The Grossman and Zhou investment strategy is not always optimal

Grossman and Zhou [1993. Optimal investment strategies for controlling drawdowns. Math. Finance 3, 241-276] proposed a strategy to maximize the asymptotic long-run growth rate of one's fortune Ft subject to its never falling below , where 0[less-than-or-equals, slant][lambda][less-than-or-equals, slant]1 is a fixed constant chosen by the investor and r is a fixed, known, non-negative, continuously compounded interest rate on invested capital. In this paper we show that the strategy proposed in Grossman and Zhou does not retain its optimal long-run growth property when generalized to the discrete-time setting.