Abstract
Two insurance companies I1,I2 with reserves R1(t),R2(t) compete for customers, such that in a suitable stochastic differential game the smaller company I2 with R2(0)<R1(0) aims at minimizing R1(t)−R2(t) by using the premium p2 as control and the larger I1 at maximizing by using p1. The dependence of reserves on premia is derived by modelling the customer’s problem explicitly, accounting for market frictions V, reflecting differences in cost of search and switching, information acquisition and processing, or preferences. Assuming V to be random across customers, the optimal simultaneous choice p1∗,p2∗ of premiums is derived and shown to provide a Nash equilibrium for beta distributed V. The analysis is based on the diffusion approximation to a standard Cramér–Lundberg risk process extended to allow investment in a risk-free asset.
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