### 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