Module Specifications..
Current Academic Year 2023  2024
Please note that this information is subject to change.
 
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Description This course is designed to give students strong practical and theoretical foundations in Monte Carlo methods with an emphasis on applications relevant to insurance and finance. Each week will consist of 2 hours of lectures/tutorials and 2 hours of computer labs. The emphasis will be on using mathematical theory to justify our approaches to solving practical programming problems which have realworld relevance. The primary applications for our methods will be frequency and severity pricing in insurance, interest rate modelling, and option pricing. This module covers practical parts for the topic "Stochastic Processes" from the IFoA subject CS2. Students are expected to know the statements of important results and interpret them but lengthy and technical proofs will not be required. Programming will be done in R; this is not primarily a coding module and the only requirement is that students write clear readable code.  
Learning Outcomes 1. Apply basic results from probability theory to study the efficiency and appropriateness of Monte Carlo methods 2. Identify suitable methods for generating pseudorandom numbers 3. Use uniform random numbers to simulate random numbers of a given distribution 4. Analyse and interpret stochastic differential equations; approximate their solutions with appropriate discrete time processes for quantitative purposes 5. Write R code to simulate random variables and solutions to stochastic differential equations, empirically test pseudorandom number generators, and price options 6. Write R code to simulate Markov processes to perform applications in finance and insurance 7. Apply the theory of the Markov process to practical tasks in insurance and finance  
All module information is indicative and subject to change. For further information,students are advised to refer to the University's Marks and Standards and Programme Specific Regulations at: http://www.dcu.ie/registry/examinations/index.shtml 

Indicative Content and Learning Activities
Review of Probability Theory Sigmaalgebras, random variables, distribution functions, limit theorems, modes of convergence Generating Random Numbers Uniform random numbers, linear congruential generators, Chisquare and KolmogorovSmirnov tests, inverse transform method, acceptance rejection method, applications to Markov chains and jump processes Introduction to the Monte Carlo Method Strong law of large numbers, confidence intervals, error estimation Introduction to Stochastic Differential Equations Brownian motion – theory and simulation, motivation for SDEs, the Ito integral, Ito’s lemma, existence and uniqueness of solutions Discretising and Simulating Stochastic Differential Equations EulerMaruyama and Milstein Schemes, strong and weak orders of convergence Applications in Finance Option pricing, interest rate modelling, frequency and severity pricing in insurance  
 
Indicative Reading List
 
Other Resources None  
MS455 is for the Bachelor level students and is distinguished from the (Masters level course) MS555 in the endofsemester assessment: MS555 students are presented with longer and more challenging questions which test their depth of knowledge more rigorously.  
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