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Let $X(t),t\in [a,b]$ be a stochastic process with $\mathbb E[X(t)]\equiv 0$ and uncorrelated increments, $f$ a continuously differentiable function.

With the above conditions, the following equality holds for a partition of $[a,b]$: $$ \int_a^b f(t)dX(t) = \lim_{n\rightarrow \infty} \sum_{k=0}^{n-1} f(t_i)(X(t_{i-1}) - X(t_i)) \text{ in }L^2 \quad (*)$$

If I'm not mistaken, an intuitive interpretation is that the function $f$ is randomly weighted by increments determined by the process $X(t)$. (pretty much like the Stieltjes integral just with a random second function)

Now two questions:

1) If I drop the above conditions, such that the equality (*) is no longer valid, is there still an intuitive picture about this kind of integration?

2) What would be a real-world example where this kind of integration is used to model? Or even better, what would be a class of models in the context of which this kind of integration naturally arises?

Added: With the conditions as in (*), what would then be the interpretation of integration by parts, i.e. changing from integrating with the process to integrating over the process?

$$\int_a^b f(t)dX(t) = f(b)X(b) - f(a)X(a) - \int_a^b X(t)f'(t)dt$$

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Are you familiar with Ito integral? –  Ilya Mar 29 '13 at 9:25
    
Yes, but I find the Ito-formulation in terms of (semi-)martingales rather unintuitive and was wondering whether there was an explanation with the $L^2$-theory of such integrals without referring to martingales. If on the other hand, there is an intuitive explanation in the language of Ito calculus, then I would also be happy to understand a little more about that. –  madison54 Mar 29 '13 at 9:59
    
Oksendal introduces Ito integrals for Brownian motion using almost only $L^2$ stuff, there's no much about martingales there - have you seen that? –  Ilya Mar 29 '13 at 10:02
    
I haven't yet. Could you give me the exact title? He has written several books in the field of stochastic calculus. Thanks! –  madison54 Mar 29 '13 at 10:09
    
Stochastic Differential Equations: Introduction with Applications. –  Ilya Mar 29 '13 at 10:13
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