After successful completion of the course, students are able to ...

• Explain the different notions of a solution of a stochastic differential equation (SDE) and illustrate the basic effects by examples.
• Derive the solution in the linear case and discuss its uniqueness.
• Define the Ornstein–Uhlenbeck process and derive its basic properties.
• Apply the extended Grönwall inequality and illustrate the necessity of some of its assumptions by corresponding counterexamples.
• Investigate SDEs for existence and uniqueness under Lipschitz and boundedness conditions, and estimate the moments of the solution.
• Explain, why continuous local martingale with given covariance process can be represented as a stochastic integral with respect to Brownian motion.
• Explain the connection between a weak solution of a SDE and a solution of the local martingale problem.
• Introduce the Prokhorov metric, illustrate it with simple examples, and relate it to the coupling of random variables and convergence in probability.
• Reflect and explain the theory as well as to apply the learned skills in practice.
• Explain the selected topics and the corresponding main results.

Stochastic differential equations (examples, terminology, solution in the linear case), Ornstein–Uhlenbeck process, extended Grönwall inequality, existence and uniqueness of strong solutions under Lipschitz and boundedness conditions, moment estimates, representation of continuous local martingales with given covariation process, local martingale problem, Prokhorov metric. Student's choice of selected topics: (1) Renewal of Lévy processes after a stopping time, (2) Burkholder–Davis–Gundy inequalities, (3) Random time changes and the Dambis–Dubins–Schwarz theorem, (4) Permutation-invariant events and the Hewitt–Savage zero–one law, (5) Doob's backward martingale convergence theorem (6) Martingale structure and strong consistence of U-statistics, (7) Joint distribution of standard Brownian motion and its supremum, (8) Arcsin law for the last time standard Brownian motion attains its supremum, (9) Sequentially relatively compact sets and the Arzelà–Ascoli theorem, (10) (Local) martingales arising via Vandemonde's determinant

The basic contents and concepts are presented by the head of the LVA and illustrated and discussed with the help of examples. Selected topics are presented by the participants, based on the lecture notes of the course.

The performance is assessed by an examination at the end of the semester.
See: https://fam.tuwien.ac.at/lehre/pr/

• Bernt Øksendal: Stochastic Differential Equations: An Introduction with Applications. 6. Edition, Springer-Verlag, 2007, ISBN 978-3-54004-758-2.
• Daniel Revuz and Marc Yor: Continuous Martingales and Brownian Motion, 3. Edition, Springer-Verlag, 1999, ISBN 3-540-64325-7.
• Olav Kallenberg: Foundations of Modern Probability. 2. Edition, Springer-Verlag, 2002, ISBN 0-387-953113-2.

• Ioannis Karatzas und Steven E. Shreve: Brownian Motion and Stochastic Calculus. 2. Edition, Springer-Verlag, ISBN 0-38797-655-8.