Numerical methods for stochastic differential equations

在工程、生物或金融应用中使用的模型固有地包括参数不确定性或其他种类的随机变化，其可以使用随机微分方程在数学上表达。 由于很少有类型的随机微分方程可以解析求解，因此找到有效的方法来近似数值解是很重要的。 我的项目的重点是继承原来的随机微分方程的定性性质的新的数值方法的发展。我也有兴趣研究所提出的数值方案的误差，以及分析由数值方案定义的随机过程的性质。随机哈密顿系统和朗之万型方程用于许多模型，从经典力学，粒子加速器的动力学，化学和生物学，随机Hamilton系统保持了辛结构，因此构造具有相似性质的数值格式是非常重要的。我们发展了一种系统的方法来构造随机Hamilton系统的新的辛数值格式。数值模拟结果表明，辛格式比非辛格式具有更高的精度。在随机情况下，这一事实没有理论证明，并且扩展确定性情况下使用的方法具有挑战性。 本计画的目标之一是研究随机哈密顿系统辛格式的误差。研究随机微分方程数值解的经典假设要求全局Lipschitz系数。这些条件不满足随机非线性振荡器和其他高度非线性系统所产生的金融数学或生物数学。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会收敛。由于我研究的是一个具有局部Lipschitz系数的随机Hamilton系统和一个全隐格式，因此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，显式数值格式是优选的，因为它们需要比隐式格式更少的计算时间，但是，除非我们考虑特殊的随机哈密顿系统，辛格式是隐式的。在确定性的情况下，指数拟合方法构造的微分方程的周期或振荡的解决方案。我打算扩展这种方法在随机情况下，并制定明确的辛龙格库塔-Nyström方法的随机振荡器。该提案包括具有挑战性的理论分析和应用，因此它将有助于从学术界到工业界的知识转移。

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.