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.
I) BACKGROUND: sigma-algebras, filtrations, conditional expectation, martingale property, variations of a function, quadratic variation.
II) RANDOM WALK: random walk, re-scaled 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.
|I. Karatzas and S. Shreve||Brownian motion and stochastic calculus|
|D. Revuz and M. Yor||Continuous martingales and Brownian motion|
|M.Yor et al||Exponential Functionals of Brownian Motion and related Processes||Springer||2010|
|D. Williams||Probability with martingales|
|B. Øksendal||Stochastic Differential Equations|
|N. Ikeda and S. Watanabe||Stochastic Differential Equations and Diffusion Processes|
|P. Protter||Stochastic integration and differential equations|
Oral exam with written exercises:
the exam is based on open-ended questions as well as on the discussion of written exercises to be carried out during the test itself. Open questions and exercises are aimed at verifying the knowledge related to the topics developed in the course program, hence concerning problems of both the basic theory of stochastic processes, and of stochastic differential equations.