This course address problems of optimal stopping, presenting one of the most used methodology: the so-called Hamilton-Jacobi-Bellman equations (HJB, for short). We focus, especially, in some financial applications like, for example, finance options and real options.
Optimal stopping problems are free-boundary problems, and thus one needs to invoke a verification theorem to prove that a solution of the HJB equation is also solution of the optimisation problem. We use the mathematics presented along the course to solve the problem of derivation of the optimal time for a firm to invest in a new project or market, in the sense that the firm maximizes its value.
At the end of this course the participants will be able to:
- solve optimal stopping problems with terminal costs (like the example of investment problems and american options);
- use tools to solve other optimal stopping problems involving integral profits (like the exit problem);
- work, with success, the stochastic optimisation problems.
The course will cover the following subjects:
- Definition of the value function;
- Some stochastic calculus concepts (like Brownian motion, stopping times, Itô's formula);
- HJB equations;
- Verification theorem;
- Comparative statics.