Optimal stopping for Shepp's urn with risk aversion

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 , when the gain function cannot be positive, or , when it is sufficient to have , 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.