Numerical approximation of stochastic differential equations with non-globally Lipschitz continuous coefficients

随机微分方程（SDEs）被用于所有领域的建模动态随机噪声。由于应用的偏微分方程通常不承认显式的解决方案，这是至关重要的求解偏微分方程的数值。大多数应用的随机微分方程具有超线性增长系数，因此，不满足大部分文献的假设。我们最近表明，算法开发的情况下，全球Lipschitz系数一般不转移到非全球Lipschitz的情况下，没有修改。为此，我们研究适当修改的显式欧拉方法的收敛性。更确切地说，我们开发了一个彻底的理论的数值方法，递归地定义为一个一般功能的前一个国家，的时间增量和增量的布朗运动。收敛理论将适用于大多数的随机常微分方程的局部Lipschitz连续系数有限的时刻。建立收敛的顺序将需要额外的假设，如局部平滑。我们的主要方法是将成功的李雅普诺夫方法引入到数值逼近理论中。此外，我们将有限维结果推广到随机偏微分方程。特别地，我们研究了指数欧拉方法的一个修正版本，它最近被提出用于加性噪声的情况，并且具有相当好的收敛阶.该项目的另一个目标是将我们的结果作为专著“具有非全局Lipschitz连续系数的随机微分方程的数值逼近”发表。

Stochastic differential equations (SDEs) are used in all areas for modeling dynamics with stochastic noise. As applied SDEs typically admit no explicit solution, it is crucial to solve SDEs numerically. The majority of applied SDEs have superlinearly growing coefficients and, therefore, do not satisfy the assumptions of the bulk of the literature. We have recently shown that algorithms developed for the case of global Lipschitz coefficients do in general not transfer to the non-global Lipschitz case without modifications. For this reason, we investigate the convergence behavior of suitably modified explicit Euler methods. More precisely we develop a thorough theory of numerical methods which are recursively defined as a general function of the previous state, of the time increment and of the increment of the Brownian motion. The convergence theory will apply to most of the stochastic ordinary differential equations with locally Lipschitz continuous coefficients having finite moments. Establishing the order of convergence will require additional assumptions such as local smoothness. Our main approach is to bring forward the successful Lyapunov technique to the theory of numerical approximations. Moreover, we extend our finite-dimensional results to stochastic partial differential equations. In particular we study a modified version of the exponential Euler method which hasrecently been proposed for the case of additive noise and which has a rather good order of convergence. An additional objective of this project is to publish our results as monograph „Numerical approximation of stochastic differential equations with non-globally Lipschitz continuous coefficients".