Stochastic Calculus with Applications to Non-Linear Filtering (KR202020) - MATH97061/MATH97172

Course aim:

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.

Modules:

• 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

Dynamical Systems (MAMCDS)

Course aim:

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

Modules:

• 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.

Introduction to Stochastic Calculus with Application to Nonlinear Filtering (M4P67)

Course aim:

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.

Modules:

•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)

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)

M2PM1 Analysis II (2010 - 2015 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 (2001-2009 - 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. <\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.

---

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.

---

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.

---

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/p>

•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.

---