Numerical Methods in Stochastic Processes

Ss. Cyril and Methodius University in Skopje

The Aim of this course is to provide the student with the knowledge of numerical modeling for stochastic problems and the ability to analyze theoretical properties and design mathematical software based on the proposed schemes.
Upon successful completion of this course, the student should be able to:

–          define and comprehend various types of stochastic processes;

–          discretize a given SDE and check resulting approximation properties;

–          implement the Monte Carlo and Multilevel Monte Carlo methods for discretization of SDEs;

–          analyze the behavior of stochastic processes through simulations;

–          control the discretization errors arising in a Monte Carlo method for the weak approximation of SDEs;

–          understand the most relevant numerical methods for the approximation of stochastic differential problems and the    design of accurate and efficient mathematical software;

–          demonstrate skills in choosing the most suitable discretization in relation to the problem to be solved.

–          A brief introduction to Brownian motion;

–          Simulations of Brownian motion and its variants;

–          A brief introduction to Ito calculus and and Stratonovich stochastic integrals;

–          Stochastic differential equations: motivation, modelling, existence and uniqueness, strong and weak solutions;

–          Ito formula,

–          Explicit methods: Euler-Maruyama and Milstein;

–          Implicit methods: stochastic theta methods;

–          Strong and weak convergence;

–          Mean-square and asymptotic linear stability. Nonlinear stability analysis;

–          Strong numerical approximations for stochastic differential equations: Galerkin approximations. Noise approximations;

–          Week approximation. Mild-Ito formula;

–          Stochastic simulations and multi-level Monte-Carlo methods.

Numerical methods for stochastic processes provide a mathematical framework for analyzing and understanding randomness and uncertainty in various systems. These methods can be applied to enhance the reliability, security, and performance of communication systems, error-correcting codes, and blockchain technologies. The interdisciplinary nature of these connections highlights the importance of mathematical and computational tools in addressing challenges in diverse fields.

In the context of data science and machine learning, stochastic processes are relevant in various areas, including Monte Carlo simulations, time series, optimization under uncertainty, Bayesian inference etc. Some machine learning models incorporate stochastic components, such as dropout in neural networks or certain types of probabilistic models. Numerical methods are used to train and simulate the behavior of these models. The connection between Numerical methods in stochastic processes and Data science and machine learning lies in their shared interest in dealing with uncertainty, randomness, and the need to make predictions or decisions based on incomplete information. Numerical methods provide the computational tools to analyze, simulate, and optimize models that involve stochastic processes, making them valuable in the broader context of data-driven decision-making.

Although Numerical methods for stochastic processes and Computer networks are distinct fields, they can be closely related in the context of modeling, simulating, and analyzing the behavior of computer networks, especially when dealing with the inherent randomness and uncertainties present in network environments. Queueing theory, which deals with the study of queues or waiting lines, is often employed to analyze network performance. Queueing models can be based on stochastic processes. Numerical methods can be used to solve complex queueing models that arise in the analysis of network systems. Numerical optimization techniques may be applied to optimize network protocols considering the stochastic nature of the network environment.

A blended learning method consisting of:

–          lectures, which provide theoretical foundations and key concepts;

–          hands-on programming labs for practical implementation of numerical algorithms;

–          analyze and discuss real-world applications and explore the role of numerical methods in solving specific problems;

–          team activities to foster collaboration and peer learning.

Teaching methods in generally increased emphasis on independent learning, critical thinking, and the application of knowledge to solve complex problems. The methods mentioned above aim to foster a deeper understanding of the subject matter and prepare students for advanced academic and professional challenges.