### Abstract

Original language | English |
---|---|

Pages (from-to) | 370-389 |

Number of pages | 20 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 53 |

Issue number | 1 |

DOIs | |

Publication status | Published - 3 Feb 2015 |

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### Keywords

- almost sure exponential stability
- linear growth condition
- Lipschitz condition
- stochastic theta method
- moment exponential stability

### Cite this

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**Almost sure exponential stability in the numerical simulation of stochastic differential equations.** / Mao, Xuerong.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Almost sure exponential stability in the numerical simulation of stochastic differential equations

AU - Mao, Xuerong

PY - 2015/2/3

Y1 - 2015/2/3

N2 - This paper is mainly concerned with whether the almost sure exponential stability of stochastic differential equations (SDEs) is shared with that of a numerical method. Under the global Lipschtiz condition, we first show that the SDE is pth moment exponentially stable (for p 2 (0; 1)) if and only if the stochastic theta method is pth moment exponentially stable for a sufficiently small step size. We then show that the pth moment exponential stability of the SDE or the stochastic theta method implies the almost sure exponential stability of the SDE or the stochastic theta method, respectively. Hence, our new theory enables us to study the almost sure exponential stability of the SDEs using the stochastic theta method, instead of the method of the Lyapunov functions. That is, we can now carry out careful numerical simulations using the stochastic theta method with a sufficiently small step size t. If the stochastic theta method is pth moment exponentially stable for a sufficiently small p 2 (0; 1), we can then infer that the underlying SDE is almost surely exponentially stable. Our new theory also enables us to show the ability of the stochastic theta method to reproduce the almost sure exponential stability of the SDEs. In particular, we give positive answers to two open problems (P1) and (P2) listed on page 2.

AB - This paper is mainly concerned with whether the almost sure exponential stability of stochastic differential equations (SDEs) is shared with that of a numerical method. Under the global Lipschtiz condition, we first show that the SDE is pth moment exponentially stable (for p 2 (0; 1)) if and only if the stochastic theta method is pth moment exponentially stable for a sufficiently small step size. We then show that the pth moment exponential stability of the SDE or the stochastic theta method implies the almost sure exponential stability of the SDE or the stochastic theta method, respectively. Hence, our new theory enables us to study the almost sure exponential stability of the SDEs using the stochastic theta method, instead of the method of the Lyapunov functions. That is, we can now carry out careful numerical simulations using the stochastic theta method with a sufficiently small step size t. If the stochastic theta method is pth moment exponentially stable for a sufficiently small p 2 (0; 1), we can then infer that the underlying SDE is almost surely exponentially stable. Our new theory also enables us to show the ability of the stochastic theta method to reproduce the almost sure exponential stability of the SDEs. In particular, we give positive answers to two open problems (P1) and (P2) listed on page 2.

KW - almost sure exponential stability

KW - linear growth condition

KW - Lipschitz condition

KW - stochastic theta method

KW - moment exponential stability

UR - http://epubs.siam.org/loi/sjnaam

U2 - 10.1137/140966198

DO - 10.1137/140966198

M3 - Article

VL - 53

SP - 370

EP - 389

JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

SN - 0036-1429

IS - 1

ER -