Mini-course on optimal stopping of diffusions and Lévy processes
Time:14:00:21/03/2016 to 16:30:23/03/2016
Venue/Location: C2-714, Viện Nghiên cứu cao cấp về Toán.
Invited Speakers: Prof. Paavo Salminen – Abo Akademi University, Finland.
Content:Let $X=(X_t)_{t\geq 0}$ be a continuous time, real-valued, strong Markov process and $\mathcal{F}=(\mathcal{F}_t)_{t\geq 0}$ the natural filtration generated by X. Given a non-negative smooth reward function G the optimal stopping problem is to find a stopping time $\t^*$ such that
$\sup_{\tau\in\mathcal{M}}\mathbb{E}_x(G(X_{\t^*}))=:V(x),$
where M is the set of all stopping times with respect to the filtration $\mathcal{F}$ and $\mathbb{E}_x$ enotes the expectation associated with X when initated at x. The function V is called the value function and τ ∗ an optimal stopping time of the problem. Important applications of the theory of optimal stopping are in sequential inference (statistics) and pricing of American options (financial mathematics).
In the first part of the course (2 lectures) we
+ discuss the concept of excessive function,
+ show that the value function is the smallest excessive majorant (Snell envelope) of the reward function,
+ study via examples two verification theorems for finding the solution when the underlying is a diffusion; the first one is based on the principle of smooth fit and the second on the representation theory of excessive functions.
References [3] – [6].
In the second part of the course (2 lectures) we focus on some recent developments and
+ analyze optimal stopping of diffusions which possess skew points,
+ present a verification theorem utlizing the representation of excessive functions as expected suprema; this approach is applicable for both diffusions and L´evy processes, in fact, even for fairly general Hunt processes.
References [1] and [2].
Download Lecture here