zbMATH — the first resource for mathematics

Iterated Brownian motion and its intrinsic skeletal structure. (English) Zbl 0943.60081
Dalang, Robert C. (ed.) et al., Seminar on Stochastic analysis, random fields and applications. Centro Stefano Franscini, Ascona, Italy, September 1996. Basel: Birkhäuser. Prog. Probab. 45, 201-210 (1999).
Iterated Brownian motion (IBM), defined by inserting one Brownian motion in an independent, two-sided, second Brownian motion, is a process, which fails to be Markovian or a semi-martingale. The paper reviews some recent results on the stochastic analysis of this process. In particular, the authors work out clearly one of the key ideas in the analysis of IBM: By stopping the inner Brownian motion at certain levels and sampling the outer motion at these times one can define the intrinsic skeletal structure of iterated Brownian motion, which permits to separate the two motions from each other and makes IBM amenable to analysis. The paper also contains some interesting speculations on a duality between IBM and Brownian motion in random scenery, as defined by H. Kesten and F. Spitzer [Z. Wahrscheinlichkeitstheorie Verw. Geb. 50, 5-25 (1979; Zbl 0396.60037)].
For the entire collection see [Zbl 0914.00071].

60J65 Brownian motion
60H05 Stochastic integrals