Dan Crisan

Professor of Mathematics

Imperial College London

Department of Mathematics

180 Queen's Gate

London SW7 2AZ, UK


0207 594 8489

Current Courses

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)

2014 Doodle link for the course

New Doodle

2014 Doodle link revision class 1 may 2014

Past Courses

M2PM1 Analysis II (2010 - Imperial College, London)

•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.

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.

•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.

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.

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


•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/p>

•Stochastic Finance

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.