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I have learned that there are several types of convergence of random variables and they are related somehow, but not yet for stochastic processes. I was wondering if the following is some type of convergence for stochastic processes $\Omega \times [0,T] \to \mathbb{R}$ with probability space $(\Omega, \mathcal{F}, \mathrm{P})$:

$$ \lim_{n\to \infty} \mathrm{E}_\mathrm{P} \left(\int_{[0,T]} |f_n(\omega,t) - f(\omega,t)|^2 dt \right) = 0 $$

I encountered this in the definition of Ito integral, but didn't find more about this type of convergence among others. I am also hoping to know relations between different convergence types and their connections to convergence types of random variables. Thanks and regards!

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As far as I know the above is just the $L^2(dP,dt)$ convergence. Other type of convergence used for constructing stochastic integrals is for example ucp convergence. Protter is a good reference on that. – Julian Wergieluk Nov 27 '11 at 17:44
Hi there is also the (Emery's) Semi-martingale topology that induce a metric, and the Skorokhod topology over càdlàg process that has also a metric space structure. Regards – TheBridge Nov 28 '11 at 8:33

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