2
$\begingroup$

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

  • 0
    Is $c_p$ allowed to also depend on $t$?2012-10-03
  • 0
    Got something from the answer below?2013-09-11
  • 0
    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)$2014-06-30

1 Answers 1