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.