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I'm facing with a problem of notation and I hope stack could help me!

Let $X(t)$ be a time-continuous stochastic process, with pdf $p_X(x, t)$. Let $g(x, t)$ be a generic function. Now, consider the following:

$$ Y(t) = \int_{0}^{t} g(X(s),s)dX(s)$$ where $Y(t)$ is itself a time-continuous stochastic process. In which way I must interpret/deal with this integral?

I mean, how do I perform the integration, since integration domain is over time while I only have $dX(t)$?

I feel like I'm missing something, and most likely I must perform a "change of variable" by using the pdf $p_X(x, t)$.

Can someone bring me some light?

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The LHS cannot be a function of $t$. Do you mean $Y(t_2)$? Likewise, $g(X,t)$ makes no sense. Do you mean $g(X(t),t)$? –  Did Dec 8 '12 at 13:32
    
thanks for the advice, I corrected it. –  the_candyman Dec 8 '12 at 14:34
    
Still g(X,s) is mysterious. –  Did Dec 8 '12 at 14:46
    
@did: Well, implicitely, $g(X,s) = g(X(s), s)$. I think that it is clear from the context and I don't understand the downvote –  the_candyman Dec 9 '12 at 11:06
    
Your identity is finally correct, this is nice. (In case your last comment implies that I downvoted your question, please note that: (1.) you have no way of knowing this, (2.) I did not downvote, (3.) I do resent very much having had to say that I did not.) –  Did Dec 9 '12 at 14:50

1 Answer 1

up vote 1 down vote accepted

This is a stochastic integral. In order to grasp the notion, I think it is a good idea to follow a whole course on that subject, maybe even a broader course on stochastic processes in continuous time and martingales.

I recommend Kuo's stochastic integration textbook for an introduction specifically targetting such integrals. There's also Øksendal's Stochastic differential equations textbook. The first chapter of Kuo's book has a very nice pedagogical explanation of the heuristics of that integral.


As an extra bonus, I also noticed that Mörters and Peres' book on Brownian motion is available online on Mörters webpage at this link.

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Thanks for the suggestion. Since I'm quite expert on time-discrete stochastic process, I was wondering that dealing with the time-continuous counterpart was a little bit similar. –  the_candyman Dec 8 '12 at 13:00
    
It is much more complicated in continuous time. –  Learner Dec 8 '12 at 13:01
    
Got it. There exists more handy reference (like a review paper or a didactic pdf) about that? –  the_candyman Dec 8 '12 at 13:04
2  
There are these lecture notes people.bath.ac.uk/maspm/stoa.ps but I feel they are not as pedagogical as the the references above. I really recommend studying one of the books because there are a lot of pitfalls and I find the topic hard when beginning. –  Learner Dec 8 '12 at 13:11

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