Tree-indexed processes: a high level crossing analysis

Consider a branching diffusion process on R 1 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 F k , n ( u ) be the probability P ( R ( u , n ) < k ) , k = 1 , 2 , … . We study the limit lim n → ∞ F k , n ( u ) = F k ( u ) . More precisely, a natural equation for the probabilities F k ( u ) is introduced and the structure of the set of solutions is analysed. We interpret F k ( 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.