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 PROBABILITY ESSENTIALS FOR FINANCIAL CALCULUS
Code MATH480
Coordinator Dr A Pantelous
Mathematical Sciences
A.Pantelous@liverpool.ac.uk
Year CATS Level Semester CATS Value
Session 2016-17 Level 7 FHEQ First Semester 15

Aims
  • To equip the students with the essential probabilistic concepts, to be used further in advanced stochastic and financial calculus.

  • To equip the students with the understanding of measure theory, probability measures and integration with respect to probability measures.

  • To acquaint students with random variables, sums of random variables, central limit theorem and law of large numbers.

  • To give the student the ability to analyse the different type of convergences of random variables.

  • To introduce the students to the concepts of conditional expectation, martingale and stopping times, building blocks of applied probability.

    		•

    Learning Outcomes
    1. Ability to fully understand the concept of measure spaces and probability measures. 

    Ability to understand the concept of random variables and their properties.

    Ability to analyse the convergence of a sequence or of a sum of random variables.

    Ability to use the concepts of conditional expectations and martingales in applications pertaining to financial mathematics.

    Syllabus
    1  Measure Space and Probability Space:
    Baby set theory, algebra and σ-algebra, σ-algebra generated by a collection of subsets, monotone class theorem (set form), measurable space, measure and probability (definitions, properties), measure spaces, probability space.

    Measurable Function and Random Variables: Measurable mappings; measurable functions, random variables, monotone class theorem (function form), independence, standard machines.

    Integrals and Expections: Definitions; properties; Fatou lemma; monotone convergence theorem; dominant convergence theorems; product measure space; Fubini theorem.

    Convergence of Random Variables: All kinds of convergence concepts of random variables; almost sure convergence and null sets; convergence in probability (in measure); convergence in p’th means; weak convergence; relationships between all kinds of convergence.

    Conditional Expectations and Conditional Probabilities: Conditional expectations given σ-algebras (definitions, existence, uniqueness); conditional expectations given random variables; conditional probability; properties of conditional expectations, relationship with elementary cases; convergence of conditional expectations, inequality of conditional expectations.

    Discrete-Time Martingales: Definition of discrete-time martingales; properties of martingales; optional stopping theorem; UI martingales; some examples and applications.

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

    Pre-requisites before taking this module (other modules and/or general educational/academic requirements):
    MATH264; MATH101; MATH103; MATH162  

    Co-requisite modules:
     

    Modules for which this module is a pre-requisite:
    MATH481; MATH482; MATH483; ; ; ; ; ;  

    Programme(s) (including Year of Study) to which this module is available on a required basis:
    Programme:FMMF Year:1 Programme:BLFR Year:1

    Programme(s) (including Year of Study) to which this module is available on an optional basis:
    Programme:G101 Year:3,4 Programme:MMAS Year:1 Programme:EGRU Year:1

    Assessment
    EXAM Duration Timing
    (Semester)    		 % of
    final
    mark    		 Resit/resubmission
    opportunity    		 Penalty for late
    submission    		 Notes
    Unseen Written Exam  2.5 hours  First semester  100  Yes  Standard UoL penalty applies  Assessment 1 Notes (applying to all assessments) Final exam rubric: Full marks will be awarded for complete answers to all questions  
    CONTINUOUS Duration Timing
    (Semester)    		 % of
    final
    mark    		 Resit/resubmission
    opportunity    		 Penalty for late
    submission    		 Notes