Integral convergence for continuous stochastic process

Question

Let $X$ be a continuous (bounded) stochastic process with $X_0=0$. Need to show that

$$\mathbb{E}\left[a^{-1}\int_0^a X^2_tdt\right]\to 0 \text{ as }a\to 0.$$

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By continuity of $t\mapsto X_t$, for each $\omega$,

$$a^{-1}\int_0^a X^2(t,\omega)dt=X^2(s,\omega)$$

for some $s\in(0,a)$ and $X(s,\omega)\to 0$ as $a\to 0$.

Alternatively,

$$a^{-1}\int_0^a X^2(t,\omega)dt\le \sup_{0\le s\le a} X^2(s,\omega)\to 0 \text{ as }a\to 0.$$

Then need to apply the Dominated convergence theorem to pass the limit inside the expectation ...

Is this reasoning correct?

Answer 1

Since $X$ is assumed to be bounded, $Y(\omega) :=\sup_{0\leqslant s\leqslant 1} X^2(s,\omega)$ is finite, and if we assume that $Y$ is integrable (which depends on the used definition of boundedness), we obtain by the dominated convergence theorem that for each sequence $a_n\downarrow 0$, $$\lim_{n\to\infty}\mathbb E\left[\sup_{0\leqslant s\leqslant a_n}X_s^2  \right] =0.$$