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.

翻译：近年来，描述数学、物理、工程和金融中众多模型的（确定性与随机性）动力系统变得日益复杂。仅局限于确定性算法的数值分析似乎不足以处理此类系统，例如，维数灾难会影响确定性方法。因此，我们可以观察到蒙特卡洛算法，以及与之密切相关的、基于随机微分方程的随机模拟的日益普及。在本讲义中，我们介绍与蒙特卡洛方法相关的主要思想及其理论性质。我们将它们应用于诸如积分和确定/随机微分方程解的逼近等问题。我们还将讨论用Python编程语言实现示例算法及其在期权定价中的应用。本讲义的部分内容已在2020年、2021年和2023年夏季学期期间，用于波兰克拉科夫AGH科技大学的博士生课程。