Students will carry out a research project in groups of no more than four, under the close supervision of one or more faculty members. Each student should have a distinct, well-defined role in the project. The group will report its findings in the form of a written thesis, and each student in the group will be required to make an oral presentation on their role in the work.

Stochastic processes: definition and classification; modeling with stochastic processes, Markov chains and applications; theoretical aspects  of stochastic simulation; counting processes and applications; queues  and applications; practical aspects of stochastic simulation.

Partial differential equations and classifications; well-posed problems,  classical solutions, initial and boundary value problems; first order linear  and quasi-linear partial differential equations; method of characteristics; conservation laws; classification of general second order operators; the wave equation, D’Alembert method of solution; Laplace’s equation, the maximum principle; the heat equation; separation of variables; boundary  value problems and Sturm-Liouville theory.

Metric spaces, examples; continuity; completeness; topological spaces;  Hausdorff spaces; connectedness.

Review of Macroeconomics; Characteristics of the various types of  investments used to fund financial security programmes; traditional  techniques of financial analysis used in selecting and managing

 nvestment portfolios.

 The course builds on the material in courses MS28D and MS38H,  introducing further tools and techniques of asset/liability management, general product design, as well as issues of pricing and valuation and  asset management.

Review of earlier statistical work; individual risk theory; other frequency distributors; mixed distributions; stoploss insurance; ruin theory.

Multiple life functions, multiple decrement model; insurance models  including expenses; nonforfeiture, benefits and dividends; valuation theory for pension plans.

Differentiablity, analyticity; contour integrals, Cauchy's theorem and its consequences; Taylor series, Laurent series; residue calculus.

Prime numbers; unique Factorization in Z and k[x]; arithmetic functions, m, d, w and lattice points; congruence; Chinese remainder theorem; quadratic reciprocity law; algebraic numbers and algebraic integers;  transcendental numbers; finite fields; diophantine equations; distribution of prime numbers; Chybeshev theorem; the Riemann-Zeta Function.

First order differential equations, separable and homogeneous types; Pfaffian forms in two variables, Bernoulli and Riccati types; existence anduniqueness theorem for the initial-value problem.



 Higher order equations, theory of the Wronskian and linear independence  of solutions of higher order linear equations, the Euler equation.

 First order linear systems, matrix formulation of first order systems for  both normal and defective matrices, funamental matrices, matrix-valued  functions and computation of the exponential of a matrix.