### Abstract

Language | English |
---|---|

Pages | 1041-1063 |

Number of pages | 22 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 40 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2002 |

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### Keywords

- finite-time convergence
- nonlinearity
- computer science
- applied mathematics

### Cite this

*SIAM Journal on Numerical Analysis*,

*40*(3), 1041-1063. https://doi.org/10.1137/S0036142901389530

}

*SIAM Journal on Numerical Analysis*, vol. 40, no. 3, pp. 1041-1063. https://doi.org/10.1137/S0036142901389530

**Strong convergence of Euler-type methods for nonlinear stochastic differential equations.** / Higham, D.J.; Mao, X.; Stuart, A.M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Strong convergence of Euler-type methods for nonlinear stochastic differential equations

AU - Higham, D.J.

AU - Mao, X.

AU - Stuart, A.M.

PY - 2002

Y1 - 2002

N2 - Traditional finite-time convergence theory for numerical methods applied to stochastic differential equations (SDEs) requires a global Lipschitz assumption on the drift and diffusion coefficients. In practice, many important SDE models satisfy only a local Lipschitz property and, since Brownian paths can make arbitrarily large excursions, the global Lipschitz-based theory is not directly relevant. In this work we prove strong convergence results under less restrictive conditions. First, we give a convergence result for Euler--Maruyama requiring only that the SDE is locally Lipschitz and that the pth moments of the exact and numerical solution are bounded for some p >2. As an application of this general theory we show that an implicit variant of Euler--Maruyama converges if the diffusion coefficient is globally Lipschitz, but the drift coefficient satisfies only a one-sided Lipschitz condition; this is achieved by showing that the implicit method has bounded moments and may be viewed as an Euler--Maruyama approximation to a perturbed SDE of the same form. Second, we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial.

AB - Traditional finite-time convergence theory for numerical methods applied to stochastic differential equations (SDEs) requires a global Lipschitz assumption on the drift and diffusion coefficients. In practice, many important SDE models satisfy only a local Lipschitz property and, since Brownian paths can make arbitrarily large excursions, the global Lipschitz-based theory is not directly relevant. In this work we prove strong convergence results under less restrictive conditions. First, we give a convergence result for Euler--Maruyama requiring only that the SDE is locally Lipschitz and that the pth moments of the exact and numerical solution are bounded for some p >2. As an application of this general theory we show that an implicit variant of Euler--Maruyama converges if the diffusion coefficient is globally Lipschitz, but the drift coefficient satisfies only a one-sided Lipschitz condition; this is achieved by showing that the implicit method has bounded moments and may be viewed as an Euler--Maruyama approximation to a perturbed SDE of the same form. Second, we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial.

KW - finite-time convergence

KW - nonlinearity

KW - computer science

KW - applied mathematics

UR - http://www.maths.strath.ac.uk/~aas96106/38953.pdf

UR - http://personal.strath.ac.uk/d.j.higham/Plist/P44.pdf

U2 - 10.1137/S0036142901389530

DO - 10.1137/S0036142901389530

M3 - Article

VL - 40

SP - 1041

EP - 1063

JO - SIAM Journal on Numerical Analysis

T2 - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

SN - 0036-1429

IS - 3

ER -