Optimal stopping for Shepp's urn with risk aversion

Robert W. Chen, Ilie Grigorescu, Min Kang

Research output: Contribution to journalArticlepeer-review


An (m,p) urn contains m balls of value − 1 and p balls of value +1. A player starts with fortune k and in each game draws a ball without replacement with the fortune increasing by one unit if the ball is positive and decreasing by one unit if the ball is negative, having to stop when k = 0 (risk aversion). Let V(m,p,k) be the expected value of the game. We are studying the question of the minimum k such that the net gain function of the game V(m,p,k) − k is positive, in both the discrete and the continuous (Brownian bridge) settings. Monotonicity in various parameters m, p, k is established for both the value and the net gain functions of the game. For the cut-off value k, since the case m − p <0 is trivial, for p → ∞, either (Formula presented.) , when the gain function cannot be positive, or (Formula presented.) , when it is sufficient to have (Formula presented.) , where α is a constant. We also determine an approximate optimal strategy with exponentially small probability of failure in terms of p. The problem goes back to Shepp [8], who determined the constant α in the unrestricted case when the net gain does not depend on k. A new proof of his result is given in the continuous setting.

Original languageEnglish (US)
Pages (from-to)702-722
Number of pages21
Issue number4
StatePublished - Jul 4 2015


  • Brownian bridge
  • Chow and Robbins S/n problem
  • Shepp's urn
  • optimal stopping
  • optimal strategy

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation


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