M2PM1 Analysis II (2010 - Imperial College, London)
∑ Differentiable functions. Linearity of derivatives.
∑ Maxima and strict maxima. Rolleís theorem.
∑ Mean value theorem. Higher derivatives. Taylorís 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.
Stochastic Filtering (2009 - Imperial College, London)
∑ 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)
∑ 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.
∑ 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.
M1P1 Analysis (2003-2009 - Imperial College, London)
∑ 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.
(2000-2005 - Imperial College,
∑ 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.
Applied Probability (Michaelmas 1998, 1999 - University of Cambridge)
∑ Poisson Random Measures
∑ Renewal Processes
Stochastic Calculus and Applications (Lent 1999, 2000 - University of Cambridge)
∑ Martingales and Brownian Motion
∑ Stochastic Integration
∑ Stochastic Differential Equations
∑ Stochastic Filtering
Stochastic Filtering (Fall 1996, Fall 1997 - Imperial College, London)
∑ Discrete Linear Filtering
∑ Continuous Linear Filtering: Kalman-Bucy Filter
∑ Non-Linear Filtering
∑ Finite Dimensional Filters: Benes Filter
∑ Lie Algebra Connections
∑ Numerical Algorithms
Measure Valued Processes (Spring 1997 - Imperial College, London)
∑ Convergence of Probability Measures
∑ The Martingale Problem. Existence and Uniqueness
∑ Branching Particle Systems
∑ Branching Markov Processes: SuperBrownian Motion
∑ Genetic Models