Foundations of Monte Carlo methods and stochastic simulations -- From Monte Carlo Lebesgue integration to weak approximation of SDEs

TL;DR

This paper reviews Monte Carlo methods and stochastic simulations, focusing on their theoretical foundations, applications to differential equations, and implementation in Python, especially for option pricing.

Contribution

It provides a comprehensive overview of Monte Carlo techniques, their theoretical properties, and practical implementation, bridging deterministic and stochastic approaches in numerical analysis.

Findings

Monte Carlo methods effectively address high-dimensional problems.

Stochastic simulations are crucial for solving differential equations.

Python implementations demonstrate practical applications in finance.

Abstract

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.