论文标题
蒙特卡洛方法和随机模拟的基础 - 从蒙特卡洛·勒布斯格整合到弱近似SDES
Foundations of Monte Carlo methods and stochastic simulations -- From Monte Carlo Lebesgue integration to weak approximation of SDEs
论文作者
论文摘要
近年来，动态系统（确定性和随机性质）描述了数学，物理，工程和财务的许多模型，变得越来越复杂。数值分析仅缩小到确定性算法似乎不足对于此类系统不足，因为例如，维度的诅咒会影响确定性方法。因此，我们可以观察到基于随机微分方程的随机仿真与它们密切相关的蒙特卡洛算法的普及。在这些讲义中，我们提出了与蒙特卡洛方法及其理论特性有关的主要思想。我们将它们应用于确定性/随机微分方程解决方案的集成和近似等问题。我们还讨论了Python编程语言中示例算法的实现及其在期权定价中的应用。 这些笔记的一部分已在2020年，2021年，2023年的夏季学期，波兰克拉科夫的AGH科学技术大学的博士学位学生中使用。
In recent years dynamical systems (of deterministic and stochastic nature), describing many models in mathematics, physics, engineering and finances, become more and more complex. Numerical analysis narrowed only to deterministic algorithms seems to be insufficient for such systems, since, for example, curse of dimensionality affects deterministic methods. Therefore, we can observe increasing popularity of Monte Carlo algorithms and, closely related with them, stochastic simulations based on stochastic differential equations. In these lecture notes we present main ideas concerned with Monte Carlo methods and their theoretical properties. We apply them to such problems as integration and approximation of solutions of deterministic/stochastic differential equations. We also discuss implementation of exemplary algorithms in Python programming language and their application to option pricing. Part of these notes has been used during lectures for PhD students at AGH University of Science and Technology, Krakow, Poland, at summer semesters in the years 2020, 2021, 2023.
