After successful completion of the course, students are able to

• explain and apply basic facts from stochastic analysis such as Ito Integral, Ito Formula and stochastic differential equations,

• apply the dynamic programming principle,

• proof the Hamilton-Jacobi-Bellman equation and verfication theorems,

• prove the existence of the local time of the Brownian movement and to solve singular control problems,

• solve problems from financial and insurance mathematics such as optimal investments,

• minimize ruin probabilities,

• apply the martingale method in stochastic optimization and

• motivate Viscosity Solutions.

a short revision of the elements of stochastic analysis as e.g. Ito Integral, Ito's formula and stochastic differential equations; dynamic programming principle, Hamilton-Jacobi-Bellman equation; verification theorems; local time of Brownian motion and singular control theory; examples from actuarial and financial mathematics as e.g. optimal investment problems, minimizing ruin probabilities, etc.; martingal methods in stochastic optimization; introduction into the theory of viscosity solutions

presentation about the theoretical foundations of the mentioned chapters. Moreover, presentations by the students.

oral exam and talk