Asymptotic stability of a jump-diffusion equation and its numerical approximation

Graeme, D Chalmers, Desmond J. Higham

Research output: Contribution to journalArticle

18 Citations (Scopus)

Abstract

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.
LanguageEnglish
Pages1141-1155
Number of pages14
JournalSIAM Journal on Scientific Computing
Volume31
Issue number2
DOIs
Publication statusPublished - 17 Dec 2008

Fingerprint

Jump Diffusion
Linear Stability
Asymptotic stability
Numerical Approximation
Diffusion equation
Asymptotic Stability
θ-method
Stochastic Equations
Discretization
Euler's method
Computational Experiments
Nonlinear Problem
Euler
Siméon Denis Poisson
Jump
Numerical Simulation
Evaluate
Evaluation
Computer simulation
Experiments

Keywords

  • asymptotic stability
  • backward Euler
  • Euler-Maruyama
  • jump-diusion
  • Poisson process
  • stochastic dierential equation
  • theta method
  • trapezoidal rule

Cite this

Chalmers, Graeme, D ; Higham, Desmond J. / Asymptotic stability of a jump-diffusion equation and its numerical approximation. In: SIAM Journal on Scientific Computing. 2008 ; Vol. 31, No. 2. pp. 1141-1155.
@article{7cbbab9f400642758ea39ad857735e3b,
title = "Asymptotic stability of a jump-diffusion equation and its numerical approximation",
abstract = "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.",
keywords = "asymptotic stability, backward Euler, Euler-Maruyama, jump-diusion, Poisson process, stochastic dierential equation, theta method, trapezoidal rule",
author = "Chalmers, {Graeme, D} and Higham, {Desmond J.}",
year = "2008",
month = "12",
day = "17",
doi = "10.1137/070699469",
language = "English",
volume = "31",
pages = "1141--1155",
journal = "SIAM Journal on Scientific Computing",
issn = "1064-8275",
number = "2",

}

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 journalArticle

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 -