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

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Is $c_p$ allowed to also depend on $t$? – Ben Derrett Oct 3 '12 at 16:25
Got something from the answer below? – Did Sep 11 '13 at 16:48
There's a typo. The oringinal problem is to prove $P(c_p(X_1^{\ast})^p\geq a)\leq P(S_p\geq a)$ – Danielsen Jun 30 '14 at 14:47

The claimed result is doubtful: assume that $X_t=t$ for every $t$, almost surely. Then, $(X_t)_t$ is indeed self-similar, $\mathbb P(S_p\geqslant a)=0$ for every $a$ large enough while $\mathbb P(c\cdot(X^*_t)^p\geqslant a)=1$ for $t$ large enough, for every fixed positive $c$ and $a$.