A Poisson process Xt changes its rate at an unknown and unobservable time θ
from λ0 to λ1. Detecting the change time as quickly as possible in an optimal
way is described in literature as the Poisson disorder problem. We provide a more
realistic generalization of the disorder problem for Poisson process by introducing
fixed and continuous costs for being able to observe the arrival process. As a
result, in addition to finding the optimal alarm time, we also characterize an
optimal way of observing the arrival process. We illustrate the structure of the
solution spaces with the help of some numerical examples.

A stock price Xt evolves according to jump diffusion process with certain parameters.
An asset manager who holds a strangle option on that stock, wants
to maximize his/her expected payoff over the infinite time horizon. We derive
an optimal exercise rule for asset manager when the underlying stock is dividend
paying and non-dividend paying. We conclude that optimal stopping strategy
changes according to stock’s dividend rate. We also illustrate the solution on
numerical examples.

Suppose that a Wiener process gains a known drift rate at some unobservable
disorder time with some zero-modified exponential distribution. The process is
observed only at some intervals that we control. Beginning and end points and
the lengths of the observation intervals are controlled optimally. We pay cost for
observing the process and for switching on the observation. We show that Bayes
optimal alarm times minimizing the expected total cost of false alarms, detection
delay cost and observation costs exist. Optimal alarms may occur during the
observations or between the observation times when the odds-ratio process hits a
set. We derive the sufficient conditions for the existence of the optimal stopping
and switching rules and describe the numerical methods to calculate optimal
value function.