Numerical methods for stochastic differential equations

随机微分方程的数值方法

• 批准号: RGPIN-2018-04449
• 负责人: Anton, Cristina
• 金额: $ 1.17万
• 依托单位: Grant MacEwan University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=712974
• 关键词: Numerical; methods; stochastic; differential; equations

Models used in engineering, biological or financial applications inherently include parameter uncertainties, or other kinds of random variations, that can be expressed mathematically using stochastic differential equations. Since very few types of stochastic differential equations can be solved analytically, it is important to find efficient methods to approximate the solutions numerically. The focus of my project is the development of new numerical methods that inherit the qualitative properties of the original stochastic differential equations. I am also interested in the study of the error of the proposed numerical schemes, and the analysis of the properties of the stochastic processes defined by the numerical schemes. Stochastic Hamiltonian systems and Langevin type equations are used in many models from classical mechanics, the dynamics of particle accelerators, chemistry, and biology, and they appear also in the numerical study of nonlinear stochastic partial differential equations.Stochastic Hamiltonian systems preserve the symplectic structure, so it is important to construct numerical schemes with similar properties. We have developed a systematic method to construct new symplectic numerical schemes for stochastic Hamiltonian systems. Numerical simulations show that symplectic schemes are more accurate than non-symplectic methods for long-time simulations. There is no theoretical proof of this fact in the stochastic case, and it is challenging to extend the approach used in the deterministic case. One of the goals of this project is to study the error associated with the symplectic schemes for stochastic Hamiltonian systems. Classical assumptions for the study of numerical solution of stochastic differential equations require globally Lipschitz coefficients. These conditions are not met for stochastic non-linear oscillators and other highly non-linear systems arising from financial mathematics or bio-mathematics. Relaxing these assumptions is a subtle problem because some numerical schemes might not be convergent. Since I investigate a stochastic Hamiltonian system with locally Lipschitz coefficients and a fully implicit scheme, the study of the error is a challenging problem. In Monte Carlo simulations, explicit numerical schemes are preferable because they require less computing time than implicit schemes, but, unless we consider special stochastic Hamiltonian systems, symplectic schemes are implicit. In the deterministic case, exponentially fitted methods are constructed for differential equations with periodic or oscillating solutions. I intend to extend this approach in the stochastic case and to develop explicit symplectic Runge-Kutta-Nyström methods for stochastic oscillators. This proposal includes both challenging theoretical analyses and applications, so it will contribute to the knowledge transfer from academia to industry.

工程、生物或金融应用中使用的模型本身就包含参数不确定性或其他类型的随机变量，这些变量可以用随机微分方程进行数学表达。由于很少有类型的随机微分方程解的解是解析的，因此找到有效的方法来数值逼近解是很重要的。我的项目的重点是发展新的数值方法，继承原始随机微分方程的定性性质。我还感兴趣的是研究所提出的数值格式的误差，以及由数值格式定义的随机过程的性质的分析。 随机哈密顿系统和朗之万型方程被用于经典力学、粒子加速器动力学、化学和生物学的许多模型中，它们也出现在非线性随机偏微分方程组的数值研究中。随机哈密顿系统保留了辛结构，因此构造具有相似性质的数值格式是很重要的。我们发展了一种系统的方法来构造随机哈密顿系统的新的辛数值格式。数值模拟表明，对于长时间的模拟，辛格式比非辛方法具有更高的精度。在随机情况下，这一事实没有理论上的证明，而在确定性情况下，推广使用的方法是具有挑战性的。这个项目的目的之一是研究随机哈密顿系统的辛格式的误差。 研究随机微分方程数值解的经典假设需要全局Lipschitz系数。这些条件不适用于随机非线性振荡器和其他源自金融数学或生物数学的高度非线性系统。放松这些假设是一个微妙的问题，因为一些数值格式可能不收敛。由于我研究的是具有局部Lipschitz系数和全隐格式的随机哈密顿系统，所以对误差的研究是一个具有挑战性的问题。 在蒙特卡罗模拟中，显式数值格式比隐式格式需要更少的计算时间，但是，除非我们考虑特殊的随机哈密顿系统，否则辛格式是隐式的。在确定性情况下，构造了具有周期解或振荡解的微分方程解的指数拟合法。我打算将这种方法推广到随机情形，并发展随机振子的显式辛Runge-Kutta-Nyström方法。 这一建议既包括具有挑战性的理论分析，也包括应用，因此它将有助于知识从学术界向产业界的转移。