Current courses:

M2PM1 Analysis II (2010 - Imperial College, London)

Modules:

· Differentiable functions. Linearity of derivatives.

· Maxima and strict maxima. Rolle’s theorem.

· Mean value theorem. Higher derivatives. Taylor’s theorem.

· Riemann integration. Integrability of continuous and monotonic functions.

· The fundamental theorem of calculus.

· Euclidian metric. Open, closed and compact sets.

· Elementary measure theory.

· Functions of many variables. 

Exercise sheets and solutions

Stochastic Filtering (2009 - Imperial College, London)

Modules:

· Preliminaries: Conditional Expectation, Brownian motion, Ito integral, Solutions of SDEs, Girsanov's Theorem

· The Filtering Framework

· The Filtering Equations: The change of probability measure approach, The Innovation Process

· Uniqueness of the solutions to the Zakai and Kushner-Stratonovitch Equations.

· Finite Dimesional Filters. The Kalman Bucy Filter. The Benes Filter.

· Numerical methods for solving the filtering problem.

· Particle filters (Sequential Monte Carlo Methods)

Numerical Stochastics (2001 - Imperial College, London)

Modules:

· Preliminaries: Random number generators. Statistical tests and estimation Brownian motion. Binomial, incremental and dyadic approximations to Brownian motion.

· Stochastic time discrete approximation: The Euler approximation. Pathwise approximations, Approximations of moments. Strong convergence and consistency. Weak convergence and consistency. Numerical stability.

· Strong Approximations: Strong Taylor Approximations. The Euler Scheme. The Milstein Scheme. Higher order schemes.

· Weak Approximations: The Euler Scheme, Leading Error Coefficients (Talay Tubaro)

· Explicit and implicit approximations: Explicit order 1.0 schemes, Implicit strong Runge-Kutta schemes, Explicit order 2.0 weak schemes.

· Variance Reduction Methods: The Measure Trasformation Method, Variance Reduced Estimators. Particular Cases, Unbiased Estimators.

Exercises and Solutions

Past Courses:

M1P1 Analysis (2003-2009 - Imperial College, London)

Modules:

· Sequences and limits of sequences.

· Basic theorems and rules about taking limit.

· The general principle of convergence.

· Series and convergence tests.

· Rules for calculating with convergent series.

· Introduction to continuous functions.

· Exercise sheets and solutions

MSE101 Mathematics (2000-2005 - Imperial College, London)

Modules:

· Functions: definition, trigonometric, exponential and logarithmic functions; odd, even and inverse functions.

· Limits: definition, basic properties, continuous and discontinuous functions.

· Differentiation: definition and properties, implicit differentiation, higher derivatives, Leibniz's formula, stationary points and points of inflection.

· Integration: definite and indefinite integrals; the fundamental theorem, improper integrals; integration by substitution and by parts, partial fractions, applications.

· Series expansions: convergence of power series, Taylor and Maclaurin series, l'Hopital's rule, ratio and comparison tests.

· Complex numbers: definition, the complex plane, polar representation, de Moivre's theorem, ln z, exp z.

· Hyperbolic functions: definitions, inverse functions, series expansions, relations between hyperbolic functions and the trigonometric functions.

Exercise sheets and solutions

Applied Probability (Michaelmas 1998, 1999 - University of Cambridge)

Modules:

· Poisson Random Measures

· Renewal Processes

· Queues

Stochastic Calculus and Applications (Lent 1999, 2000 - University of Cambridge)

Modules:

· Martingales and Brownian Motion

· Stochastic Integration

· Stochastic Differential Equations

· Stochastic Filtering

· Stochastic Finance

Stochastic Filtering (Fall 1996, Fall 1997 - Imperial College, London)

Modules:

· Discrete Linear Filtering

· Continuous Linear Filtering: Kalman-Bucy Filter

· Non-Linear Filtering

· Finite Dimensional Filters: Benes Filter

· Lie Algebra Connections

· Numerical Algorithms

· Applications

Measure Valued Processes (Spring 1997 - Imperial College, London)

Modules:

· Convergence of Probability Measures

· The Martingale Problem. Existence and Uniqueness

· Branching Particle Systems

· Branching Markov Processes: SuperBrownian Motion

· Genetic Models