A multidimensional ruin problem and an associated notion of duality

Article Type

Research Article

Publication Title

Stochastic Models

Abstract

We consider a (Formula presented.) dimensional general Cramer–Lundberg type network, with initial capital (Formula presented.) operating under a risk diversifying treaty; this is described in terms of the appropriate Skorokhod problem (SP) in (Formula presented.) determined by a reflection matrix R. Ruin of the insurance network is defined as the marginal deficit of each company being positive (and hence surplus of each company being 0) at the same time. The corresponding ruin problem is studied using the associated regulated random walk {(Y(a)n, Z(a)n)} with {Z(a)n} denoting the reflected part, and {Y(a)n} the pushing part. A dual process is introduced through time reversal at sample path level over finite time horizon; the stochastic analogue is again a regulated random walk in (Formula presented.) starting at 0; let {Wn} denote the reflected part of the dual regulated random walk. It is shown that ruin for the insurance network corresponds to {Wn} hitting the open upper orthant with vertex at R− 1a before hitting the boundary of (Formula presented.) even at the sample path level. Under natural hypotheses, we show that the ruin probability is (Formula presented.) for some n ⩾ 1) (Formula presented.) boundary hitting time of storage process) (Formula presented.) A notion of (Formula presented.) dimensional ladder height distribution is defined, and a Pollaczek–Khinchine formula is derived. An expression for the ladder height distribution is given. (One-dimensional ladder height distributions are known for about 50 years.) Some examples are presented.

First Page

539

Last Page

574

DOI

10.1080/15326349.2016.1175949

Publication Date

10-1-2016

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