A singularly perturbed ruin problem for a two-dimensional Brownian motion in the positive quadrant

We consider the following problem: the drift of the wealth process of two companies, modelled by a two-dimensional Brownian motion, is controllable such that the total drift adds up to a constant. The aim is to maximize the probability that both companies survive. We assume that the volatility of on...

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Bibliographic Details
Published inJournal of applied probability Vol. 62; no. 1; pp. 269 - 283
Main Author Grandits, Peter
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.03.2025
Subjects
Original Article
Ruin probability
optimal control problem
singular perturbation
49J20
free boundary problem
91B70
35R35
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Summary:We consider the following problem: the drift of the wealth process of two companies, modelled by a two-dimensional Brownian motion, is controllable such that the total drift adds up to a constant. The aim is to maximize the probability that both companies survive. We assume that the volatility of one company is small with respect to the other, and use methods from singular perturbation theory to construct a formal approximation of the value function. Moreover, we validate this formal result by explicitly constructing a strategy that provides a target functional, approximating the value function uniformly on the whole state space.
ISSN:0021-9002
1475-6072
DOI:10.1017/jpr.2024.68