# MMath Mathematics with Financial Mathematics

Year of entry: 2023

## Course unit details:Stochastic Calculus

Unit code MATH47101 15 Level 4 Semester 1 Department of Mathematics No

### Overview

The stochastic integral (Ito's integral) with respect to a continuous semimartingale is introduced and its properties are studied. The fundamental theorem of stochastic calculus (Ito's formula) is proved and its utility is demonstrated by various examples. Stochastic differential equations driven by a Wiener process are studied.

### Pre/co-requisites

Unit title Unit code Requirement type Description
Foundations of Modern Probability MATH20722 Pre-Requisite Recommended

[MATH20722 Foundation of Modern Probability or equivalent]

Students are not permitted to take, for credit, MATH47101 in an undergraduate programme and then MATH67101 in a postgraduate programme at the University of Manchester, as the courses are identical.

### Aims

The course unit aims to provide the basic knowledge necessary to pursue further studies/applications where stochastic calculus plays a fundamental role (e.g. Financial Mathematics).

### Learning outcomes

On successful completion of this course unit students will be able to:

• define the Wiener process (standard Brownian motion) and calculate the expected values of its basic functionals;
• define the (Ito’s) stochastic integral with respect to a continuous semimartingale and state Ito's formula;
• apply Ito's formula to smooth functions of continuous semimartingales and derive their semimartingale decompositions;
• state basic facts and theorems of stochastic calculus and use them in a variety of applied settings (e.g. mathematical finance);
• define the stochastic differential equation driven by a Wiener process and use the basic existence and uniqueness theorem.

### Syllabus

• The Wiener process (standard Brownian motion): Review of various constructions. Basic properties and theorems. Brownian paths are of unbounded variation. [6 lectures]
• The Ito's integral with respect to a Wiener process: Definition and basic properties. Continuous local martingales. The quadratic variation process. The Kunita-Watanabe inequality. Continuous semimartingales. The Ito's integral with respect to a continuous semimartingale: Definition and basic properties. Stochastic dominated convergence theorem. 
• The Ito's formula: Statement and proof. Integration by parts formula. The Levy characterization theorem. The Cameron-Martin-Girsanov theorem (change of measure). The Dambis-Dubins-Schwarz theorem (change of time). 
• The Ito's-Clark theorem. The martingale representation theorem. Optimal prediction of the maximum process. 
• Stochastic differential equations. Examples: Brownian motion with drift, geometric Brownian motion, Bessel process, squared Bessel process, the Ornstein-Uhlenbeck process, branching diffusion, Brownian bridge. The existence and uniqueness of solutions in the case of Lipschitz coefficients. 

### Assessment methods

End of semester examination: weighting 100%

### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

• Rogers, L. C. G. and Williams, D., Diffusions, Markov Processes and Martingales, Vol. 1 & 2, Cambridge University Press 2000.
• Revuz, D. and Yor, M., Continuous Martingales and Brownian Motion, Springer 1999.
• Karatzas, I. and Shreve, S. E., Brownian Motion and Stochastic Calculus, Springer 1991.
• Durrett, R., Stochastic Calculus, CRC Press LCL 1996.

### Study hours

Scheduled activity hours
Lectures 33
Tutorials 11
Independent study hours
Independent study 106

### Teaching staff

Staff member Role
Goran Peskir Unit coordinator