Tree-indexed processes

A high level crossing analysis

Mark Kelbert, Iouri M. Soukhov

Research output: Contribution to journalArticle

5 Citations (Scopus)

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 language English (US) 127-139 13 Journal of Applied Mathematics and Stochastic Analysis 16 2 https://doi.org/10.1155/S1048953303000091 Published - Jan 1 2003

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Level Crossing
Risk Theory
Ruin Probability
Branching process
Diffusion Process
Branching
Count
Branch
Analogue
Standards

All Science Journal Classification (ASJC) codes

• Statistics and Probability
• Modeling and Simulation
• Applied Mathematics

Cite this

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title = "Tree-indexed processes: A high level crossing analysis",
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.",
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In: Journal of Applied Mathematics and Stochastic Analysis, Vol. 16, No. 2, 01.01.2003, p. 127-139.

Research output: Contribution to journalArticle

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AU - Soukhov, Iouri M.

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