Optimal shopping when the sales are on - A Markovian full-information best-choice problem

Mahmut Parlar, David Perry, Wolfgang Stadje

Research output: Contribution to journalArticlepeer-review


We study a full-information best-choice problem viewed in a shopping context. A certain commodity can be found at certain random times with stochastically fluctuating prices. While the prices may have a tendency to decrease, the instants at which items are offered become less frequent and it is possible that the item currently found will be the last one. The prospective customer's objective is to buy at the right time so as to minimize the expected price of the acquired item. We propose a two-dimensional Markov chain model with a rather general continuous-time point process structure and dependence of the random prices on the availability times of the items. The value function v of the associated optimal stopping problem is characterized as the smallest solution of a two-dimensional integral equation; this allows us to find the optimal policy under certain conditions. In particular, we consider a nonhomogeneous Poisson model for which more specific results can be obtained. We derive a differential equation of which v is the uniformly smallest nonnegative solution. This way v is determined up to a boundary condition at infinity. We provide criteria for identifying a solution as the value function and also for the natural stopping rule to be optimal. Several examples are given.

Original languageEnglish
Pages (from-to)351-371
Number of pages21
JournalStochastic Models
Issue number3
StatePublished - Jul 2007


  • Best-choice problem
  • Boundary condition at infinity
  • Full information
  • Integral equation
  • Nonhomogeneous Poisson process
  • Two-dimensional continuous-time Markov chain
  • Value function

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Applied Mathematics


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