A diffusion approximation for the ruin function of a risk process with compounding assets

Abstract

The traditional theory of collective risk is concerned with fluctuations in the capital reserve {Y(t): t ⩾O} of an insurance company. The classical model represents {Y(t)} as a positive constant x (initial capital) plus a deterministic linear function (cumulative income) minus a compound Poisson process (cumulative claims). The central problem is to determine the ruin probability ψ(x) that capital ever falls to zero. It is known that, under reasonable assumptions, one can approximate {Y(t)} by an appropriate Wiener process and hence ψ(.) by the corresponding exponential function of (Brownian) first passage probabilities. This paper considers the classical model modified by the assumption that interest is earned continuously on current capital at rate β > O. It is argued that Y(t) can in this case be approximated by a diffusion process Y*(t) which is closely related to the classical Ornstein-Uhlenbeck process. The diffusion {Y*(t)}, which we call compounding Brownian motion, reduces to the ordinary Wiener process when β = O. The first passage probabilities for Y*(t) are found to form a truncated normal distribution, which approximates the ruin function ψ(.) for the model with compounding assets. The approximate expression for ψ(.) is compared against the exact expression for a special case in which the latter is known. Assuming parameter values for which one would anticipate a good approximation, the two expressions are found to agree extremely well over a wide range of initial asset levels.