Discontinuity set of the expected value of a continuous process 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)
 A: 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:

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*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.

