### Abstract

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

Pages | 362-375 |

Number of pages | 14 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 296 |

Early online date | 13 Oct 2015 |

Publication status | Published - Apr 2016 |

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### Keywords

- stochastic differential equation
- local Lipschitz condition
- Khasminskii-type condition
- truncated Euler-Maruyama method
- convergence rate

### Cite this

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**Convergence rates of the truncated Euler-Maruyama method for stochastic differential equations.** / Mao, Xuerong.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Convergence rates of the truncated Euler-Maruyama method for stochastic differential equations

AU - Mao, Xuerong

PY - 2016/4

Y1 - 2016/4

N2 - Influenced by Higham, Mao and Stuart [9], several numerical methods have been developed to study the strong convergence of the numerical solutions to stochastic differential equations (SDEs) under the local Lipschitz condition. These numerical methods include the tamed Euler–Maruyama (EM) method, the tamed Milstein method, the stopped EM, the backward EM, the backward forward EM, etc. Recently, we developed a new explicit method in [23], called the truncated EM method, for the nonlinear SDE dx(t) = f (x(t))dt + g(x(t))dB(t) and established the strong convergence theory under the local Lip- schitz condition plus the Khasminskii-type condition xT f (x) + p−1 |g(x)|2 ≤ K(1 + |x|2). However, due to the page limit there, we did not study the convergence rates for the method, which is the aim of this paper. We will, under some additional conditions, discuss the rates of Lq -convergence of the truncated EM method for 2 ≤ q < p and show that the order of Lq -convergence can be arbitrarily close to q/2.

AB - Influenced by Higham, Mao and Stuart [9], several numerical methods have been developed to study the strong convergence of the numerical solutions to stochastic differential equations (SDEs) under the local Lipschitz condition. These numerical methods include the tamed Euler–Maruyama (EM) method, the tamed Milstein method, the stopped EM, the backward EM, the backward forward EM, etc. Recently, we developed a new explicit method in [23], called the truncated EM method, for the nonlinear SDE dx(t) = f (x(t))dt + g(x(t))dB(t) and established the strong convergence theory under the local Lip- schitz condition plus the Khasminskii-type condition xT f (x) + p−1 |g(x)|2 ≤ K(1 + |x|2). However, due to the page limit there, we did not study the convergence rates for the method, which is the aim of this paper. We will, under some additional conditions, discuss the rates of Lq -convergence of the truncated EM method for 2 ≤ q < p and show that the order of Lq -convergence can be arbitrarily close to q/2.

KW - stochastic differential equation

KW - local Lipschitz condition

KW - Khasminskii-type condition

KW - truncated Euler-Maruyama method

KW - convergence rate

UR - http://www.sciencedirect.com/science/journal/03770427

M3 - Article

VL - 296

SP - 362

EP - 375

JO - Journal of Computational and Applied Mathematics

T2 - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

ER -