  We consider a stochastic model of investment on an asset of a stock market
for a prudent investor. She decides to buy permanent goods with a fraction $\a$
of the maximum amount of money owned in her life in order that her economic
level never decreases. The optimal strategy is obtained by maximizing the
exponential growth rate for a fixed $\a$. We derive analytical expressions for
the typical exponential growth rate of the capital and its fluctuations by
solving an one-dimensional random walk with drift.
