A bold strategy is not always optimal in the presence of inflation

A gambler, with an initial fortune less than 1, wants to buy a house which sells today for 1. Due to inflation, the price of the house tomorrow will be 1 + ca, where ca is a nonnegative constant, and will continue to go up at this rate, becoming (1 + a)n' on the nth day. Once each day, he can stake any amount of fortune in his possession, but no more than he possesses, on a primitive casino. It is well known that, in a subfair primitive casino without the presence of inflation, the gambler should play boldly. The presence of inflation would motivate the gambler to recognize the time value of his fortune and to try to reach his goal as quickly as possible; intuitively, we would conjecture that the gambler should again play boldly. However, in this note we will show that, unexpectedly, bold play is not necessarily optimal.