## Expected value of integral = integral of expected value?

 Quote by EnumaElish This is all fine, and thank you; but in many problems the "y variable" is seen as parametric and not explicitly specified as a variable. For example in stochastic finance, f(t) might denote an asset price at any given moment, given its history, f(t-1), ..., f(0); but the notation is not f(t, f(t-1), ..., f(0)). Alternatively, EY might denote EY[Etf(t) | Y] where Y = {f(t-1), ..., f(0)} is a conditional, and not an explicit part of the list of arguments for the function f.
Whether explicitly written or implicitly understood, you still need a dependence on two variables to make sense of the question as posed.

There are ways to do that, including simply stating early on the implicit dependence that will be presumed throughout the discussion. But when one writes an expression with no further context being described, one must simply assume that the writer meant what was said. One can presume the reader to be clairvoyant, but then one is likely to be disappointed.

One of the biggest difficulties with many applications of probability theory is the failure of the author to make clear what he is talking about, or what the probability space is. A lot of confusion can be cleared up with just a bit of attention to rigor in setting up the problem.

For instance, in your example, you are dealing with a time series, and in that context, dependencies on past history can be easily incorporated into the mathematical model. Nothing wrong or mysterious with that. But that is quite a different thing from just a set of values of some function at various discrete values of the argument, with no further description of the context.

 Tags average, brownian, expected value, integral, stochastic

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