Optimal stopping for Shepp's urn with risk aversion

摘要

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 -> infinity, either m - p >= alpha root 2p, when the gain function cannot be positive, or m - p < alpha root 2p, when it is sufficient to have k similar to root p log p, where alpha 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 a 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.