Abstract
| Language | English |
|---|---|
| Pages | 1141-1155 |
| Number of pages | 14 |
| Journal | SIAM Journal on Scientific Computing |
| Volume | 31 |
| Issue number | 2 |
| DOIs | |
| Publication status | Published - 17 Dec 2008 |
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Keywords
- asymptotic stability
- backward Euler
- Euler-Maruyama
- jump-diusion
- Poisson process
- stochastic dierential equation
- theta method
- trapezoidal rule
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Asymptotic stability of a jump-diffusion equation and its numerical approximation. / Chalmers, Graeme, D; Higham, Desmond J.
In: SIAM Journal on Scientific Computing, Vol. 31, No. 2, 17.12.2008, p. 1141-1155.Research output: Contribution to journal › Article
TY - JOUR
T1 - Asymptotic stability of a jump-diffusion equation and its numerical approximation
AU - Chalmers, Graeme, D
AU - Higham, Desmond J.
PY - 2008/12/17
Y1 - 2008/12/17
N2 - Asymptotic linear stability is studied for stochastic dierential equations (SDEs) that incorporate Poisson-driven jumps and their numerical simulation using theta-method discretisations. The property is shown to have a simple explicit characterisation for the SDE, whereas for the discretisation a condition is found that is amenable to numerical evaluation. This allows us to evaluate the asymptotic stability behaviour of the methods. One surprising observation is that there exist problem parameters for which an explicit, forward Euler method has better stability properties than its trapezoidal and backward Euler counterparts. Other computational experiments indicate that all theta methods reproduce the correct asymptotic linear stability for suffciently small step sizes. By using a recent result of Appleby, Berkolaiko and Rodkina, we give a rigorous verication that both linear stability and instability are reproduced for small step sizes. This property is known not to hold for general, nonlinear problems.
AB - Asymptotic linear stability is studied for stochastic dierential equations (SDEs) that incorporate Poisson-driven jumps and their numerical simulation using theta-method discretisations. The property is shown to have a simple explicit characterisation for the SDE, whereas for the discretisation a condition is found that is amenable to numerical evaluation. This allows us to evaluate the asymptotic stability behaviour of the methods. One surprising observation is that there exist problem parameters for which an explicit, forward Euler method has better stability properties than its trapezoidal and backward Euler counterparts. Other computational experiments indicate that all theta methods reproduce the correct asymptotic linear stability for suffciently small step sizes. By using a recent result of Appleby, Berkolaiko and Rodkina, we give a rigorous verication that both linear stability and instability are reproduced for small step sizes. This property is known not to hold for general, nonlinear problems.
KW - asymptotic stability
KW - backward Euler
KW - Euler-Maruyama
KW - jump-diusion
KW - Poisson process
KW - stochastic dierential equation
KW - theta method
KW - trapezoidal rule
UR - http://siamdl.aip.org/sisc
UR - http://dx.doi.org/10.1137/070699469
U2 - 10.1137/070699469
DO - 10.1137/070699469
M3 - Article
VL - 31
SP - 1141
EP - 1155
JO - SIAM Journal on Scientific Computing
T2 - SIAM Journal on Scientific Computing
JF - SIAM Journal on Scientific Computing
SN - 1064-8275
IS - 2
ER -