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Department of Mathematics
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Numerical methods for stochastic differential equations

Advisor

Mahboub Baccouch

Description

A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process. SDE models play a prominent role in a range of application areas, including biology, physics, chemistry, epidemiology, mechanics, microelectronics, economics, and finance, etc., when uncertainties or random influences (called noises), are taken into account. Furthermore, SDE are used to model diverse phenomena such as fluctuating stock prices or physical system subject to thermal fluctuations. Typically, SDEs incorporate white noise which can be thought of as the derivative of Brownian motion (or the Wiener process); however, it should be mentioned that other types of random fluctuations are possible, such as jump processes.

A complete understanding of SDE theory requires familiarity with advanced probability and stochastic processes. The numerical analysis of SDEs differs significantly from that of ordinary differential equations due to peculiarities of stochastic calculus. However, it is possible to appreciate the basics of how to simulate SDEs numerically with just a background knowledge of Euler's method for deterministic ordinary differential equations and an intuitive understanding of random variables. Furthermore, experience with numerical methods gives a useful first step toward the underlying theory of SDEs.

Numerical solution of SDEs is a young field relatively speaking. Almost all algorithms that are used for the solution of ordinary differential equations will work very poorly for SDEs, having very poor numerical convergence. A textbook describing many different algorithms is Kloeden & Platen (1995). Methods include the Euler-Maruyama method, Milstein method and Runge-Kutta method (SDE).

In this project, we propose to apply simple numerical methods to an SDE and discuss concepts such as convergence and linear stability from a practical viewpoint. We will study some properties of solutions of SDEs arising in applications by using the numerical methods. We will discuss some numerical schemes that come from the Ito-Taylor expansion including their order of convergence. Also, we will use some schemes to solve the other SDE e.g., stochastic Duffing equation, the stochastic Lorenz equation, the stochastic pendulum equation, and so on.

Tasks

  1. Review some of the basic theory of the stochastic calculus and the Ito-Taylor expansion for stochastic differential equations.
  2. Discuss some numerical schemes that come from the Ito-Taylor expansion including their order of convergence.
  3. Find a Numerical solution using Matlab.
  4. Apply these methods to some stochastic equation.
  5. Compare the numerical results with the deterministic cases, and comment.
  6. Use higher-order numerical methods to find numerical solutions.
  7. Write up results and present at Math Awareness Month.

Prerequisites

Math 2350 and familiarity with probability are required. Math 3300 recommended, but not required. You will be using the software Matlab.

References

Numerical Solution of Stochastic Differential Equations (Stochastic Modelling and Applied Probability), by Peter E. Kloeden, Eckhard Platen, Springer, ISBN-13: 978-3540540625