Degree Programme in Financial Mathematics and Economics

• First Year
• Second Year
• Third Year
• Fourth Year
• Third and Fourth Year Optional Courses

MA180 Mathematics

MA181 Analysis: Functions of one real variable: Construction of R. countable and uncountable sets. Sequences and limits, limits of sums and products, the geometric and Dirichlet series, continuous and discontinuous functions, the intermediate value theorem, inverse functions. Mean-value theorem, Taylor?s theorem. Sequences and Series, tests, power series, products of series, exponential and logarithmic functions. Riemann Integration.

MA183 Algebra: Elementary number theory: primes, factorisation, the division algorithm, greatest common divisors, arithmetic modulo m, solution of congruences, applications. Transformations of the plane: linear transformations, isometrics, symmetries. Introduction to groups, rings, fields: examples from earlier topics, cyclic groups, Lagrange?s theorem, polynomials. Basic matrix algebra: multiplication, adjoint, determinant, inverse, eigenvalues and eigenvectors.

EC101/2 Economics

Applied Economics: The aim in applied economics is to discuss how the tools of economic theory can be used to analyse various issues and problems in everyday economic life. The material covered includes basic economic concepts, and topics related to microeconomic and macroeconomic issues.

Microeconomics: This course is an introduction to the principles of microeconomics. It studies the decisions of individual households and firms. It also analyses how individual markets operate.

Macroeconomics: This is an introductory course in macroeconomics. The interrelationship of the various actors in the aggregate economy is studied. Also, the derivation of certain national aggregates is studied using simple models of the macroeconomy.

CS103 Computer Science

Introduction to programming: Programming in a high level language (such as C), algorithms, variables, expressions, syntax, implementation of programs on machines, loops, procedures, function, modular programming, recursion, introduction to systems software, compilers, batch and on-line processing modes.

MA110 Statistics & Probability

Explanation of statistics through practical examples of its applications.

Data summarisation and presentation: Numerical measures of location and spread for both ungrouped and grouped data; graphical methods including histograms, stem-and-leaf and box plots.

Probability: The role of probability theory in modelling random phenomena and in statistical decision making; sample spaces and events; some basic probability formulae; conditional probability and independence; Bayes formula; counting techniques; discrete and continuous random variables; hypergeometric and binomial distributions; normal distributions; the distribution of the sample mean when sampling from a normal distribution; the Central Limit Theorem with applications including normal approximations to binomial distributions.

Statistical Inference: Concepts of point and interval estimation; concepts in hypothesis testing including Type I and Type II errors and power; confidence intervals and hypothesis tests about a single population mean, a single population proportion, the difference between two population means, a single population variance and the ratio of two population variances; the analysis of enumerative data, including chi-squared goodness-of-fit and contingency table tests; correlation and linear regression analysis, including least squares estimation of the parameters of the simple linear regression model, inferences about these parameters, and prediction.

MA111 Mathematics of Finance I

Simple and compound interest, annuities certain and variable, perpetuities, amortisation schedules, sinking funds.

MP191 Mathematical Methods I

Linear differential and difference equations. Applications to the modelling of population, economic, mechanical and financial systems. Introduction to the phase plane method.

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Second Year

MA286 Functions of Several Real Variables

Revision of the main definitions and properties of sequences and series of real numbers. Lim inf and lim sup, power series, differentiation term by term, Abel?s limit theorem, Taylor series. Continuity and differentiability of a function f : R^m® Rⁿ, partial derivatives, directional derivatives, the Chain rule, the mean-value theorems, maxima and minima, the inverse function theorem.

MA284 Discrete Mathematics

Enumeration: product rule, sum rule and sieve principle, selections and distributions, pigeonhole principle. Graphs, the fundamentals (including various notions of ?path? and ?tree?), plus a study of some of the following topics: colouring problems, bipartite graphs, Hamiltonian graphs, planar graphs and tournaments. Algorithms and applications are emphasised throughout.

EC215 Microeconomics:

Review of microeconomic aspects of the first year course in introductory economics and in greater detail the following: Demand analysis: Individual consumer behaviour, market demand, cross demand, elasticity. Utility approach: indifference curve analysis. Production functions, cost of production, isoquants, applications of supply and demand analysis. Market structure: purely competitive market, market equilibrium, the theory of the firm, monopoly pricing and output decisions under monopoly and under perfect competition, imperfect markets, monopolistic competition. Income distribution: factor markets and determination of factor prices. General equilibrium. Welfare economics.

MA227 Probability

Probability spaces; random variables and vectors, their distributions and moments; functions of random variables; sampling distributions; limit theorems.

CS204 Algorithms

Theory of computation: Turing machines, complexity, computability, decidability.

Design and analysis of algorithms: set operations, tables, stacks, queues, trees, searching and sorting, file organisation.

MP291 Mathematical Methods II

Ordinary differential equations including the Laplace transform. Phase plane analysis. Linear stability theory. Fourier series. Introduction to partial differential equations. Optimisation with constraints including the Lagrange multiplier method.

MA287 Functions of One Complex Variable

Functions of a complex variable: differentiability, the Cauchy-Riemann equations, harmonic conjugates, line integrals, log z and e^z, Cauchy?s integral theorem, Cauchy?s formula. Cauchy?s inequalities, the Laurent series of a function, poles, residues, contour integration, Rouche?s theorem.

MA283 Linear Algebra

Vector spaces, bases, dimension, linear maps, matrix representation of linear maps, matrix algebra, kernels and images, least squares fitting, inner product spaces, the Gram-Schmidt process, Fourier series, dual spaces, the rank of a matrix, determinants, eigenvalues and eigenvectors, the characteristic polynomial, quadratic forms, diagonalisation of a symmetric or Hermitian liner map, triangularisation of a linear map, the Hamilton-Cayley theorem, linear programming.

EC217 Macroeconomics

Basic concepts of national income. Output and expenditure. Aggregate supply and demand. Income and output. Equilibrium and disequilibrium. Saving-investment relationship. Consumption function. Factors influencing consumption demand. The multiplier. The determinants of investment. Liquidity preference and theory of interest. The principle of acceleration. The Government sector and National income and output. Foreign trade and the national income. Terms of trade. Balance of payments. Exchange rates. Incomes, output, employment, prices. The classical theory. Keynesian theories. Money. Supply and demand. General price level. Index numbers. The inflationary process. Economic growth, investment and employment. Cyclical fluctuations. Monetary and fiscal policies.

EC218 Mathematical Economics

Applications of mathematical methods in constructing and analysing economic models, with an emphasis on methods of constrained optimisation. Topics may include comparative static analysis, economic dynamics and game-theoretic methods in economics.

MA228 Statistical Inference

Concepts and criteria in point and interval estimation and in hypothesis testing; applications to one- and two-sample problems involving quantitative variables, enumerative data analysis, and regression.

CS205 Languages and Operating Systems

Operating Systems: Introduction to VMS, UNIX, MSDOS. Database management systems: architecture of DBMS, data sublanguages, commercially available DBMSs.

Study of programming languages.

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Third Year

MA341 Metric Spaces

Metric spaces: continuity, convergence, compactness, sequential compactness, the Lebesque number of a covering completeness, Cantor's theorem, Baire's theorem.

MA343 Groups I

Group axioms, cyclic groups, permutation groups. Normal subgroups, homomorphisms, isomorphism theorems. Direct products, finite Abelian groups. Automorphisms, groups of automorphisms.

EC302 Advanced Microeconomics

This course builds on the foundation of the second year course in microeconomics. A series of topics will be analysed of a more technical nature than previously encountered. Students will be introduced to some of the latest development in microeconomic theory.

EC311 Economics of Financial Markets

The objective of the course is to appraise students of the concepts of financial markets (bond, equity and foreign exchange) and financial instruments as well as interest rate determination for various financial instruments. In addition, economic theories of short term and long term interest rate determination will be covered.

MA313 Applied Statistics

Sampling techniques; various designs and analyses for survey data; applied regression analyses; simple linear regression, multiple linear regression, diagnostics and remedial measures for violations of assumptions; experimental design and analysis of variance; principles of experimental designs, completely randomised, randomised block and Latin square designs and their analyses; use of computer software.

MA311 Annuities and Life Insurance

Elements of probability and applications to life contingencies; force of mortality; construction of mortality tables; types of life annuities and life insurance; commutation functions; net premiums.

MP391 Mathematical Modelling

A selection of diverse areas are modelled mathematically and computationally, including population models, network maximal flow/minimal cost models, diffusion processes, linear and non-linear programming, queuing theory via Markov processes, random number generators and Monte Carlo methods, Steiner tree problems, continuous and discrete applications to financial and economic models.

MG Industrial Management ------ (Semester II also)

Management and organisation theory. Marketing, personnel and financial functions, financial aspects of production, statement of account, working capital, cost accounting, budgets, budgetary control, standard costs, variance analysis, performance analysis, financial decision making, management and modelling, industrial relations. The cultural context of industrial relations, 19^th century developments, the 1906 Trade Disputes Acts, picketing, trade union organisation, the ICTU, employer organisations, the Labour Court and its functioning, the Employer Labour Conference, Rights Commissioners, substantive and procedural agreements, unofficial action, inter-union disputes, recognition, the law and industrial relations, diagnosis and proposals of the Commission of Inquiry on Industrial Relations.

MA342 Topology

Topology: Continuity and homeomorphism, subspaces, bases and sub-bases, product spaces, compactness, connected and path-connected spaces, paths in a topological space, the fundamental group of the circle.

MA344 Groups II

Group actions, automorphism groups of graphs, applications to enumeration. Sylow's Theorems, groups of small order, simple groups. Frattini subgroup. Semi-groups, machines.

EC304 Advanced Macroeconomics

This builds on previous macroeconomics courses and provides a rigorous survey of selected key themes in modern macroeconomics, which may include growth and technological change, business cycles, interactions between the real and monetary sectors of the economy and institutional aspects of monetary and fiscal policy.

EC321 Money and Banking

The objective of the course is to discuss the significance of financial intermediaries in modern financial structures and the issues arising from bank regulation and deregulation. In addition, theories of money supply, money demand and the impact of monetary policy on economic activity and inflation will be discussed.

MA310 Actuarial Mathematics I

Net premium reserves; multiple life functions; multiple decrement models; valuation theory for pension plans.

EC323 Advanced Econometrics

Building on previous statistcs courses in this programme, this course introduces students to the concepts and problems of multicollinearity, heteroscedasticity, autocorrelation and model specification in the context of applied economic modelling. Other topics may include simultaneous equation models and dummy variables.

(See also MG?Industrial Management outline above)
 

(In addition, one optional course from list below in Semester II)

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Fourth Year

MA416 Rings

Commutative rings: ideals, prime ideals, maximal ideals, integral domains, unique factorisation domains, principal ideal domains, Euclidean domains, polynomial rings, rings of fractions with respect to a multiplicative set.

MA490 Measure Theory

The Lebesgue integral: the deficiencies of the Riemann integral, Lebesgue measure, measurable functions, the Lebesgue integral. Convergence theorems, functions of bounded variatiton and absolutely continuous functions, Vitali?s Covering Theorem, integration and differentiation. General measure and integration theory: outer measures, measures, measurable functions, modes of convergence.

MA418 Differential Equations with Financial Derivatives

Introduction to Continuous Stochastic Processes. General probability spaces and information structures. The Wiener process. Stochastic processes as solutions of Stochastic Differential Equations. Ito process, Ito?s lemmas an anlogue of the chain rule. Application to the Block-Scholes Model. Derivation of the Block-Scholes Partial Differential Equation.

MP491 Non Linear Systems

Non-linear stability and phase plane methods. Chaotic dynamical systems, bifurcation theory, period doubling and strange attractors. Models with chaotic behaviour in mechanics, economics and financial systems are studied mathematically and computationally.

EC410/1 Seminar in Economics of Financial Markets I and II

The aim of this course, over both Semesters, is to provide an opportunity for students to integrate the diverse material in other courses in the context of developments in financial markets and institutions and related policy debates. This may in particular involve further explorations in the literature of the economics of financial markets, and contributions from a number of sources, including financial market participants. These seminars may also provide a platform from which the project/minor theses which students undertake in this programme will be advanced.

MA(?) Actuarial Mathematics II

Insurance models and economics of insurance; nonforfeiture benefits; dividends; risk models, independent increment processes; Markov processes.

CS421/3 Neural Networks I and II

Training and optimization. Perceptrons, linear and sigmoid neurons. Single and multi layer networks. Various architectures. Model Identification Non-linear time series, parameter estimation and predictions.

(In addition, one optional course from list below, as well as project/minor thesis work).

MA491 Fields and Codes

Finite field extensions, splitting fields of polynomials, normal extension, separable extensions, Galois extensions, the fundamental theorem of Galois Theory. Solubility of equations by radicals, ruler and compass constructions. Coding theory, polynomial codes, Hamming codes.

MA494 Stochastic Processes

Discrete Models of Financial Markets. Information structures, trading strategies. Completeness of markets. Adapted processes, conditional expectations, martingales. Discrete versions of the stochastic integral, Ito?s lemma, Girsanov?s theorem. Application to option pricing models.

MA Actuarial Mathematics III

Premium calculations; retentions and reserves; stability; dividend policy; utility; applications of risk theory.

EC320 International Monetary Economics

This course aims to introduce students to international economic transactions, as summarised in the balance of payments. Having taken the course, students should be in a position to follow contemporary discussion of exchange rates, the current account, and the overall balance of payments, and to appreciate the impact of policy changes on such variables.

(In addition, project/minor thesis work).

See also entries above for CS421/3 Neural Networks I and II and for EC410/1 Seminar in Economics of Financial Markets I and II.

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One optional course (two hours per week, 4 credits) is to be taken in semester two of Third Year and in semester one of Fourth Year. Options must be chosen in consultation with the relevant Departments. Not all options may be available in any year and additions may be made to this list.

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