  Even in the face of deteriorating and highly volatile demand, firms often
invest in, rather than discard, aging technologies. In order to study this
phenomenon, we model the firm's profit stream as a Brownian motion with
negative drift. At each point in time, the firm can continue operations, or it
can stop and exit the project. In addition, there is a one-time option to make
an investment which boosts the project's profit rate. Using stochastic
analysis, we show that the optimal policy always exists and that it is
characterized by three thresholds. There are investment and exit thresholds
before investment, and there is a threshold for exit after investment. We also
effect a comparative statics analysis of the thresholds with respect to the
drift and the volatility of the Brownian motion. When the profit boost upon
investment is sufficiently large, we find a novel result: the investment
threshold decreases in volatility.
