### Module Details

 The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
 Title STOCHASTIC ANALYSIS AND ITS APPLICATIONS Code MATH483 Coordinator Professor OO Menoukeu Pamen Mathematical Sciences O.Menoukeu-Pamen@liverpool.ac.uk Year CATS Level Semester CATS Value Session 2023-24 Level 7 FHEQ Second Semester 15

### Aims

This module aims to demonstrate the advanced mathematical techniques underlying financial markets and the practical use of financial derivative products to analyse various problems arising in financial markets. Emphases are on the stochastic techniques, probability theory, Markov processes and stochastic calculus, together with the related applications

### Learning Outcomes

(LO1) Critically analyse current problems and research issues in the fields of probability and stochastic processes, stochastic analysis and financial mathematics.

(LO2) Formulate stochastic calculus for the purpose of modelling particular financial questions.

(LO3) Be able to read, understand and communicate research literature in the fields of probability, stochastic analysis and financial mathematics.

(LO4) Be able to recognise potential research opportunities and research directions.

### Syllabus

Brownian Motion: Definition of Brownian Motion; Brownian bridge; The Reflection Principle and Scaling.

Martingales : Change of measure, Radon-Nikodym derivative; conditional expectation (definition and properties); martingales (discrete and continuous time); stopping times; applications of stopping times; Doob’s inequality.

Stochastic calculus : Ito’s processes and stochastic differential. Definition and properties of Ito’s integral; Ito’s formula; integration by parts; stochastic Fubini theorem; Girsanov theorem; the Brownian martingale representation theorem.

Stochastic differential equations : Markov property; stochastic differential equations (SDEs); diffusions and the PDE connection; Feynman-Kac representation.

Applications :

Application to Optimal Control : optimal stopping problem; Hamilton-Jacob-Bellman equations.

Application to Mathematical Finance : risk-neutral pricing; exotic options; derivative securities.< /p>

Application to Biology : epidemic, competition and predation processes; population genetics process; expected time to extinction and first passage time.

### Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Final Assessment  90    50
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Class test  90    50