Journal of Applied Mathematics and Stochastic Analysis
Volume 16 (2003), Issue 2, Pages 127-139

Tree-indexed processes: a high level crossing analysis

Mark Kelbert1 and Yuri Suhov2

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.


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 limnFk,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.