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

随机微分方程（SDEs）用于所有领域的随机噪声动力学建模。由于实际应用的SDEs通常不允许显式解，因此对SDEs进行数值求解至关重要。大多数应用的sde具有超线性增长的系数，因此不满足大部分文献的假设。我们最近已经证明，为全局利普希茨系数的情况下开发的算法在没有修改的情况下通常不能转移到非全局利普希茨情况。为此，我们研究了适当修正的显式欧拉方法的收敛性。更准确地说，我们发展了一套完整的数值方法理论，这些方法递归地定义为前一状态、时间增量和布朗运动增量的一般函数。收敛理论适用于大多数局部具有有限矩的Lipschitz连续系数的随机常微分方程。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亚普诺夫技术引入数值逼近理论。此外，我们将有限维的结果推广到随机偏微分方程。我们特别研究了指数欧拉方法的一个改进版本，该方法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“非全局Lipschitz连续系数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".