Unit of study_

# MATH4512: Stochastic Analysis

Capturing random phenomena is a challenging problem in many disciplines from biology, chemistry and physics through engineering to economics and finance. There is a wide spectrum of problems in these fields, which are described using random processes that evolve with time. Hence it is of crucial importance that applied mathematicians are equipped with tools used to analyse and quantify random phenomena. This unit will introduce an important class of stochastic processes, using the theory of martingales. You will study concepts such as the Ito stochastic integral with respect to a continuous martingale and related stochastic differential equations. Special attention will be given to the classical notion of the Brownian motion, which is the most celebrated and widely used example of a continuous martingale. By completing this unit, you will learn how to rigorously describe and tackle the evolution of random phenomena using continuous time stochastic processes. You will also gain a deep knowledge about stochastic integration, which is an indispensable tool to study problems arising, for example, in Financial Mathematics.

Code MATH4512 Mathematics and Statistics Academic Operations 6
 Prerequisites: ? None None None Students should have a sound knowledge of probability theory and stochastic processes from, for example, STAT2X11 and STAT3021 or equivalent

At the completion of this unit, you should be able to:

• LO1. Demonstrate an understanding of the concepts of continuous time processes, stopping times and martingales.​
• LO2. ​Apply the properties of Brownian motion (such as martingale property, Markov property, behaviour under the equivalent change of measure and predictable representation property). ​
• LO3. Describe the construction of Ito integral and apply Ito integration correctly in relevant examples.​
• LO4. Identify the similarities and differences between strong and weak solutions to SDEs and apply existence and uniqueness results to some specific SDEs.​
• LO5. Check the martingale property of stochastic processes using Ito's lemma.
• LO6. Use the Feymann Kac formula to analyse real world problems including arbitrage pricing.
• LO7. Identify, formulate and solve original practical problems that can be addressed using mathematical techniques learnt in this unit and examine the implementations and provide an interpretation of the results.​
• LO8. Independently research sources provided by lecturers such as journal articles and working papers and evaluate the real-world plausibility and effectiveness of various approaches and tools.​

## Unit outlines

Unit outlines will be available 2 weeks before the first day of teaching for the relevant session.