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Destabilising nonnormal stochastic differential equations

  • *Corresponding author: Carmela Scalone

    *Corresponding author: Carmela Scalone
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  • In this article we address the stability of linear stochastic differential equations. In particular, we focus our attention on non-normality in stochastic differential equations. Following Higham and Mao we study a test problem for non-normal stochastic differential equations, that is stable without noise, and prove a property conjectured by Higham and Mao, that is that an exponentially small (in the dimension) noise term is able to destabilise in a mean-square sense the solution of the SDE.

    Mathematics Subject Classification: 65C30, 60H10, 65Fxx.

    Citation:

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  • Table 1.  Computed values for the spectral abscissa of the perturbed matrix $ A_{\rm HM} + \varepsilon E $

    $ n $ $ \varepsilon $ $ \alpha(A_{\rm HM}+\varepsilon E) $
    $ 5 $ $ 1.5^{-5} $ $ 1.607340 \cdot 10^{-1} $
    $ 10 $ $ 1.5^{-10} $ $ 2.440439 \cdot 10^{-1} $
    $ 20 $ $ 1.5^{-20} $ $ 2.879151 \cdot 10^{-1} $
    $ 40 $ $ 1.5^{-40} $ $ 3.104275 \cdot 10^{-1} $
     | Show Table
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    Table 2.  Values of the spectral abscissa $ \alpha $ of the stability matrix $ S $ for the threshold perturbations $ \gamma $

    $ \gamma $ $ \alpha $
    $ 1.78 $ $ 0.0274 $
    $ 1.79 $ $ 0.0158 $
    $ 1.80 $ $ 0.0043 $
    $ 1.81 $ $ -0.0071 $
    $ 1.82 $ $ -0.0183 $
    $ 1.83 $ $ -0.0294 $
    $ 1.84 $ $ -0.0404 $
     | Show Table
    DownLoad: CSV

    Table 3.  Values of the spectral radius $ \rho $ of the stability matrix of the explicit Euler-Maruyama method, in correspondence of the perturbation $ \varepsilon E $, with $ \varepsilon=\gamma^{-n} $ and $ E=e_n e_1^T $, for different stepsize $ h $

    $ h \setminus \gamma $ $ {1.78} $ $ {1.79} $ $ {1.8} $ $ {1.81} $ $ {1.82} $ $ {1.83} $ $ {1.84} $
    $ 1 $ $ 1.1779 $ $ 1.1648 $ $ 1.1519 $ $ 1.1392 $ $ 1.1267 $ $ 1.1144 $ $ 1.1024 $
    $ 0.5 $ $ 1.0228 $ $ 1.0169 $ $ 1.0111 $ $ 1.0054 $ $ 0.9997 $ $ 0.9942 $ $ 0.9887 $
    $ 0.1 $ $ 1.0030 $ $ 1.0018 $ $ 1.0007 $ $ 0.9996 $ $ 0.9984 $ $ 0.9973 $ $ 0.9962 $
    $ 0.01 $ $ 1.0003 $ $ 1.0002 $ $ 1 $ $ 0.9999 $ $ 0.9998 $ $ 0.9997 $ $ 0.9996 $
     | Show Table
    DownLoad: CSV

    Table 4.  Values of the spectral radius $ \rho $ of the stability matrix of the implicit Euler-Maruyama method, in correspondence of the perturbation $ \varepsilon E $, with $ \varepsilon=\gamma^{-n} $ and $ E=e_n e_1^T $, for different stepsize $ h $

    $ h \setminus \gamma $ $ {1.78} $ $ {1.79} $ $ {1.8} $ $ {1.81} $ $ {1.82} $ $ {1.83} $ $ {1.84} $
    $ 2 $ $ 0.9987 $ $ 0.09777 $ $ 0.9577 $ $ 0.9384 $ $ 0.9199 $ $ 0.9021 $ $ 0.8848 $
    $ 1.5 $ $ 1.0055 $ $ 0.9893 $ $ 0.9736 $ $ 0.9585 $ $ 0.9438 $ $ 0.9297 $ $ 0.9159 $
    $ 1 $ $ 1.0093 $ $ 0.9981 $ $ 0.9873 $ $ 0.9767 $ $ 0.9664 $ $ 0.9564 $ $ 0.9467 $
    $ 0.5 $ $ 1.0084 $ $ 1.0026 $ $ 0.9970 $ $ 0.9915 $ $ 0.9861 $ $ 0.9809 $ $ 0.9757 $
    $ 0.1 $ $ 1.0025 $ $ 1.0013 $ $ 1.0002 $ $ 0.9991 $ $ 0.9979 $ $ 0.9968 $ $ 0.9958 $
    $ 0.01 $ $ 1.0003 $ $ 1.0002 $ $ 1 $ $ 0.9999 $ $ 0.9998 $ $ 0.9997 $ $ 0.9996 $
     | Show Table
    DownLoad: CSV
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