Duality for multidimensional ruin problem

Abstract

We consider a $d-$dimensional insurance network, with initial capital $a\in\R^d_+,$ operating under a risk diversifying treaty; this is described in terms of a regulated random walk $\{Z^{(a)}_n\}$ via Skorokhod problem in $\R^d_+$ with reflection matrix $R;$ $\{Y^{(a)}_n\}$ denotes the corresponding pushing process. Ruin (in a strong sense) of $\{Z^{(a)}_n\}$ is defined as the marginal deficit of each company being positive (and hence zero surplus) at some time $n.$ A dual storage network is introduced through time reversal at sample path level over finite time horizon; the stochastic analogue is again a regulated random walk $\{W_n\}$ in $\R^d_+$ starting at $0.$ It is shown that ruin for $\{Z^{(a)}_n\}$ corresponds to $\{W_n\}$ hitting open upper orthant determined by $R^{-1}a$ before hitting the boundary of $\R^d_+,$ even at the sample path level. Under natural hypotheses, we show that $\P($ ruin of $\{Z^{(a)}_n\}$ in finite time) $=\lim_{n\r\iy}\P(W_n\gg R^{-1}a: n<$ boundary hitting time of storage process) $=\lim_{n\r\iy}\P(Y^{(0)}_n \gg R^{-1}a:\Delta Y^{(0)}_n\gg 0).$ A notion of $d-$dimensional ladder height distribution is defined, and a Pollaczek-Khinchine formula derived; an expression for the ladder height distribution is presented. Our method is applicable to ruin problem for a continuous time $d-$dimensional Cramer-Lundberg type network, where the companies act independently in the absence of treaty.