I have been very confused by the idea of infinitesimal random variables, namely letting $\{Z(\omega,t)\}_{t\in\mathbb{R}}$ be a stochastic process. What do we mean by $dZ$. Is this meant by $$\lim_{h\rightarrow0}Z(t+h)-Z(t)$$. First if this is so, what does this type of limit represent (since cant use usual concept of limit as is in reals). I am confused because do we consider the quantity $dZ$ a random variable. I am just confused in general with this type of notation in terms of random variables. I know for reals quantities such as $dx$ con be made sense of as extending the reals to the hyperreals but I am lost when thinking of these concepts in probability theory.

I was also wondering about non standard analysis application to probability theory.

Also I am quite possibly just completley lost and misunderstanding notation and such.


To your specific query of what expressions like $\text dY_t$ mean, I offer the following. If you are referring to such entities as they arise in, say, the study of Ito processes, such as $$ \text dY_t = \mu_t\text dt + \sigma_t \text dW_t,\ \ Y_0=y $$

such entities don't mean anything except only as meaningful shorthand for equivalent statements in terms of well-defined Stochastic and Riemann integrals, in this case $$ Y_t-y = \int_{0}^{t}\mu_u\text du + \int_{0}^{t}\sigma_u \text dW_u\,. $$

This notational shorthand is simply a practical convenience: by following (rigorously justifiable) rules for operations involving these symbols, one can keep the notation in computations involving stochastic integrals from becoming unwieldy while, at the same time, obtaining correct results. In some sense, it is partly because of the deceptive ease with which these computations can be (and certainly are) made that one might be lulled into thinking of their similar looking classical "cousins" (such as $\text dy$). But don't be fooled; $\text dY_t$ is not a differential/infinitesimal in any rigorous sense. Neither does it make sense to, say, "differentiate" a random variable.

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    $\begingroup$ Thank you very much for your answer. I asked my professor this question today and he gave a very similar explanation. I was wondering if you know anything about infinitesimal calculus and its application to probability theory $\endgroup$ – Kamster Apr 9 '15 at 7:14
  • $\begingroup$ @Kamster, You are welcome. I am certainly aware of one non-standard analysis formulation of Stochastic Calculus, but I do not know its inner workings in any great detail. The school of "Kolmogorov" is good enough for me, thanks. ;) $\endgroup$ – ki3i Apr 9 '15 at 11:26

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