Iterated Brownian motion and stable \((1/4)\) subordinator
Let \(B_t\), \(B^+_t\), \(B^-_t\), \(t \geq 0\), be three independent linear Brownian motions started from 0. Denote \(S_t = \sup \{B_s :0 \leq s \leq t\}\) and \(\sigma_t = \inf \{u : S_u > t\}\). Consider also the supremum processes \(S^+\), \(S^-\) of \(B^+\) and \(B^-\), respectively, denote \(\sigma^+\), \(\sigma^-\) the right-continuous inverses of \(B^+\) and \(B^-\), respectively. The author notices that the compound process \(\sigma \circ \sigma^+_t\) is a stable subordinator with index \(\alpha = 1/4\) and proposes a very easy method to investigate the path behaviour of iterated Brownian motion \(X_t\). Here the process \(X_t\), \(t \geq 0\), is given by expressions \(X_t = B^+ (B_t)\) if \(B_t \geq 0\) and \(X_t = B^-(- B_t)\) if \(B_t < 0\). Based on known properties of stable processes with \(\alpha \in (0,1)\) the LIL for \(X_t\) is of the form: \[ \limsup t^{-1/4} (\log |\log t |)^{- 3/4} X_t = 2^{5/4} 3^{-3/4}. \] Both as \(t \to 0\) and \(t \to \infty\) are proved. Also the integral tests to study the rate of grows for \(S^+ \circ S\) and \(X_t\) are proposed and modulas of continuity of the supremum of iterated Brownian motion are obtained.
scientific article; zbMATH DE number 896064