Ruin problems with assets and liabilities of diffusion type
type: Item
instance of: scholarly article
title: Ruin problems with assets and liabilities of diffusion type
zbMATH Open document ID: 0962.60075
DOI: 10.1016/S0304-4149(98)00103-3
author: Ragnar Norberg
published in: Stochastic Processes and their Applications
publication date: 2001-01-17T00:00:00Z
review text: The author studies the properties of ruin probabilities for a risky business when its cash balance process \(X\) is a Markov diffusion with drift, i.e. satisfies an Itô s.d.e. \(dX_{t}=\mu (t,X_{t}) dt+\sigma (t,X_{t}) dW_{t}\), \(W\) being a standard Brownian motion, with coefficients dependent only on time and the current level of cash balances. The analyzed ruin functions are of two types: the probability of ruin at a certain future date (transition probability) and the probability of ruin at any time prior to a given future date (crossing probability), both conditioned on the present cash balance level. Accordingly, there exist two types of ruin event martingales, which are the above named probabilities conditioned on the presently available information. Using Itô's formula and the properties of martingales, the author derives partial differential equations for both ruin function types. These p.d.e. happen to be the same but with different initial and boundary conditions for the transition and crossing cases. Solutions to the derived p.d.e. are obtained in a closed form when the cash balance process is time-homogeneous (coefficients \(\mu \) and \(\sigma \) are functions of \(X\) only) and the time horizon of possible ruin is set to infinity. Important special cases are, in particular, the ones of constant coefficients in the Itô equations for the net claims and the rate of return processes (the latter are the two variables determining the cash balances). Under these assumptions, the author derives a superexponential upper bound for the crossing probability in the infinite ruin time case.
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60J70
Mathematics Subject Classification ID: 60G35
zbMATH DE Number: 1554315
zbMATH Keywords: actuarial risk theory
zbMATH Keywords: finance
zbMATH Keywords: martingales
zbMATH Keywords: functionals of Brownian motions
reviewed by: Alexis Derviz
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1554315