Tree-indexed processes: A high level crossing analysis

Mark Kelbert, Yuri Suhov

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

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→∞ F k,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.

Original languageEnglish (US)
Pages (from-to)127-139
Number of pages13
JournalJournal of Applied Mathematics and Stochastic Analysis
Volume16
Issue number2
DOIs
StatePublished - 2003

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Modeling and Simulation
  • Applied Mathematics

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