I am trying to prove that $E[\sup \limits_{t \leq L_n} |R_n(t)-t|^p]=O(n^{-p/2})$ where $\rho(n)$ is a solution of the following Stochastic differential equation \begin{equation} \rho_n^2(t)=2 \int_0^t \rho_n(s)dB_s+nt \end{equation} Please note that in the following we consider only the law of the stochastic process $\rho_n$ , one integer $n$ at a time each of which is generated by the the afore given SDE We define $$R_n(t)= \big(\rho_n(t)\big)^2\big/n$$
Lemma:Let $r >1$, then for any $p>0$, there exists a constant $C_{r,p}$ such that for every continuous local martingale $M$ starting from $0$ any for any random variable $L$ one has $$ E[(M^*_{L})^p] \leq C_{r,p} \| \langle M \rangle_L^{\frac{p}{2}} \|_r \text{ and } E[\langle M \rangle_L^{\frac{p}{2}} ] \leq C_{r,p} \| (M^*_L)^p \|_r$$
Where $L$ is a random time(a measurable map taking values in the positive reals) and $M^*_L=\sup \limits_{0 \leq t \leq L} |M_t|$ The random time $L_n$ is defined as $$L_n =sup\{t: \rho_n(t) =\sqrt{n}\}$$ Now I need to show using the above lemma that
a) For every $p >0$ $E[\sup \limits_{t \leq L_n} |R_n(t)-t|^p]=O(n^{-p/2})$ b)The previous result maybe refined as : $$E\big[ \sup \limits_{t \leq L_n} |\sqrt{n}(R_n(t)-t) -2 \int_0^t \sqrt{s} d \beta_s |^p \big]=\mathcal{O}(n^{-p/4})$$ c)In particular , if $V_n=T_n$ or $L_n$ then as $n \to \infty $: $$ V_n \to 1 \text{ and } \sqrt{n}(1-V_n) \to 2 \int_0^1 \sqrt{s} d \beta_s$$ where the convergence takes place both a.s and in $L^p$.
Finally , $n(L_n-T_n)$ converges in law to $\sup \{t :2 \beta_t +t=0\}$
Attempt: In proving a) I need to show that $\exists n_0 \in \mathbb{N}$ and $M>0$ such that $\forall n >n_0$ we have that $$E[\sup \limits_{t \leq L_n} |R_n(t)-t|^p] \leq Mn^{-p/2}$$ Simplifying we get $$ \frac{2^p}{n^p}E\bigg[\sup \limits_{t \leq L_n} \bigg(\int_0^t\rho_n(s)dB_s\bigg)^p\bigg] \leq Mn^{-p/2} $$ Which is is equivalent to $$2^p E\bigg[\sup \limits_{t \leq L_n} \bigg(\int_0^t\rho_n(s)dB_s\bigg)^p\bigg] \leq Mn^{p/2}$$
From the above lemma we have $$2^p E\bigg[\sup \limits_{t \leq L_n} \bigg(\int_0^t\rho_n(s)dB_s\bigg)^p\bigg] \leq 2^p C_{r,p}\|\big\langle \int_0^t \rho_n(s) dB_s \big\rangle_L^{\frac{p}{2}}\|_r=2^p C_{r,p} \|\int_0^{L_n} \rho_n(s)^2 ds \|_r$$
I have no idea how to proceed, I tried several things but I can't find a way to prove $a)$. I think the part b) of the problem is related to part a) and the proof probably should be similar Can someone please help me solve the first problem or give some hint on how I could solve it?