Adaptive control strategies and dependence of finite time ruin on the premium loading

Abstract

Abstract. The paper is devoted to risk theory insight into the problem of asset-liability and solvency adaptive management. Two adaptive control strategies in the multiperiodic insurance risk model composed of chained classical risk models are introduced and their performance in terms of probability of ruin is examined. The analysis is based on an explicit expression of the probability of ruin within finite time in terms of Bessel functions. Dependence of that probability on the premium loading, either positive or negative, is basic technical result of independent interest.

Risk theory insight into a zone-adaptive control strategy

“…Recall (see Definition 4.1 in Malinovskii (2006b)) the target value u λ,µ (α, t) of the risk reserve corresponding to a level 0 < α < 1 which is a positive solution of the "neutral-loading" equation…”

“…The second advantage is the solvency aspect. It concerns the upper bounds on the probability of ruin ψ t (u z,t , τ z,t ) considered as a function of z ∈ R, obtained in Malinovskii (2006b). Since for z = 0 that probability equals ψ t (u λ,µ (α, t); 0) = α, the bounds demonstrate how much ψ t (u z,t , τ z,t ) deviates from the prescribed value of solvency α for different z.…”

“…In this paper, we continue (see Malinovskii (2006a,b)) to deal exclusively with the classical risk model, bearing however in mind both perspectives of weakening its technical assumptions and extending the results to the case of incomplete information, subject to certain risk scenarios. The basic technical tool is therefore the explicit expression for the classical finitetime ruin probability discussed in Malinovskii (2006b), and its representation in terms of the asymptotic series, as time increases. The problem of asymptotic behaviour of the finitetime ruin probability, as time is large, but when the starting capital is arbitrary, is known in risk theory (see, e.g., Teugels (1982) and Section 6 in Asmussen (1984Asmussen ( -1985).…”

Zone-Adaptive Control Strategy for a Multiperiodic Model of Risk

“…It defines the initial risk reserve sufficient to make the probability of ruin equal to α without resort to premium loading, which may be voted fair by customers. In the diffusion model the "probability of ruin" solvency criterion is as simple to deal with as a short-cut "ultimate risk reserve" solvency criterion (see [6], p. [16][17]: solution of the equation…”

“…Paramount in (1) is the annual probability mechanism of insurance. It is modelled in [16] and [17] by Lundberg's collective risk model, while in this paper it is taken simplistic: the annual risk reserve of an insurance company at time t is assumed to be…”