most recent 30 from http://mathoverflow.net 2013-06-19T10:18:40Z http://mathoverflow.net/feeds/question/40500 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/40500/brownian-motion-martingales-markov-chains-rosetta-stone Spencer 2010-09-29T17:30:12Z 2010-10-04T14:33:40Z

<blockquote> <p>What are the most fundamental/useful/interesting ways in which the concepts of Brownian motion, martingales and markov chains are related?</p> </blockquote> <p>I'm a graduate student doing a crash course in probability and stochastic analysis. At the moment, the world of probability is a confusing blur, but I'm starting with a grounding in the basic theory of markov chains, martingales and Brownian motion. While I've done a fair amount of analysis, I have almost no experience in these other matters and while understanding the definitions on their own isn't too difficult, the big picture is a long way away.</p> <p>I would like to <strong>gather together results and heuristics</strong>, each of which links together two or more of Brownian motion, martingales and Markov chains in some way. Answers which <strong>relate probability to real or complex analysis</strong> would also be welcome, such as "Result X about martingales is much like the basic fact Y about sequences".</p> <p>The thread may go on to contain a Big List in which each answer is the posters' favourite as yet unspecified result of the form "This expression related to a markov chain is always a martingale because blah. It represents the intuitive idea that blah".</p> <p>Because I know little, I can't gauge the worthiness of this question very well so apologies in advance if it is deemed untenable by the MO police.</p>

http://mathoverflow.net/questions/40500/brownian-motion-martingales-markov-chains-rosetta-stone/40721#40721 The Bridge 2010-10-01T07:57:53Z 2010-10-04T13:43:07Z

<p>Hi, </p> <p>Regarding Martingales you can see them as fair games This means that if the (martingale) process represents your (random) wealth, you should not be able to design a strategy to increase your current wealth, no matter what the outcome of the sample space is.</p> <p>Brownian Motion can be seen as a limit of rather simple random walks but I'm sure that you know about this. </p> <p>Markov processes "disconnect" Future and Past of the process conditionnally on the present value of the process. Where "disconnect" means that functions of past and of future values of the process are independent conditionnally on the present value of the process.</p> <p>Does it make things more clear ?</p>

http://mathoverflow.net/questions/40500/brownian-motion-martingales-markov-chains-rosetta-stone/40738#40738 Simon Lyons 2010-10-01T11:42:13Z 2010-10-01T11:42:13Z

<p>Levy's characterisation of Brownian motion:</p> <p>If X is a continuous martingale and X has quadratic variation process $[ X]_t = t$ then $X$ is a standard Brownian motion.</p>

http://mathoverflow.net/questions/40500/brownian-motion-martingales-markov-chains-rosetta-stone/40740#40740 Simon Lyons 2010-10-01T12:01:31Z 2010-10-01T12:01:31Z

<p>If X is a continuous martingale of finite variation such that $X_0 = 0$, then $P(X_t = 0 \ \ \forall t) = 1$. </p>

http://mathoverflow.net/questions/40500/brownian-motion-martingales-markov-chains-rosetta-stone/41025#41025 Steven Heston 2010-10-04T14:33:40Z 2010-10-04T14:33:40Z

<p>Girsanov's Theorem.</p>