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 Dr OO Menoukeu Pamen
Mathematical Sciences
O.Menoukeu-Pamen@liverpool.ac.uk
Year CATS Level Semester CATS Value
Session 2020-21 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) A critical awareness of current problems and research issues in the fields of probability and stochastic processes, stochastic analysis and financial mathematics.

(LO2) The ability to formulate stochastic cuclulus for the purpose of modelling particular financial questions.

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

(LO4) The ability 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. · Application to Biology : epidemic, competition and predation processes; population genetics process; expected time to extinction and first passage time.

Recommended Texts
Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module.

Pre-requisites before taking this module (other modules and/or general educational/academic requirements):
 

Co-requisite modules:
 

Modules for which this module is a pre-requisite:
 

Programme(s) (including Year of Study) to which this module is available on a required basis:
 

Programme(s) (including Year of Study) to which this module is available on an optional basis:
 

Assessment
EXAM Duration Timing
(Semester)		 % of
final
mark		 Resit/resubmission
opportunity		 Penalty for late
submission		 Notes
final assessment open book and remote Assessment Schedule (When) :Second  1 hour time on task    50       
CONTINUOUS Duration Timing
(Semester)		 % of
final
mark		 Resit/resubmission
opportunity		 Penalty for late
submission		 Notes
Class test open book and remote  around 60-90 minutes    50