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Preserving exponential mean-square stability in the simulation of hybrid stochastic differential equations

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Abstract

Positive results are derived concerning the long time dynamics of fixed step size numerical simulations of stochastic differential equation systems with Markovian switching. Euler–Maruyama and implicit theta-method discretisations are shown to capture exponential mean-square stability for all sufficiently small time-steps under appropriate conditions. Moreover, the decay rate, as measured by the second moment Lyapunov exponent, can be reproduced arbitrarily accurately. New finite-time convergence results are derived as an intermediate step in this analysis. We also show, however, that the mean-square A-stability of the theta method does not carry through to this switching scenario. The proof techniques are quite general and hence have the potential to be applied to other numerical methods.

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Authors and Affiliations

  1. Department of Mathematics, University of Strathclyde, Glasgow, G1 1XH, Scotland, UK

    Desmond J. Higham

  2. Department of Statistics and Modelling Science, University of Strathclyde, Glasgow, G1 1XH, Scotland, UK

    Xuerong Mao

  3. Department of Mathematics, University of Wales Swansea, Swansea, SA2 8PP, UK

    Chenggui Yuan

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  1. Desmond J. Higham
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  2. Xuerong Mao
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  3. Chenggui Yuan
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Correspondence to Desmond J. Higham.

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Higham, D.J., Mao, X. & Yuan, C. Preserving exponential mean-square stability in the simulation of hybrid stochastic differential equations. Numer. Math. 108, 295–325 (2007). https://doi.org/10.1007/s00211-007-0113-y

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  • DOI: https://doi.org/10.1007/s00211-007-0113-y

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