Introduction to stochastic analysis 1000-135WAS
1. Elements of the general theory of stochastic processes: Examples of stochastic processes, Wiener process. Processes as measures on the trajectory space.
Finite dimensional distributions. Equivalence of processes. Kolmogorov's consistency theorem (without proof). Regularity of trajectories.
2. Continuos-time martingales: Stopping times with respect to continuos-time filtrations. Optional sampling theorem and Doob's inequalities. Almost sure convergence of martingales. Uniform integrability and convergence of martingales in L^p. Regularity of martingale trajectories. The Doob-Meyer decomposition (without proof).
3. Stochastic integrals: Definition and basic properties of the Stieltjes integral (without proofs). Ito's integral with respect to Wiener process. Stochastic integrals with respect to continuous martingales.
4. Ito's formula: Quadratic variation process for stochastic integral. Integration by parts for continuous semimartingales. Ito's formula and its multidimensional version. Levy's characterization of Wiener process.
5. Stochastic differential equations: Strong solutions of SDE. Existence and uniqueness of solutions for Lipschitz coefficients. Connections with partial differential equations. Linear stochastic equations. Feynmann-Kac formula.
6. Girsanov's theorem.
Type of course
Prerequisites
Prerequisites (description)
Course coordinators
Term 2024Z: | Term 2023Z: |
Learning outcomes
The student knows:
1. basic properties of the Wiener process and martingales in continuous time.
2. the notion of local martingale and its quadratic variation, the Doob-Meyer theorem,
3. the definition of Itô integral over the Wiener process and continuous semimartingales and their basic properties; the Itô formula and can determine quadratic variation of a stochastic integral,
4. the theorem on existence and uniqueness of strong solutions to stochastic differential equations with lipschitz coefficients; can solve linear stochastic equations
Assessment criteria
Final mark is based on the total sum of points gained in colloquiums and examination.
Bibliography
1. I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus. Springer-Verlag 1997.
2. D. Revuz, M. Yor, Continuous Martingales and Brownian Motion. Springer-Verlag 1999.
3. A.D. Wentzell, Wykłady z teorii procesów stochastycznych. PWN 1980
4.P. Protter, Stochastic integration and differential equations., Springer-Verlag 1995.
5. 5. R. Latała, Wstęp do Analizy Stochastycznej, (in Polish) https://www.mimuw.edu.pl/~rlatala/testy/proc/was.pdf
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
- Bachelor's degree, first cycle programme, Mathematics
- Master's degree, second cycle programme, Mathematics
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: