The famous Snell envelope or something like it ! ! !

I was just reading about the famous Snell envelope in Financial mathematics. When I met this "Snelll" feature I was totally and completely confused weather I was reading mathematics, probability or something else.

Eventually it led me both to the lately used and announced as the important feature, the theory of martingales, and to the theory of operations research. Namely, I was reading about optimal stopping problems of stochastic processes and found some of the most interesting calculations and mixture of the twp fields.

It made me so curious that I ended up reading some new paper about not one, but multiple optimal stopping times problem in the American option trading system. The idea is to find the supremum of both conditional and unconditional mathematical expectation of the reward function at some random time. Utterly very very interesting theory...... think about it

Martingales.......?

How one would have guessed that martingales play one of the most important roles in so many applications of probability theory, stochastic modeling etc..... To take Financial mathematics as an example I have to say that the theory of arbitrage, portfolio theory or risk theory ...etc....could not exist without the theory of martingales. To agree on what is said on most probability books I have to say that martingales play the role of constant functions in probability theory. The whole Ito calculus or stochastic calculus could not be derived or less alone worked with, without the martingales, supermartingales and submartingales. Potential theory, in pure analisys or in probability theory, explore in some way the theory of martingales, local and stopping times and the numbered theorems of the convergence criteria for martingales. All of this should be enough to convince someone to start reading about Martingales and Smartingales.....:)

SSP & Princeton March 2006

From March 22 till March 25 2006 there was a "Seminar on Stochastic Processes 2006 &Special Day on Markov Processes in honor of Erhan Çinlar on his 65th Birthday” held at the Princeton University.

It is a great honor to say that I attended this seminar, as a graduate student at the University of Belgrade, Faculty of Mathematics. To be able to hear lectures and to participate in discussions, with such big "names" in the world of probability theory, was an amazing opportunity to me as a young researcher. Just to give you an insight of the renamed professors: Erhan Cinlar, Ron Getoor, Jean Jacod, John Walsh, Haya Kaspi, Walter Schachermayer, Frank den Hollander, Steve Evans....Beside the leading probabilists there were also young researchers eager to present their current research work and problems which they have met. This gave the Seminar spirit of motivation for everybody, for those already with great experience and for the others with much less experience.

The topics of the lectures given ranged from duality and potential theory for different Markov Processes to the financial mathematics and some statistical limit theorems problem. In other words it ranged from stochastic differential and partial differential equations to the pure functional analysis. On the other hand on discussions that took place in the afternoons the scope of the problems presented were much broader. It included different kinds of Brownian motion behaviors (University of Wisconsin-Madison), large deviation theory (Cornell University), the behavior of maxima of alpha stable processes (Cornell University), self-financing strategies and discrete approximation of Markov risk (Columbia University) etc...... It was really inspiring to hear such broad range of problems that are being done by these young probabilists. Nevertheless I was by far the youngest in both years and research work; not to mention that I came to Princeton all the way back from Serbia, Belgrade where we have such a small number of people doing any research work in the probability theory.

I would like to thank professor Rene Carmona for selecting me among the candidates applied, for financial aid, without which it would be impossible to attend this seminar. You can read the archive of the seminar and read the official web page of the seminar on this page http://orfe.princeton.edu/~rcarmona/SSP2006/index.php

Financial Mathematics - history notes

In Finance Stochastic Processes were introduced in 1900. and then forgotten. In the 1960's and 1970's they were investigated again. In that time the known Black-Scholed model of a financial market was developed. This model was based on the theory of and the continuus time analysis of the prices of the assets. Later was discovered that this Brownian Motion is not the perfect and the most appropriate solution for the changes in prices of the assets. Some ten years later the Browninan MotionGeometric Brwninan Motion was shown to be the best model solution.

These results made some mathematicians to look deeper into the financial market theory. The result of the later inhanced analysis was "The fundamental theorem of Asset Pricing" which gave some necesary boundaries for the solution of finding the best option price in funture betting. It was shown that these conditions were already satisfied both in Brownian Motion and in Geometrical Brownian Motion.

The necessary condition for the market was the "no-arbitrage" quality which is considered the most valuable quality for the market to be "fair". In short it states that there should not be possibilities for the non-risky profits in the market. Later this quality of the market was conected to the martingale representation of that particulat financial market. The fundamental Theorem of Asset Pricing states that if the market has the no-arbitrage quality then it exists another probability measure equivalent to the first one, in which the market is martingale.

Introduction

The various topics, ranging from Functional analysis, Stochastic Processes , to Hidden Markov Models and Statistical learning applications will be discussed here..........