Let $X\_t$ be a continuous real valued stochastic process on $\mathbb R\_+$. Then it is not necessarily true that $E[X\_t]$ is continuous in $t$.


**Question:**


What is known about the discontinuity set of $E[X\_t]$? Namely:


* Can it have removable discontinuities?
* Can it have essential discontinuities?
* Can it be discontinuous everywhere?
* What is the analytic class of the discontinuity set, if if is known? (eg, $G\_{\sigma \delta}$, $F\_\delta$, etc)



Discontinuity set of the expected value of a continuous process
Without further assumptions on $X\_t$, this is a question "what type of functions one gets if one averages continuous functions". Let $\phi\_n(x) = x$ when $0 \leqslant x \leqslant n$, $\phi\_n(x) = n$ if $x > n$, and $\phi\_n(x) = 0$ when $x < 0$. By the dominated convergence theorem,
\[ f\_n^+(t) := \mathbb E \phi\_n(X\_t) \qquad \text{and} \qquad f\_n^-(t) := \mathbb E \phi\_n(-X\_t) \]
are continuous function of $t$, $(f\_n^\pm(t))$ is a non-decreasing sequence, and by the monotone convergence theorem,
\[ f^\pm(t) := \lim\_{n \to \infty} f\_n^\pm(t) = \mathbb E \max\{\pm X\_t, 0\} \]
in the pointwise sense. Thus, $f^\pm$ are lower semi-continuous. Furthermore, whenever $X\_t$ is integrable, we have $f^\pm(t) < \infty$ and
\[ \mathbb E X\_t = f\_n^+(t) - f\_n^-(t) . \]
More generally, whenever $\mathbb E X\_t$ is well-defined, then $f^+(t) < \infty$ or $f^-(t) < \infty$, and again
\[ \mathbb E X\_t = f\_n^+(t) - f\_n^-(t) . \]
In particular, $\mathbb E X\_t$ is a Baire class 1 function, which is a difference of two lower semi-continuous functions (that is, Hausdorff's "type $d$" function, according to the reference listed at the end of this answer).




---


Conversely, it is not very difficult to see that any such function arises in the above way. Indeed: suppose that $f$ is non-negative and lower semi-continuous. Then $f$ is a pointwise limit of a non-decreasing sequence of non-negative continuous functions $f\_n$. Let us partition the probability space into a sequence of events $E\_n$ of positive probabilities. Now we define $X\_t = h\_n(t)$ on $E\_n$, where $h\_n$ is a (non-negative) continuous function chosen inductively in such a way that
\[ \mathbb E(X\_t \mathbb 1\_{E\_1 \cup \ldots \cup E\_n}) = f\_n(t) \]
The monotone convergence theorem tells us that
\[ \mathbb X\_t = \lim\_{n \to \infty} \mathbb E(X\_t \mathbb 1\_{E\_1 \cup \ldots \cup E\_n}) = \lim\_{n \to \infty} g\_n(t) = f(t) , \]
so $\mathbb E X\_t$ can be an arbitrary non-negative lower semicontinuous function $f$.


In order to get a difference of two non-negative lower semicontinuous functions $f^+$ and $f^-$, we just repeat the above construction twice: we construct two processes $X^+\_t$ and $X^-\_t$ with expectations $f^+(t)$ and $f^-(t)$, and we simply let $X\_t = X^+\_t - X^-\_t$.




---


I did not check this carefully, but I think the above result can be found in:


* Mark Finkelstein, Robert James Whitley, *Integrals of continuous functions*, Pacific J. Math. 67(2) (1976): 365–372, [DOI:http://dx.doi.org/10.2140/pjm.1976.67.365](https://doi.org/10.2140/pjm.1976.67.365).


