Best stochastic modeling books according to redditors
We found 2 Reddit comments discussing the best stochastic modeling books. We ranked the 2 resulting products by number of redditors who mentioned them. Here are the top 20.
We found 2 Reddit comments discussing the best stochastic modeling books. We ranked the 2 resulting products by number of redditors who mentioned them. Here are the top 20.
Why is it not an honest integral? The ito integral with respect to brownian motion IS a measure. Granted, it's abstract-nonsense kind of stuff (it rises from Kolmogorov's extension theorem) but the idea is pretty solid and definitely rigorous.
In fact, a lot of integration in this area is tricky because the measure HAS to be bad. As you probably know, there's no "lebesgue measure" on infinite dimensional banach spaces, and in fact, we have much worse things e.g. these measures behave TERRIBLY under dilations (pushforward of weiner measure under dilation is singular with respect to the original measure!!!). Of course, these things are not that hard to believe when you think about it for a bit. After all, if $B_r$ is the ball of radius $r$ in $R^n$, then $m(B_2) = 2^n m(B_1)$ so you can imagine whath appens as $n$ goes to infinity.
There's a good section on Gaussian measures on Martin Hairer's notes http://www.hairer.org/notes/SPDEs.pdf . Alternatively, you could try to looking at books which deal with analysis on path space. People try to study things like log-Sobolev inequalities with functions on path space. Notably, Hsu's book http://www.amazon.com/Stochastic-Analysis-Manifolds-Graduate-Mathematics/dp/0821808028 deals with this. However, I'm not sure if this is exactly what you're looking for.
This book gives an amazing treatment on this subject: http://www.amazon.com/gp/aw/d/0486497976?cache=533d3b750d53d3ee6967d1366708bce0#ref=mp_s_a_1_2&qid=1393783467&sr=8-2