MF0: Stochastic integrals: an introduction to Ito calculus
Lecturer: Professor Rama Cont
Course objectives:
This course is an introduction to the Ito calculus, a calculus applicable to functions of stochastic processes with irregular paths, which has many applications in finance, engineering and physics. The course shall focus on the mathematical foundations of stochastic calculus and the theory of stochastic integration, using a less conventional approach which emphasizes pathwise, rather than probabilistic, methods.
The first part of the course will focus on pathwise integration with respect to functions of finite quadratic variation, without using any probabilistic tools.
The second part of the course will explore the application of these results to stochastic integrations with respect to semimartingales, a setting which covers most examples of stochastic processes of interest in applications - including jump processes and diffusion processes.
Knowledge of measure theory, probability theory and martingales is assumed as a prerequisite.
This course is intended for:
- students enroled in the MRes in Mathematical Sciences (Stochastics & Finance stream) at Imperial College
- MRes students enroled in the EPSRC CDT in Financial Analytics and Financial Computing
- PhD students in the Dept of Mathematics, Imperial College
- PhD students in Mathematical Finance from institutions participating in the London Graduate School of Mathematical Finance
Venue
Lectures will take place in Lecture Room 3, Level 4, Centre for Doctoral Training, Imperial College located in Imperial College’s South Kensington Campus. Entrance is through the Sherfield Building, Level 2.
Lecture dates :
Tuesday Oct 14, 9:30-12 :00, Thursday Oct 16, 9:30-12:00
Tuesday Oct 21, 9:30-12 :00, Tuesday Oct 28, 9:30-12:00
Tuesday Nov 4, 9:30-12 :00, Tuesday Nov 18, 9:30-12:00, Tuesday Nov 25, 9:30-12:00
Tuesday Dec 2, 9:30-12 :00, Tuesday Dec 9, 9:30-12:00
Lecture 1: Reimann-Stieltjes integration with respect to finite variation processes
Cadlag functions. Cadlag processes. Modifications.
Stochastic processes, measurability and filtrations. Predictable and optional processes.
Functions and processes of finite variation. Jordan decomposition of a FV function.
Pathwise Riemann Stieltjes integration for processes of finite variation.
The pathwise change of variable formula for finite variation processes.
Failure of the Riemann Stieltjes integration for functions of infinite variation.
Quadratic variation with respect to a partition.
Failure of the change of variable formula for paths of non-zero quadratic variation.
Lecture 2: The Ito formula: a pathwise approach
Quadratic variation along a sequence of partitions. Quadratic Riemann sums.
A pathwise change of variable formula for functions with finite quadratic variation.
Ito, Kiyosi (1946) On a stochastic integral equation, Proc. Japan Acad. Volume 22, Number 2 (1946), 32-35.
Föllmer, Hans Calcul d'Ito sans probabilités. Séminaire de probabilités de Strasbourg, 15 (1981), p. 143-150.
MF0: Lecture 2: Slides
Lecture 3 : Simple predictable processes. Ito's stochastic integral
Simple predictable processes. Stochastic integral for simple predictable integrands.
Martingale preserving property. Uniform convergence in probability (UCP).
Semimartingales: definition, properties, examples.
Stochastic integral of a left-continuous adapted process with respect to a semimartingale.
Stochastic integrals as limits of non-anticipative Riemann sums.
Lecture 4: Ito Processes. Quadratic Variation
Brownian stochastic integrals. L2 extension. Ito isometry formula.
Ito processes.
Quadratic variation of a semimartingale: definition, properties, decomposition.
Quadratic covariation. Associativity formula. Kunita-Watanabe inequality.
Lecture 5 : The Ito formula: a pathwise approach
Quadratic variation along a sequence of partitions. Quadratic Riemann sums.
A pathwise change of variable formula.
The Ito formula for semimartingales.
Applications of the Ito formula. Stochastic exponentials.
MF0: Lecture 5: Slides
References:
Ito, Kiyosi (1946) On a stochastic integral equation, Proc. Japan Acad. Volume 22, Number 2 (1946), 32-35.
Föllmer, Hans Calcul d'Ito sans probabilit&a mp;a mp;a mp;a mp;e acute;s. Séminaire de probabilités de Strasbourg, 15 (1981), p. 143-150.
Lecture 6 : Applications of the Ito formula
Levy's theorem.
Watanabe's characterization of the Poisson process. Intensity of a point process.
Representation of Brownian martingales as stochastic integrals.
Continuous martingales as time-changed Brownian motion: Dubins-Schwarz theorem.
Stochastic exponential of a martingale. Criteria for the martingale property of a stochastic exponential.
MF0: Lecture 6: Slides
Lecture 7: Change of Measure: The Girsanov-Meyer theorem
Equivalent measures and densities.
Change of measure on a filtered probability space.
Exponential martingales.
The Girsanov-Meyer theorem. Applications: Cameron-Martin theorem, change of measure for point processes.
Lecture 8: Poisson random measures and Levy-Ito processes
Integer-valued random measures.
Poisson random measures: definition and construction. Exponential formula.
Integration with respect to a Poisson random measures.
Compensated Poisson integrals: martingale property and L2 extension.
Lévy processes. Ito-Lévy processes.