### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 127-139 |

Number of pages | 13 |

Journal | Journal of Applied Mathematics and Stochastic Analysis |

Volume | 16 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2003 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Modeling and Simulation
- Applied Mathematics

### Cite this

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*Journal of Applied Mathematics and Stochastic Analysis*, vol. 16, no. 2, pp. 127-139. https://doi.org/10.1155/S1048953303000091

**Tree-indexed processes : A high level crossing analysis.** / Kelbert, Mark; Soukhov, Iouri M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Tree-indexed processes

T2 - A high level crossing analysis

AU - Kelbert, Mark

AU - Soukhov, Iouri M.

PY - 2003/1/1

Y1 - 2003/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=2342550974&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=2342550974&partnerID=8YFLogxK

U2 - 10.1155/S1048953303000091

DO - 10.1155/S1048953303000091

M3 - Article

VL - 16

SP - 127

EP - 139

JO - International Journal of Stochastic Analysis

JF - International Journal of Stochastic Analysis

SN - 2090-3332

IS - 2

ER -