Topic
Milstein method
About: Milstein method is a research topic. Over the lifetime, 240 publications have been published within this topic receiving 6096 citations.
Papers published on a yearly basis
Papers
1 Jan 2015
TL;DR: In this article, the main concepts and techniques necessary for someone who wishes to carry out numerical experiments involving stochastic differential equations (SDEs) are described and compared. And the convergence of Euler-Maruyama and Milstein and Taylor approximate solutions are compared.
Abstract: This paper provides an introduction to the main concepts and techniques necessary for someone who wishes to carryout numerical experiments involving Stochastic Differential Equation (SDEs). As SDEs are frictionless generally and the solutions are continuous stochastic process that represent diffusive dynamic especially in finance, it is required of us to take into account random effects and influences in real world systems which are essential in the accurate description of such situations. We include a review of Stochastic Differential equations (SDE), Geometric Brownian Motion, Euler- Maruyama, Milstein and Taylor approximate which gives a clear picture of their graphical approximate and exact solution. We finally compared the convergence of Euler-Maruyama and Milstein
2,019 citations
TL;DR: It is shown that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using Monte Carlo path simulations.
Abstract: We show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using Monte Carlo path simulations. In the simplest case of a Lipschitz payoff and a Euler discretisation, the computational cost to achieve an accuracy of O(e) is reduced from O(e-3) to O(e-2 (log e)2). The analysis is supported by numerical results showing significant computational savings.
1,964 citations
TL;DR: In this article, it was shown that for a large class of SDEs with non-globally Lipschitz continuous drift and diffusion coefficients, Euler's approximation converges neither in the strong mean-square sense nor in the numerically weak sense to the exact solution at a finite time point.
Abstract: The stochastic Euler scheme is known to converge to the exact solution of a stochastic differential equation (SDE) with globally Lipschitz continuous drift and diffusion coefficients. Recent results extend this convergence to coefficients that grow, at most, linearly. For superlinearly growing coefficients, finite-time convergence in the strong mean-square sense remains. In this article, we answer this question to the negative and prove, for a large class of SDEs with non-globally Lipschitz continuous coefficients, that Euler’s approximation converges neither in the strong mean-square sense nor in the numerically weak sense to the exact solution at a finite time point. Even worse, the difference of the exact solution and of the numerical approximation at a finite time point diverges to infinity in the strong mean-square sense and in the numerically weak sense.
299 citations
TL;DR: This paper develops a new explicit method, called the truncated EM method, for the nonlinear SDE, and establishes the strong convergence theory under the local Lipschitz condition plus the Khasminskii-type condition.
283 citations
TL;DR: In this article, a tamed version of the Milstein method for stochastic differential equations with commutative noise is proposed, which achieves higher strong convergence order than the tamed Euler method does.
Abstract: For stochastic differential equations (SDEs) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient, the classical explicit Euler scheme fails to converge strongly to the exact solution. Recently, an explicit strongly convergent numerical scheme, called the tamed Euler method, has been proposed in [8] for such SDEs. Motivated by their work, we here introduce a tamed version of the Milstein scheme for SDEs with commutative noise. The proposed method is also explicit and easily implementable, but achieves higher strong convergence order than the tamed Euler method does. In recovering the strong convergence order one of the new method, new difficulties arise and kind of a bootstrap argument is developed to overcome them. Finally, an illustrative example confirms the computational efficiency of the tamed Milstein method compared to the tamed Euler method.
216 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2022 | 4 |
| 2021 | 12 |
| 2020 | 20 |
| 2019 | 19 |
| 2018 | 16 |
| 2017 | 18 |