# Reference request for Optimal Stopping (Stochastic Analysis)

I would like to start and get into the habit of reading some publications in different areas of mathematics, to get used to the writing style / mathematical sophistication etc. that is expected.

In particular, one of the fields in which I would like to do this is Stochastic Analysis on Optimal Stopping Times. I am familiar with the usual big names, but I was wondering whether there are some authors that have a particular reputation for a lucid, accessible style to make it a bit easier getting a first foot on the ground ?

-

The answer on this question depends on your goals: do you want to learn about optimal stopping in dicrete or continuous time? In the latter case, are you satisfied with diffusions - or you want to study it in general?

1. Bernt Øksendal has a very nice book "Stochastic Differential Equations" which may be the best one to start reading about this topic in a continuous time setting. The first chapter contains some examples of "real-world" problems which need a stochastic treatment, among which there are optimal stopping problems (OSP). Moreover, Chapter 10 is fully devoted to the OSPs - the only point is that stochastic processes there are considered to be Ito diffusions, so their paths are continuous.

2. For the general exposition you may want to take a look on "Optimal Stopping Rules" by Albert Shiryaev (the last edition should be from 2007). This book presents the general theory for the optimal stopping of Markov processes both in discrete and continuous time. It is also quite short (220 pages) and material is given nicely and rigorously.

3. The book "Optimal stopping and free-boundary problems" by Goran Peskir and Albert Shiryaev could be considered (partly) as the extension of "Optimal Stopping Rules" and seems to contain one of the most comprehensive expositions on OSP. Besides Markov processes, the theory for general processes is included using martingale methods. The considered class of functionals to be optimized is much richer (including supremum and integral functionals). There are several cool examples both for diffusions and for jump processes. The methods used for solution of these examples are also very interesting conceptually.

There are also many book on stochastic mathematical finance which would necessary include OSPs due to American option problems, optimal portfolio theory etc. To be honest, the area of mathematical finance is one of the main areas given applications for OSP. However, I don't know if you are interested in such applications - so please tell me if you want to know which books on OSP you should account there, I will extend the list.

With changepoint detection I guess you meant something close to the disorder problem: namely, given a stochastic process $X$ and some random time $\theta$ you would like to find a stopping time $\tau^*$ adapted to the natural filtration of the original process, $\mathbb F^X$ such that say $$|\tau^*-\sigma| = \min\limits_\tau |\tau-\sigma|$$ where $\tau$ is any $\mathbb F^X$-adapted stopping time and $\sigma$ is unobservable from $\mathbb F^X$. Well, you can start with "Optimal Stopping and Free-boundary problems" where such problem is treated both for the Brownian motion and a Poisson process with the full solution given.

Next, the problem for the Poisson process was extended in two ways: extending the functional to be minimized and considering the compound Poisson process with not only intensity but also the distribution of jumps changing at some random time $\theta$. I know on research in this area done by Ioannis Karataz, Erhan Bayraktar and Savas Dayanik in the following three papers: the first, the second and the third. I think it is enough to start with.

-
Thanks a lot. Would you say Shiryaev also has an accessible style in his papers ? (I m already working with Oksendal, but looking to read a little beyond, so what I m really after are papers that are as intelligible as possible for a beginner !)Finance Applications might be interesting, but are not my main focus. If you know some good sources in Changepoint Detection that would be very interesting! –  Beltrame Feb 29 '12 at 17:19
@Beltrame Sure, you can read papers and books by Shiryaev - and they are pretty awesome. You only should be ready that they are not written as introductory texts, so you may need a bit of a patience and knowledge to benefit from them. The language I like very much, don't be afraid of them - just they are not on the introductory level. Changepoint detecting (I guess you mean the disorder problem) is perfectly discussed in book 3. Also, there were some papers generalizing it - I will look them up for you. –  Ilya Feb 29 '12 at 17:21
thks, that would be really great ! –  Beltrame Feb 29 '12 at 18:39
@Beltrame: well, on your request I've added some more material. Should be enough so far - but if you need more, please just tell me. OSPs is something that I am also interested in. I also wonder what caused your interest in that field. –  Ilya Feb 29 '12 at 20:44
great, thanks ! I think a bit of an interest in Inference more generally got me interested initially, but there are some many interesting things to learn in Maths, its sometimes hard to put a finger on how prioritisation originally occurred :) –  Beltrame Mar 1 '12 at 19:34