I am studying Stochastic processes and I am a bit confused on the general approach to their study.

In engineering textbooks a concept called "Mean Squared" Calculus is used to define things like time integration/differentiation/continuity of a Stochastic Process.

How does this interact with Ito integration? It seems Ito integration is focused (loosely speaking) on integration with respect to Brownian motion or some other stochastic process, not time integration.

In addition I haven't found any information on Mean-Squared calculus outside of engineering texts, the Stochastic Processes book(math) I have read doesn't even mention it.

So what's going on? Is Mean-Squared Calculus a mathematically recognized theory? Why don't math-books focused on Stochastic Processes use it?

  • $\begingroup$ Mean-square is synonymous with $L^2$. Is that what you are asking about? $\endgroup$ – S.Surace Apr 18 '18 at 4:58
  • $\begingroup$ I think so, limits are taken with respect to the expectation of the square, so I assume so. Most of the theory is built upon the convergence of Cauchy Sequences(Hilbert Space) which is taken for granted. $\endgroup$ – FourierFlux Apr 18 '18 at 9:06
  • $\begingroup$ Yes, that can be taken for granted because every Hilbert space is a complete metric space. Since $L^2$ convergence plays an extensive role in the theory of stochastic integration, you might be able to map the statements found in the engineering literature to the more precise mathematical statements with a bit of work. $\endgroup$ – S.Surace Apr 18 '18 at 13:00
  • $\begingroup$ Have any book recommendations? For example, engineering books talk about Mean-Square(L^2) continuity, differentiation, integration WRT to time. I know of texts which focus on integration WRT a stochastic process(Ito) but not time integration. $\endgroup$ – FourierFlux Apr 19 '18 at 5:33
  • $\begingroup$ I suggest Protter, P. E. (2013). Stochastic Integration and Differential Equations. Springer. Time integration is treated as part of integration wrt. a finite variation process, one of the components of a semimartingale. The latter is in a sense the most general integrator for a stochastic integral. $\endgroup$ – S.Surace Apr 19 '18 at 6:02

I found another good introduction to the topic in Chapter 1 of Ash, R., & Gardner, M. F. (1975). Topics in Stochastic Processes. Academic Press. It covers second order calculus in Section 1.3.

  • $\begingroup$ Hello, thanks for updating this, I am a bit late again. I'm curious: what is the significance of mean-squared continuity, for example? It' $\endgroup$ – FourierFlux May 16 '18 at 23:48
  • $\begingroup$ It is stronger than continuity in probability, but easier to work with than almost sure continuity, is characterized in terms of the continuity of the covariance function, and has a host of important consequences, e.g. the spectral decomposition of wide-sense stationary square integrable processes with a continuous covariance function. $\endgroup$ – S.Surace May 17 '18 at 9:56
  • $\begingroup$ Hmm, but the question remains - what does it really get you(how is it used)? Also, mean-squared continuity at each time t_0 doesn't give any information on the set of continuous sample paths right? A common example is passing white noise through a dampening LTI filter, you end up getting a random process which is mean-squared continuous at each point but the question remains, what does this really mean, for example in terms of physical phenomenon? $\endgroup$ – FourierFlux May 18 '18 at 20:38
  • $\begingroup$ You need a bit more than $L^2$ continuity to get the existence of a continuous modification. I think $L^2$ continuity plus some growth condition on the covariance function should be sufficient. Please have a look at en.wikipedia.org/wiki/Kolmogorov_continuity_theorem $\endgroup$ – S.Surace May 18 '18 at 21:41
  • $\begingroup$ Ok thanks, what are some other outcomes of Mean Squared continuity? I'm having a hard time trying to understand why it really matters verse not having it at all since, as you pointed out by itself it's not good enough to get continuous sample paths. $\endgroup$ – FourierFlux May 19 '18 at 5:34

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