Integro-differential equations, Markov processes and backward stochastic differential equations 7404-PMPF

1. Probabilistic interpretation of solutions of the Cauchy problem, Dirichlet problem and Cauchy-Dirichlet problem for linear equations involving the Laplace operator. Information on the Neumann problem. The Feynman-Kac formula.

2. Backward stochastic differential equations. Exsistence and uniqueness of solutions for square-integrable data.

3. Markov-type backward stochastic differential equations. Probabilistic representation of viscosity solutions of semilinear equations with nondivergence form. Mild solutions. Nonlinear Feynman-Kac formula.

4. Probabilistic representation of weak solutions of semilinear equations in divergence form and some equations involving integro-differential operators of L'evy type.

Total student workload

30 hours - lecture 30 hours - preparation for lectures and study of literature 30 hours - preparation for exam Total: 90 hours

Learning outcomes - knowledge

PhD student can formulate stochastic representation of Feynman-Kac type of basic initial-boundary value problems for partial differential equations of second order. Knows basic theorems on existence and uniqueness of solutions of backward stochastic differential equations and can give stochastic representation, in terms of such solutions, of the Cauchy problem for semilinear equations with local operators. Student is aware of new mathematical problems arising in the study by probabilistic methods of equations with nonlocal operators (P8S_UW)

Learning outcomes - skills

PhD student is able: 1) indicate problems in partial differential equations which can be effectively studied by probabilistic methods; 2) indicate, in some specific problems, advantages and limitations of probabilistic methods. (P8S_UW)

Teaching methods

lecture

Expository teaching methods

- informative (conventional) lecture

Prerequisites

Basic knowledge of stochastic analysis (continuous time martingales, stochastic integral, Ito's formula, stochastic differential equations).

Course coordinators

Andrzej Rozkosz

Assessment criteria

Oral exam

Bibliography

I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus. Springer, New York, 1988.

E. Pardoux, Backward Stochastic Differential Equations and Viscosity Solutions of Systems of Semilinear Parabolic and Elliptic PDEs of Second Order. In: Stochastic Analysis and Related Topics VI, The Geilo Workshop 1996, pp. 79--127, L. Decreusefond, J. Gjerde, B. Oksendal, A.S. Ustunel (Eds.), Birkhuser, Boston 1998.

Additional information

Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system:

Description of 7404-PMPF in USOSweb