Article Contents
Article Contents

# Destabilising nonnormal stochastic differential equations

• *Corresponding author: Carmela Scalone
• 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:

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

 $n$ $\varepsilon$ $\alpha(A_{\rm HM}+\varepsilon E)$ $5$ $1.5^{-5}$ $1.607340 \cdot 10^{-1}$ $10$ $1.5^{-10}$ $2.440439 \cdot 10^{-1}$ $20$ $1.5^{-20}$ $2.879151 \cdot 10^{-1}$ $40$ $1.5^{-40}$ $3.104275 \cdot 10^{-1}$

Table 2.  Values of the spectral abscissa $\alpha$ of the stability matrix $S$ for the threshold perturbations $\gamma$

 $\gamma$ $\alpha$ $1.78$ $0.0274$ $1.79$ $0.0158$ $1.80$ $0.0043$ $1.81$ $-0.0071$ $1.82$ $-0.0183$ $1.83$ $-0.0294$ $1.84$ $-0.0404$

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 $h$

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

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 $h$

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