I have a question in my homework:
A continuous process $X$ is said to be self-similar if for every $\lambda>0$, $(X_{\lambda t})_{t\geq 0}$ has the same law as $(\lambda X_t)_{t\geq 0}$.
Let $X$ be self-similar and positive and for $p>1$, set
$S_p=\sup_{s\geq 0}(X_s-s^p),\ \ X_t^{\ast}=\sup_{s\leq t}X_s$
Prove there exists a constant $c_p$ depending only on $p$ such that for any $a>0$
$P(c_p(X_t^{\ast})^p\geq a)\leq P(S_p\geq a)$
Thanks a lot for your help