In Lemma 3.1 and Theorem 3.2 from the article [_Numerical solution of random differential equations: A mean square approach_](https://www.sciencedirect.com/science/article/pii/S0895717706002962), 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_](https://www.amazon.com/differential-equations-engineering-Mathematics-Engineering/dp/0126548501) 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?