Let $\psi$ be a sequence of progressively measurable processes s.t. for all $t \ge 0$ we have: $\psi_t^n\to0$ a.s. Let B be a standard brownian motion.
Assume also that there exists a process $\phi\in \lambda^2_{loc}$ with $|\psi^n|\le \phi$. Where $\lambda^2_{loc}$ is the set of {$\phi:$ $\phi_s$ is progressive and $\int_0^t\phi_s^2ds< \infty$ for all t}
How could I show that for $t>0$ $$\int_0^t\psi^n_sdB_s\to0$$ in probability?
What is making it hard for me is that $\phi$ is not surely integrable and therefore I cannot apply Lebesgue theorem.
I will provide my attempt, I would really appreciate a feedback.
Let's divide the problem in two cases:
$1)$ assume $E[\int_0^t\phi^2_sds]<\infty$
Then $$\lim_{n \to \infty}E\bigg[ (\int_0^t\psi^n_sdB_s-0)^2\bigg]=\lim_{n \to \infty}E\bigg[ \int_0^t(\psi^n_s)^2ds\bigg]\to0$$ by dominated convergenge theorem, since both $\int_0^t(\psi^n_s)^2ds<\int_0^t(\phi_s)^2ds$ and $(\psi^n_s)^2<\phi_s^2$ and both $\phi_s^2$ and $\int_0^t(\phi_s)^2ds$ are integrable by our initial assumption.
$2)$ Assume now instead that $\tau_k$ is a sequence of stopping time defined as $$\tau_k:inf(t\ge0:\int_0^t\phi^2_sds\ge k)$$ Now: For a $\epsilon > 0 $ there exists a $k$ s.t. $P(t > \tau_k)<\epsilon$
For $a >0$
$$\begin{align*} P(|\int_0^t\psi^n_sdB_s - 0|>a) &= P(|\int_0^t\psi^n_sdB_s|>a;t<\tau_k)+P(|\int_0^t\psi^n_sdB_s|>a;t\ge \tau_k) \\ &\le P(|\int_0^t\psi^n_sdB_s|>a;t<\tau_k) + \epsilon \\ &\le P(|\int_0^{t \land \tau_k}\psi^n_sdB_s|>a) \to0 \end{align*} $$
and the last step follow since, as before we could write: $\lim_{n \to \infty}E\bigg[ (\int_0^{t \land \tau_k}\psi^n_sdB_s-0)^2\bigg]=\lim_{n \to \infty}E\bigg[ \int_0^{t \land \tau_k}(\psi^n_s)^2ds\bigg]\to0$
and $\int_0^{t \land \tau_k}(\psi^n_s)^2ds \le \int_0^{t \land \tau_k}\phi^2_sds$ and $(\psi^n_s)^2<\phi_s^2$ and both $\int_0^{t \land \tau_k}(\phi_s)^2s$ and $\phi_s^2$ (is it right that also $\phi_s^2$ is integrable under the inital assumption of step 2?) are integrable by our initial assumption.
So what do you think about this proof? is it correct?