Journal of Applied Mathematics and Stochastic Analysis
Volume 16 (2003), Issue 2, Pages 127-139
doi:10.1155/S1048953303000091
Tree-indexed processes: a high level crossing analysis
1EBMS, University of Wales-Swansea, Singleton Park, Swansea SA2 8PP, UK
2DPMMS, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK
Received 1 June 2002; Revised 1 April 2003
Copyright © 2003 Mark Kelbert and Yuri Suhov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Consider a branching diffusion process on R1 starting at the origin. Take a high level u>0 and count the number R(u,n) of branches reaching u by generation n. Let Fk,n(u) be the probability P(R(u,n)<k),k=1,2,…. We study the limit limn→∞Fk,n(u)=Fk(u). More precisely, a natural equation for the probabilities Fk(u) is introduced and the structure of the set of solutions is analysed. We interpret Fk(u) as a potential ruin probability in the situation of a multiple choice of a decision taken at vertices of a ‘logical tree’. It is shown that, unlike the standard risk theory, the above equation has a manifold of solutions. Also an analogue of Lundberg's bound for branching diffusion is derived.