Download Questions Stochastic Control with Financial Applications

Questions
Stochastic Control with Financial Applications
2016
Sören Christensen
The aim of the following questions is that you have a possibility to check yourself.
1. What is a general Markov process and what is a real-world interpretation for the axioms?
2. If W is a 1-dimensional Brownian motion and Mt := sups≤t Ws . Is M a Markov process
then? Can you extend the state space so that M becomes Markovian? Can you give
another example of a process that is not Markovian.
3. Explain why one can expect the Markov property to be helpful for studying stochastic
control problems.
4. What is the generator of a Feller process?
5. What is the generator of a Brownian motion with drift?
6. What is a problem of optimal stopping and why is it useful to have the Markov property
of the underlying process?
7. Why is the value function always a majorant of the reward function?
8. Why is a superharmonic majorant of g always a majorant of v?
9. Describe the relationship between smallest superharmonic majorant and value function
in detail.
10. Give an example of on optimal stopping problem without an optimal stopping time.
11. Summarize all important proofs without going too much into detail.
12. What is the intuitive reason for first hitting times of the optimal stopping set to be
optimal?
13. How can one reduce problems with a finite time horizon and with discounting to the
situation we discussed?
14. In some problem, the reward process Y Ris not adapted to the filtration, for example,
t+h
in many applications it is the form Yt = t f (Xs )ds for some fixed h > 0, a Markov
process X and a function f . How can we reduce the situation to a problem with adapted
reward process?
15. Solve the optimal stopping problem for an absorbed Brownian motion for some reward
function g.
16. Solve an example for a problem with discounting, e.g. X=SBM, g(x) = |x| using the
concave majorant-approach.
17. Solve the same example using a free boundary approach.
18. Why is it not surprising that the American put problem with a finite time horizon is
much more difficult to solve than the problem with an infinite time horizon?
19. Describe the approach for finite time horizon stopping problem in own words.