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

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

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.