Tail bounds for the joint distribution of the surplus prior to and at ruin

Abstract

Keywords: Ruin probability, Time of ruin, Surplus process, Deficit at ruin, Defective renewal equation.We consider the classical risk model where claims Y 1 , Y 2 , . . . arrive in a compound Poisson process with rate λ. The claims are independent identically distributed non-negative random variables and have common distribution function P with finite mean µ. In the case where P has a density, we denote this density by p. We further assume that the claims are independent of the claimarrivals process. Premiums are paid to the insurer continuously at a rate c per unit time. The surplus of the insurer at time t is then U (t) = u + ct − Nt k=1 Y i , where u is the initial surplus and N t is the number of claims until t. We assume throughout that c > λµ, so that ruin is not certain to occur. Moreover, we write c = (1+θ)λµ, where θ is the relative security loading. Let T denote the time of ruin, i.e. the time that the surplus becomes negative for the first time and note that T is a defective random variable. The probability of ruin is then defined by ψ(u) = P (T < ∞|U (0) = u).(1)

Refinements of two-sided bounds for renewal equations

“…The above bounds (4.7) and (4.8) for n = 1 agree with the results in Theorem 6.1 in Psarrakos and Politis (2008). Also, we may easily obtain tighter bounds by applying a larger value of n where n ≥ 2.…”

“…For the exponential bounds, Chadjiconstantinidis and Politis (2007) and generalize similar results obtained by Garrido (1998, 1999), Willmot (2002), and . Psarrakos and Politis (2008) have also derived improved tail bounds for the joint distribution of the surplus prior to and at ruin in the classical risk model. For further details of nonexponential-type bounds, see Chadjiconstantinidis and Politis (2005).…”

Asymptotic aspects of the Gerber–Shiu function in the renewal risk model using Wiener–Hopf factorization and convolution equivalence

“…Very often it is impossible to get closed-form formulas, or the established integro-differential equations are not solvable at all, or the calculation is rather involved. For bounds, we refer to Ng and Yang (2005), Psarrakos and Politis (2008), and Psarrakos (2008).…”