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Mean-square approximations of Lévy noise driven SDEs with super-linearly growing diffusion and jump coefficients

This work was supported by NSF of China (11571373, 11671405, 91630312), NSF of Hunan Province (2016JJ3137), Innovation-Driven Project of CSU (2017CX017), Shenghua Yuying Program of CSU and Hunan Provincial Innovation Foundation For Postgraduate (CX2018B051)

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  • This paper first establishes a fundamental mean-square convergence theorem for general one-step numerical approximations of Lévy noise driven stochastic differential equations with non-globally Lipschitz coefficients. Then two novel explicit schemes are designed and their convergence rates are exactly identified via the fundamental theorem. Different from existing works, we do not impose a globally Lipschitz condition on the jump coefficient but formulate appropriate assumptions to allow for its super-linear growth. However, we require that the Lévy measure is finite. New arguments are developed to handle essential difficulties in the convergence analysis, caused by the super-linear growth of the jump coefficient and the fact that higher moment bounds of the Poisson increments $ \int_t^{t+h} \int_Z \,\bar{N}(\mbox{d}s,\mbox{d}z), t \geq 0, h >0 $ contribute to magnitude not more than $ O(h) $. Numerical results are finally reported to confirm the theoretical findings.

    Mathematics Subject Classification: Primary: 60H10, 60H35; Secondary: 65C50.

    Citation:

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  • Figure 1.  Mean-square convergence rates for (87) (left) and (88) (right)

    Table 1.  CPU time of the tamed and sine methods with different stepsizes

    $h$CPU time (second)
    non-additive caseadditive case
    tamed methodsine methodtamed methodsine method
    $2^{-8}$0.8694400.7444480.7945770.632519
    $2^{-9}$1.2033000.9327201.0312260.776562
    $2^{-10}$1.6251051.1002451.2763870.906966
    $2^{-11}$2.9510171.8870722.3556081.433305
    $2^{-12}$5.7893353.5881974.3251452.473830
     | Show Table
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