Undergraduate Project 2023-24 (3H)

Stochastic differential equations (SDEs) are one of the fundamental tools for modelling financial, industrial, and scientific problems. In these equations, at least one of the terms is a stochastic process, resulting in a solution that is a stochastic process a well. To get a better feeling about SDEs and their difference with ordinary differential equation, consider the motion of a projectile in the air. If there is no wind (or the wind is well defined), we can find the exact trajectory of the projectile by solving a differential equation that comes from the laws of physics. If there is a randomly varying wind, we cannot determine the trajectory for every launch of the projectile. However, if we know the statistics of the wind (like its mean, variance and distribution), we can derive the statistics of the projectile trajectory, which is provided by the solution to a SDE.

Although some basic concepts about stochastic processes and Brownian motion are reviewed in this project, our focus is developing numerical methods for solving SDEs. Hence, this project includes fair share of programming and is more suited to students who are interested in computational methods for solving mathematical problems that arise in various applications. The students are expected to be familiar with basic numerical methods for ordinary differential equations (like Euler’s method) and to have at least an intuitive feel for random variables. That being said, no advanced knowledge of probability or stochastic processes is required. Programming experience with Python or MATLAB are needed for this project..

[1] Higham, Desmond J. “An algorithmic introduction to numerical simulation of stochastic differential equations.” SIAM review 43.3 (2001): 525-546.

[2] Evans, Lawrence C. An introduction to stochastic differential equations. Vol. 82. American Mathematical Soc., 2012.