Weak convergence in $C([0,+\infty))$ and convergence in probability

Question

I'm reading an article and I can't manage to solve something. They say "It's not hard to check that $\sup_{[0,T]}\sqrt{\epsilon}|V_{(t-\epsilon \tau)/\epsilon }-V_{t/\epsilon}|$ converges to 0 in probability as $\epsilon \to 0$ since $(\sqrt(\epsilon)V_{t/\epsilon})_{t\geq0}$ goes in law in $C([0,+\infty))$ to the continuous process $(U_t)_t$". $\tau$ is an a.s. finite stopping time. I first would like to use the continuous mapping theorem but my function $g_{\epsilon}:x\mapsto (x_{t-\epsilon \tau}-x_t)_t$ depends on $\epsilon$. Then, I thought, if it converges uniformly to 0, I'm done. But I can't show that it converges uniformly. Does somebody have an idea please ?

Answer 1

I finally find the answer !Here it is : Let $T>0$. We have $\sqrt{\epsilon}\sup_{[0,T]}|V_{t/\epsilon}-\hat{V}_{t/\epsilon}|=\sqrt{\epsilon}\sup_{[0,T]}|\hat{V}_{t/\epsilon-\tau}-\hat{V}_{t/\epsilon}|$. Let $\eta>0$, we want to show that $\mathbb{P}\left( \sqrt{\epsilon}\sup_{[0,T]}|\hat{V}_{t/\epsilon-\tau}-\hat{V}_{t/\epsilon}| \geq \eta \right) \underset{\epsilon\to 0}{\to}0$. Since $\left(\sqrt{\epsilon}\hat{V}_{t/\epsilon}\right)_{t\geq 0}$ converges in law in $C([0,+\infty))$, the family $\{\left(\sqrt{\epsilon}\hat{V}_{t/\epsilon}\right)_{t\geq 0}, \epsilon>0 \}$ is tight. Hence by Ascoli theorem (see Billingsley theorem 7.3 p.82) $\lim\limits_{\delta \to 0}\sup_{\epsilon>0}\mathbb{P}(w_{\delta}(\sqrt{\epsilon}\hat{V}_{\cdot/\epsilon})\geq \eta)=0$. Let $\gamma>0$. Let $\delta_0>0$ such that $\sup_{\epsilon>0}\mathbb{P}(w_{\delta_0}(\sqrt{\epsilon}\hat{V}_{\cdot/\epsilon})\geq \eta)\leq \gamma/2$. We have $$\mathbb{P}\left( \sqrt{\epsilon}\sup_{[0,T]}|\hat{V}_{t/\epsilon-\tau}-\hat{V}_{t/\epsilon}| \geq \eta \right)\leq \mathbb{P}\left( \sqrt{\epsilon}\sup_{[0,T]}|\hat{V}_{t/\epsilon-\tau}-\hat{V}_{t/\epsilon}| \geq \eta ,\epsilon\tau \leq \delta_0\right)+\mathbb{P}(\epsilon\tau >\delta_0).$$ Since $\tau$ is finite a.s., $\epsilon\tau$ converges to $0$ a.s. so on in probability, thus there exists $\epsilon_0$ such that for all $\epsilon\leq \epsilon_0$, $\mathbb{P}(\epsilon\tau >\delta_0)\leq \gamma/2$. On the other hand, on the event $\{\epsilon\tau \leq \delta_0\}$, for $t\in [0,T]$, $$ \sqrt{\epsilon}|\hat{V}_{t/\epsilon-\tau}-\hat{V}_{t/\epsilon}|\leq \sup_{\substack{t,s\in[0,T] \\ |t-s|\leq \delta_0}}\sqrt{\epsilon}|\hat{V}_{s/\epsilon}-\hat{V}_{t/\epsilon}|=w_{\delta_0}(\sqrt{\epsilon}\hat{V}_{\cdot/\epsilon}).$$ So $\sup_{[0,T]}\sqrt{\epsilon}|\hat{V}_{t/\epsilon-\tau}-\hat{V}_{t/\epsilon}|\mathbb{1}_{\epsilon\tau \leq \delta_0}\leq w_{\delta_0}(\sqrt{\epsilon}\hat{V}_{\cdot/\epsilon})$. Hence, $$\mathbb{P}\left( \sqrt{\epsilon}\sup_{[0,T]}|\hat{V}_{t/\epsilon-\tau}-\hat{V}_{t/\epsilon}| \geq \eta ,\epsilon\tau \leq \delta_0\right)\leq \mathbb{P}\left(\sup_{[0,T]}\sqrt{\epsilon}|\hat{V}_{t/\epsilon-\tau}-\hat{V}_{t/\epsilon}|\mathbb{1}_{\epsilon\tau \leq \delta_0} \geq \eta \right)\leq \mathbb{P}(w_{\delta_0}(\sqrt{\epsilon}\hat{V}_{\cdot/\epsilon})\geq \eta)\leq \gamma/2.$$

Answer 2

In order to get rid of the stopping time $\tau$, we can treat it in the following way: it has a small probability to be too big, and when it is small we can control it by enlarging the range of the supremum.

More formally, fix a positive $\eta$. Since $\tau$ is almost surely finite, there exists an $R\gt 0$ such that $\Pr\{\tau\gt R\}\lt\eta$. Then for all $\delta$, $$ \Pr\left\{\sup_{[0,T]}\sqrt{\epsilon}|V_{(t-\epsilon \tau)/\epsilon }-V_{t/\epsilon}|\gt \delta\right\}\leqslant \eta+\Pr\left(\left\{\sup_{[0,T]}\sqrt{\epsilon}|V_{(t-\epsilon \tau)/\epsilon }-V_{t/\epsilon}|\gt \delta\right\}\cap\{\tau \leqslant R\}\right). $$ If $\tau\leqslant R$, then $$ \sup_{t\in [0,T]}\sqrt{\epsilon}|V_{(t-\epsilon \tau)/\epsilon }-V_{t/\epsilon}| \leqslant \sup_{s\in[0,R\epsilon]}\sup_{t\in [0,T]}\sqrt{\epsilon}|V_{(t-s)/\epsilon }-V_{t/\epsilon}|. $$ Let $\epsilon_0>0$ be fixed. For $\epsilon<\epsilon_0$, we established the following: $$ \Pr\left\{\sup_{[0,T]}\sqrt{\epsilon}|V_{(t-\epsilon \tau)/\epsilon }-V_{t/\epsilon}|\gt \delta\right\}\leqslant \eta+\Pr\left(\sup_{s\in[0,R\epsilon_0]}\sup_{t\in [0,T]}\sqrt{\epsilon}|V_{(t-s)/\epsilon }-V_{t/\epsilon}|\gt \delta\} \right) $$ Now apply the continuous mapping theorem to the functional $$ F\colon x\in C[0,+\infty)\mapsto \sup_{s\in[0,R\epsilon_0]}\sup_{t\in [0,T]}\left\lvert x(t-s)-x(t)\right\rvert. $$