STOCHASTIC OPTIMIZATION APPROACH OF CAR VALUE DEPRECIATION ∗ BADER ALSHAMARY † AND OVIDIU CALIN ‡ Abstract. This paper presents a stochastic model for a car’s value and its depreciation under random repairs modeled by a Poisson process; the usage functional is defined and the optimal selling time is estimated. Exact or approximative formulas are provided where possible. The car’s value is evaluated as an asset with negative return and paying random normally distributed dividends at stochastic times which are Erlang distributed. Key words. Poisson process, selling time, depreciation, variational problem AMS subject classifications. 62P30, 60G99 1. Introduction and Motivation. This paper deals with often asked ques- tions such as When it is optimal to sell my car?, or Do I regret now that I bought a second-hand car instead of a new one? Approaching these questions leads to certain optimization problems which car dealers might find attractive to solve in order to optimize their businesses. Keeping the car for a long time is not optimal since the cumulated cost of repairs will soon worth more than a new car. Selling the car just shortly after buying it is not an optimal decision either because the value of the car depreciates the most at the beginning. We shall become more precise about the conditions of this problem. Assume one buys a second-hand car which does not have a full coverage, so if it breaks down, the owner is solely responsible for the repairs. Therefore, the car owner would like to optimize the following two opposite effects: 1. Maximize the usage of the car, keeping the car for as long as possible; 2. Minimize the incidental costs, by paying as little as possible in repairs. The value of the car goes down over time, while the cumulated value of repairs in- creases. It must be a time T in the future when the owner will decide that he is better off selling the car rather than keeping it and continuing to pour money into it. One of the proposed tasks of the present article is to describe the optimal selling time T . We shall treat this problem by setting up a stochastic differential equation for the value of the car at time t, which takes into consideration the depreciation of the car and the stochastic payments. Sections 2 and 3 introduce the main hypotheses and notions needed to construct the stochastic models of later sections. In section 4 we deal with a stochastic differential equation for the car’s value and with its solution. The time when the car becomes valueless is also computed. Section 5 is dedicated to defining and solving a variational problem defined by introducing the usage functional of a car as the expectation of the difference between the benefit and the loss encountered by keeping and using the car until time t. Section 6 extends some results of previous sections in the case when the deprecia- tion rate is stochastic. The reason behind this approach is the fact that the blue-book * This work was supported by Kuwait University Research Grant No. [SM01/11]. † Department of Mathematics, Faculty of Science, Kuwait University, P.O. Box 5969 Safat 13060, Kuwait (bader231@sci.kuniv.edu.kw). ‡ Department of Mathematics, Eastern Michigan University, Ypsilanti, Michigan 48197, USA (ocalin@emich.edu). 1