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.....:)