Professor of Mathematics

Imperial College London

Department of Mathematics

180 Queen's Gate

London SW7 2AZ, UK

d.crisan@imperial.ac.uk

0207 594 8489

The course offers a bespoke introduction to stochastic calculus required to cover the classical theoretical results of nonlinear filtering as well as some modern numerical methods for solving the filtering problem. The first part of the course will equip the students with the necessary knowledge (e.g., It\^o calculus, stochastic integration by parts, Girsanov's theorem) and skills (solving linear stochastic differential equation, analysing continuous martingales, etc.) to handle a variety of applications. The focus will be on the application of stochastic calculus to the theory and numerical solution of nonlinear filtering.

- Martingales in Continuous Time (Doob Meyer decomposition, $L_p$ bounds, Brownian motion, exponential martingales, semi-martingales, local martingales)
- Stochastic Calculus (It\^o's isometry, chain rule, integration by parts)
- Stochastic Differential Equations (well posedness, linear SDEs, the Ornstein-Uhlenbeck process, Girsanov's Theorem, Novikov's condition)
- Stochastic Filtering (definition, mathematical model for the signal process and the observation process)
- The Filtering Equations (well-posedness, the Kalman-Bucy filter)
- The innovation process

This course is intended to develop and reinforce knowledge and intuition towards SDEs via numerical approaches to their construction.

- Brownian motion: Binomial, incremental and dyadic approximation to Brownian motion
- Stochastic differential equations: Existence and uniqueness of solutions
- Stochastic time discrete approximations: The Euler approximation. Pathwise approximations, 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.
- Explicit and implicit approximations: Explicit order 1.0 schemes. Implicit strong Runge-Kutta schemes. Explicit order 2.0 weak schemes.
- Appendix : Random number generators. Statistical tests and estimation.

The course offers a bespoke introduction to stochastic calculus required to cover the classical theoretical results of nonlinear filtering as well as some modern numerical methods for solving the filtering problem. The first part of the course will equip the students with the necessary knowledge (e.g., It^o calculus, stochastic integration by parts, Girsanov's theorem) and skills (solving linear stochastic differential equation, analysing continuous martingales, etc.) to handle a variety of applications. The focus will be on the application of stochastic calculus to the theory and numerical solution of nonlinear filtering.

•Martingales on Continuous Time (Doob Meyer decomposition, Lp bounds, Brownian motion, exponential martingales, semi-martingales, local martingales)

•Stochastic Calculus (It^o's isometry, chain rule, integration by parts)

•Stochastic Differential Equations (well posedness, linear SDEs, the Ornstein-Uhlenbeck process, Girsanov's Theorem,Novikov's condition)

•Stochastic Filtering (definition, mathematical model for the signal process and the observation process)

•The Filtering Equations (well-posedness, the innovation process, the Kalman-Bucy filter)

•Numerical Methods (the Extended Kalman-filter, Sequential Monte-Carlo methods)

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)

•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.

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. <\p>

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.

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.

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.

Modules:

•Poisson Random Measures

•Renewal Processes.

•Queues

Modules:

•Martingales and Brownian Motion

•Stochastic Integration

•Stochastic Differential Equations

•Stochastic Filtering/p>

•Stochastic Finance