Optimal Spacing of “Covered” and “Exposed” Time Intervals in a Stochastic Process with High Penalty Costs: Applications to Parking and Insurance

Abstract

This paper studies a class of policies for a stochastic process that is constituted of several time intervals of total time T. The intervals can be covered (or insured) at a pay-per-use rate or exposed (uninsured) with the risk of a large penalty. A decision maker has the three options: (i) Pay the user fee for the full period, (ii) not pay at all, and (iii) sporadically pay a user fee leaving an uncovered period at the end of each covered one. The penalty risk is assumed to occur during an uncovered interval according to a Poisson process. We present the expected cost model and find the optimal coverage policy. We present conditions under which it is always optimal to pay in full or not pay at all to minimize the expected total cost. Finally, we relax two assumptions and allow for the consideration of setup costs for every time the decision maker pays the coverage fees as well as a random duration T and derive new conditions for optimal strategies. Possible application of our model is paying parking meter fees and deciding between self-insuring one’s property and buying full (or partial) insurance coverage.