Professor of Mathematics
Imperial College London
Department of Mathematics
180 Queen's Gate
London SW7 2AZ, UK
0207 594 8489
•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)2014 Doodle link for the course New Doodle 2014 Doodle link revision class 1 may 2014
•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.
•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.
•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.
•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.
•Poisson Random Measures
•Martingales and Brownian Motion
•Stochastic Differential Equations
•Discrete Linear Filtering
•Continuous Linear Filtering: Kalman-Bucy Filter
•Finite Dimensional Filters: Benes Filter
•Lie Algebra Connections