## 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) |
---|---|

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 - 2003 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Modeling and Simulation
- Applied Mathematics