### 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 2019-20 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

(LO1) 1. Ability to fully understand the concept of measure spaces and probability measures.

(LO2) Ability to understand the concept of random variables and their properties.

(LO3) Ability to analyse the convergence of a sequence or of a sum of random variables.

(LO4) Ability to use the concepts of conditional expectations and martingales in applications pertaining to financial mathematics.

(S1) Numeracy/computational skills - Problem solving

(S2) Numeracy/computational skills - Numerical methods

### Syllabus

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 (definiti ons, 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.

### Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Assessment 1 There is a resit opportunity. Standard UoL penalty applies for late submission. Assessment Schedule (When) :First semester  2.5 hours    100
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes