Section: Course Materials | B8.2 Continuous Martingales and Stochastic Calculus (2025-26)

Lecturer: Profile: Zhongmin Qian

Course information

General prerequisites:

B8.1 Probability, Measure and Martingales is a prerequisite. Consequently, Part A Integration and Part A Probability are also prerequisites.

Course term: Hilary

Course lecture information: 16 lectures

Course weight: 1

Course level: H

Assessment type: Written Examination

Course overview:

Stochastic processes - random phenomena evolving in time - are encountered in many disciplines from biology, physics, geology through to economics and finance. This course focuses on developing the mathematics needed to describe stochastic processes evolving continuously in time and introduces the basic tools of stochastic calculus which are a cornerstone of modern probability theory. The canonical example of such a stochastic process is Brownian motion, also called the Wiener process. This mathematical object was initially proposed by Bachelier, as a model for asset prices, and by Einstein to describe the displacement of a pollen particle in a fluid. The paths of Brownian motion, or of any continuous martingale, are of infinite variation (they are in fact nowhere differentiable and have non-zero quadratic variation) and one of the aims of the course is to define a theory of integration along such paths as well as a suitable version of the chain rule, given by Itô's formula.

Learning outcomes:

The students will develop an understanding of Brownian motion and continuous martingales in continuous time. They will became familiar with stochastic calculus and in particular be able to use Itô's formula.

Course synopsis:

An introduction to stochastic processes in continuous time.

Brownian motion - definition, construction and basic properties, regularity of paths.

Filtrations and stopping times, first hitting times.

Brownian motion - martingale and strong Markov properties, reflection principle.

Martingales - definitions, regularisation and convergence theorems, optional sampling theorem, maximal and Doob's \(L^p\) inequalities.

Quadratic variation, local martingales, semimartingales.

Discussion of the Stieltjes integral.

Stochastic integration and Itô's formula with applications.

Section outline

- Select activity B8.2 Lecture Notes
  
  B8.2 Lecture Notes File
- Select activity B8.2 Problem Sheet 1
  
  B8.2 Problem Sheet 1 Assignment
  
  Updated by corrected an error in Section C, Question 4
- Select activity B8.2 Problem Sheet 2
  
  B8.2 Problem Sheet 2 Assignment
  
  Covers the material on stopping times, optional stopping theorem, and martingale spaces, quadratic variation processes.
- Select activity B8.2 Problem Sheet 3
  
  B8.2 Problem Sheet 3 Assignment
  
  About Brownian motion.
- Select activity B8.2 Problem Sheet 4 (updated)
  
  B8.2 Problem Sheet 4 (updated) Assignment
  
  Stochastic integrals and Ito formula.
- Select activity Lecture Note (New Edition)
  
  Lecture Note (New Edition) File
  
  3 March -- Added the detail of the proof of Lemma 5.4, the key computation in the definition of Ito integrals for martingales.
  
  1 March -- Made several corrections and added more comments.
  
  18 Feb 2026 : Additional comments in Section 4 Brownian motion are added mainly through footnotes for helping your understanding the material better. This should be the final version of the lecture notes used this year.
  
  17 Feb 2026: Made a few modifications and additions in Section 4.1.
  
  1) I have reorganized the material about Brownian motion. O shall follow the order in this version for lectures on Brownian motion (in weeks 5 and 6).
  
  2) A few additions in appendix.

Main content blocks

Section outline