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 coll ectiva del rischio. In: Transactions of the 15th International Congress of Actuaries, Vol. II, New York, pp. 433-443] proposal, of a modified random w alk for the (accumulated) surplus reserve (S) of an insurance system with a reflecting barrier at a predefined level, we design a model similar to tha t of Baiter and Benjamin [J. Inst. Actuaries 107 (1980) 513], involving a s mooth control action. Given the basic difference equation, which describes the development of the surplus process and the delays inherent to an insura nce system, we propose a particular decision function for the determination of the premium (P). For this purpose, we use the recent claim (C) experien ce and a negative feedback mechanism based on the latest known surplus valu e. The model assumes that the delay factor (S) is a free control parameter with a constant accumulation factor (R) for the surplus reserve. We investi gate 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 det ermine appropriate values for the feedback factor (epsilon) under the speci fic 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(infi nity)) beyond which instability is certain irrespective of the choice of th e feedback factor (epsilon). (C) 2001 Elsevier Science B.V. All rights rese rved.