Stochastic differential equations (2017/2018)

Course code
4S001444
Name of lecturer
Luca Di Persio
Coordinator
Luca Di Persio
Number of ECTS credits allocated
6
MAT/06 - PROBABILITY AND STATISTICS
Language of instruction
English
Period
II sem. dal Mar 1, 2018 al Jun 15, 2018.

Learning outcomes

This course will provide an introduction to the theory of Stochastic Differential Equations (SDEs), mainly based on the Brownian motion type of noise

The purpose of this course is to introduce and analyse probability models that capture
the stochastic features of the system under study to predict the short and long term
effects that this randomness will have on the systems under consideration. The study
of probability models for continuous-time stochastic processes involves a broad range of mathematical and computational tools. This course will strike a balance between the mathematics and the applications. The main applications will be mathematical finance, biology and populations evolution, also with respect to their descriptions in terms of the associated SDEs.
Topics include: construction of Brownian motion; martingales in continuous time; stochastic integral; Ito calculus; stochastic differential equations; Girsanov theorem; martingale representation; the Feynman-Kac formula and Lévy processes.

Some references
• Introduction to Stochastic Calculus Applied to Finance by D. Lamberton and B. Lapeyre
• Diffusions, Markov Processes and Martingales by L. Rogers and D. Williams, vol 2.
• Stochastic Differential Equations and Diffusion Processes by N. Ikeda and S. Watanabe
• Stochastic differential equations, by B. Øksendal.
• Brownian motion and stochastic calculus, by I. Karatzas and S. Shreve.
• Continuous martingales and Brownian motion, by D. Revuz and M. Yor.
• Stochastic integration and differential equations, by P. Protter.
• Probability with martingales, by D. Williams.

Syllabus

Course Plan

I) BACKGROUND: sigma-algebras, filtrations, conditional expectation, martingale property, variations of a function, quadratic variation.

II) RANDOM WALK: random walk, rescaled random walk, martingale property.

III) BROWNIAN MOTION: definition of Brownian motion, function of a Brownian motion, martingale property, exponential martingale, applications in biology and finance, examples and exercises.

IIIa) SHORT INTRODUCTION TO JUMP PROCESSES: motivation, Poisson processes, properties, discrete time case, introduction to Galton Watson model.

III) WIENER INTEGRAL: motivation, case of step function, definition of Wiener integral, properties, law, martingale, quadratic variation, applications in biology and finance, examples and exercises.

IV) STOCHASTIC INTEGRALS: motivation, case of step function, definition of stochastic integral, properties, martingale, quadratic variation, variance, finite variation processes, Ito processes, applications in biology and finance, examples and exercises.

V) ITO CALCULUS: motivation, Itō-Doeblin formula for Brownian motion, Itō-Doeblin formula for function depending on time,Itō-Doeblin formula for Ito processes, applications in biology and finance, examples and exercises.

VI) SDEs: motivations, definition, existence and uniqueness result, applications in biology and finance, examples and exercises.

VII) MULTI-DIMENSIONAL CASE: multi-dim Brownian motion, correlation, multi-dim Ito-formula,
SDE, applications in biology and finance, examples and exercises.

VIII) CHANGE OF PROBABILITY: motivations, Cameron-Martin theorem, Girsanov theorem, representation of martingale theorem, applications in biology and finance, examples and exercises.

IX) FEYNMAN KAC FORMULA: motivation, Feynman Kac formula, link PDE/SDE, Monte-Carlo methods.

X) JUMPS PROCESSES: Levy processes, characterization and properties.

 Author Title Publisher Year ISBN Note I. Karatzas and S. Shreve Brownian motion and stochastic calculus D. Revuz and M. Yor Continuous martingales and Brownian motion L. Rogers and D. Williams Diffusions, Markov Processes and Martingales (Vol 2.) D. Lamberton and B. Lapeyre Introduction to Stochastic Calculus Applied to Finance D. Williams Probability with martingales S. E. Shreve Stochastic Calculus for Finance II: Continuous-Time Models Springer, New York 2004 B. Øksendal Stochastic Differential Equations N. Ikeda and S. Watanabe Stochastic Differential Equations and Diffusion Processes P. Protter Stochastic integration and differential equations

Assessment methods and criteria

Wrriten exam. An intermediate (mid-term) exam is forecasted. Then, will be a final exam focusing on the
whole course.