Mini-course "Theory and Applications of Stochastic Differential Equations"

The aim of this mini-course is to present theoretical and practical aspects related to stochastic processes characterized by a stochastic differential equation (or SDE in short). The first part of the course will be devoted to a short recall of some basic notions of stochastic calculus (from the definition of the Brownian motion to the Itô integral). In the second part, we will present the general context and interest of a SDE- with a particular focus on Itô's diffusion processes-, and review simple applications arising in Physics and Finance.

The third part of the course will be dedicated to the general theory of SDEs addressing: the notions of strong and weak solutions; sufficient conditions ensuring existence and uniqueness of a solution; the Markovian nature and some characteristic properties of a solution; the formulation of a SDE in terms of martingale problems; and the links between stochastic differential equations and partial differential equations.

Finally, we will discuss numerical methods to approximate the solution of a SDE, and some particular class of SDEs, such as nonlinear SDEs of McKean-Vlasov type and their recent applications in statistical mechanics and economic modelling. 

Program

I           Fundamental notions of stochastic processes and stochastic calculus.

·        Some basic definitions.

·        The Brownian motion.

·        Ito integral and Ito formula.

II          Preliminaries on Stochastic Differential Equations

·        Generic form and basic properties.

·        Ordinary Differential Equations and Stochastic Differential Equations.

·        Ito diffusion processes and link with Markov processes.

III         Theory of Stochastic Differential Equations and applications

• The principle of causality and the notion of strong solution to a SDE.
• Construction and uniqueness and properties of a strong solution.
• The notion of weak solutions and its link with strong solutions.
• Examples in Physics, in Finance and some applications.
• The martingale problems related to a SDE.
• Link between PDE and SDEs.
• Density estimates and strong uniqueness results for singular SDEs.

IV        SDEs of McKean-Vlasov type and their applications.

• Historical background and link with nonlinear pdes and propagation of chaos.
• Some well-posedness results.
• Applications in Physics and recent applications in Economy.

Schedule of mini-course:
16th December - 18:00 - 21:00 room 2220
19th December - 18:00 - 21:00 room 4222
21th December - 18:00 - 21:00 room 4223         

Course is open to anyone interested. Whether you need the pass to the Higher School of Economics, do not hesitate  to contact the lab manager Julia Pavlyuk ypavlyuk@hse.ru