This is not a homework question so please be kind not to remove it right away. I am working on some research but have to justify the following argument: Assume $S_t$ is a continuous stochastic process, don't want to make an assumption about distribution, think about something like a smooth function of Brownian motion. I define another process $$Y_t=\frac{1}{t} \int_0^t S_u du$$ Now I am interested in the limit of $Y_t$ as $t$ approaches zero. I would like to know in what sense the argument would hold, the guess is that the limit is $S(0)$. Please suggest a solution or the way to approach this problem.
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You can prove it reasoning $\omega$ by $\omega$. I mean, if you are working in a probability space $\Omega$, the continuous path $(S_t)_{t\geqslant 0}$ depends on $\omega\in \Omega$. But for all $\omega$, $t\mapsto S_t(\omega)$ is continuous, so the following convergence holds : $$ Y_t(\omega) \rightarrow S_0(\omega) $$ It is enough to conclude that $Y_t$ converges almost surely to $S_0$ as $t$ goes to zero. |
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