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Continuous Time Stochastic Processes


  • Credit value: 15 credits at Level 7

Module description

This module introduces you to continuous time stochastic processes and to stochastic differential calculus, in particular the stochastic differential equations (SDE) that arise in quantitative finance, as well as many other applied areas. You will learn some of the main numerical solution techniques used to solve SDE.

Indicative module syllabus

  • Review of probability theory, notion of a stochastic process
  • Examples of stochastic processes: Brownian motion and the Poisson process
  • Stochastic differential calculus: heuristic approach to stochastic differentials, Ito's lemma, rigorous approach to Ito's stochastic integral, multivariate Ito calculus
  • Stochastic differential equations (SDE), examples from financial modelling
  • Conditional expectation and martingales
  • Connection with partial differential equations (PDE): Kolmogorov-Focker-Planck equations, Feynman-Kac formula
  • Extensions of Ito calculus: jump diffusions, more general processes
  • Monte Carlo simulation of stochastic processes, Brownian motion
  • Numerical solutions of SDE

Learning objectives

By the end of this module, you should be able to demonstrate:

  • understanding of the basic theory of continuous time stochastic processes, in particular Brownian motion and the Poisson process
  • understanding of stochastic differential calculus (Ito calculus) and the concept of a stochastic differential equation (SDE)
  • knowledge of how to numerically simulate solutions to an SDE
  • the ability to manipulate stochastic integrals and use Ito's lemma
  • the ability to use and solve SDEs.