The uniform modulus of continuity of iterated Brownian motion (Q678079)

Let \(X\) be a Brownian motion on \(R^1\) with \(B_0=0\) and \(Y\) be another Brownian motion on \(R^{+}\), independent of \(X\). The iterated Brownian motion (IBM) is defined by \(Z_t=X_{Y_t}.\) IBM was introduced by \textit{P. Deheuvels} and \textit{D. M. Mason} [in: Probability in Banach Spaces, 8. Prog. Probab. 30, 255-266 (1992; Zbl 0844.60012)]. The uniform modulus of Z is studied in the present paper. For \(\delta\in (0,1),\) let \(\omega (\delta )\) \(=\sup_{0\leq s,t\leq 1}\) \(\sup_{0\leq {}s-t{}\leq\delta}\) \({}Z_t\) \(-Z_s{}\) and \(\psi ( \delta )=\delta^{1/4}(\ln(1/\delta))^{3/4}\). It is proven that \(P(\lim_{\delta\to 0}(\omega (\delta)/\psi(\delta))=1)=1\).