The lecture covers the following topics:

Stochastic optimization problems
Optimal stopping, classical stochastic control, singular control, impulse control.

Stochastic analysis
Martingale theory, stochastic differential equations.

Classical stochastic control
Controlled stochastic differential equations, optimization criteria, martingale optimality principle, Bellman principle, Hamilton-Jacobi-Bellman equations, applications.

Viscosity solutions
Non-differentiable value functions, viscosity solutions via test functions, viscosity solutions via semijets, comparison theorems, Ishii’s lemma.

The stochastic Perron’s method
Comparison theorem, stochastic sub- and supersolutions, viscosity characterization of the value function.