In Lemma 3.1 and Theorem 3.2 from the article Numerical solution of random differential equations: A mean square approach, it is stated and proved a Mean Value Theorem for stochastic integrals and derivatives in the mean square sense. See Chapter 4 in the book Random differential equations in science and engineering for an introduction to mean square theory.

My question is about the proof of Lemma 3.1.

Statement of Lemma 3.1: Let $Y(t)$ be a mean square continuous process with finite second order moments on $T=[t_0,t_1]$. Then, there exists $\xi\in [t_0,t_1]$ such that $\int_{t_0}^t Y(s)ds=Y(\xi)(t-t_0)$, $t_0<t<t_1$.

Question: The first thing that surprises me is the fact that $\xi$ is the same for every $t_0<t<t_1$. Is this correct?

Proof of Lemma 3.1 in the article: By page 90 from the book, $\Gamma_Y(r,s)=E[Y(r)Y(s)]$ is continuous on $T\times T$. Then $\Gamma_Y(r,\cdot)$ is continuous on $T$, for each $r\in T$. By the Mean Value Theorem for Riemann integrals, $\int_{t_0}^t \Gamma_Y(r,s)ds=\Gamma_Y(r,\xi)(t-t_0)$, $\xi\in[t_0,t]$. From here, $\xi$ is considered constant.

Question: I think $\xi$ depends on $r$ and $t$, so this invalidates the proof. Is this correct?

Question: In case Lemma 3.1 is correct, for Theorem 3.2 I think it misses the hypothesis $\dot{X}(t)$ be mean square integrable, by page 104 from the book. Is this correct?