# B8.2 Continuous Martingales and Stochastic Calculus - Material for the year 2020-2021

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

16 lectures

### Assessment type:

- Written Examination

Stochastic processes - random phenomena evolving in time - are encountered in many disciplines from biology, through geology to finance. This course focuses on 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 motivating example of a stochastic process is Brownian motion, also called the Wiener process - a mathematical object initially proposed by Bachelier and Einstein, which originally modelled 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 equipped with a suitable integration by parts formula (Itô formula).

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.

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. Recall of Stieltjes integral. Stochastic integration and Itô's formula with applications.

Lecture notes will be provided, but there are also many textbooks which cover the course material with a varying degrees of detail/rigour.

These include:

- D. Revuz and M. Yor,
*Continuous martingales and Brownian motion*, Springer (Revised $3^{rd}$ ed.), 2001, Chapters 0-4. - I. Karatzas and S. Shreve,
*Brownian motion and stochastic calculus*, Springer ($2^{nd}$ ed.), 1991, Chapters 1-3. - R. Durrett,
*Stochastic Calculus: A practical introduction*, CRC Press, 1996. Sections 1.1 - 2.10. - F. Klebaner,
*Introduction to Stochastic Calculus with Applications*, 3rd edition, Imperial College Press, 2012. Chapters 1, 2, 3.1--3.11, 4.1-4.5, 7.1-7.8, 8.1-8.7. - J. M. Steele,
*Stochastic Calculus and Financial Applications*, Springer, 2010. Chapters 3 - 8. - B. Oksendal,
*Stochastic Differential Equations: An introduction with applications*, 6th edition, Springer (Universitext), 2007. Chapters 1 - 3. - S. Shreve,
*Stochastic calculus for finance*, Vol 2: Continuous-time models, Springer Finance, Springer-Verlag, New York, 2004. Chapters 3 - 4.