The combined effect of delay and feedback on the insurance pricing process: a control theory approach

Further to De Finetti’s [Su una impostazione alternativa della theoria collectiva del rischio. In: Transactions of the 15th International Congress of Actuaries, Vol. II, New York, pp. 433–443] proposal, of a modified random walk for the (accumulated) surplus reserve (S) of an insurance system with a reflecting barrier at a predefined level, we design a model similar to that of Balzer and Benjamin [J. Inst. Actuaries 107 (1980) 513], involving a smooth control action. Given the basic difference equation, which describes the development of the surplus process and the delays inherent to an insurance system, we propose a particular decision function for the determination of the premium (P). For this purpose, we use the recent claim (C) experience and a negative feedback mechanism based on the latest known surplus value. The model assumes that the delay factor (f) is a free control parameter with a constant accumulation factor (R) for the surplus reserve. We investigate the stability of the system and the optimal parameter design (in terms of the fastest response and return to the initial or steady state). We determine appropriate values for the feedback factor (ε) under the specific premium decision function using the tools of control theory. One of the results is the derivation of a critical value for the delay factor (f∞) beyond which instability is certain irrespective of the choice of the feedback factor (ε).