B-Consistency of Explicit and Implicit Milstein-Type Schemes”, Journal of Scientific Computing, vol. 70, 2017, pp. 1042-1077.
Beyn, W.-J., Isaak, E., Kruse, R.: Stochastic C-Stability and B-Consistency of Explicit and Implicit Milstein-Type Schemes. Journal of Scientific Computing. 70, 1042-1077 (2017).
Beyn, Wolf-Jürgen, Isaak, Elena, and Kruse, Raphael. “Stochastic C-Stability and B-Consistency of Explicit and Implicit Milstein-Type Schemes”. Journal of Scientific Computing 70.3 (2017): 1042-1077.