We study the infinite time ruin probability
for an insurance company in the classical
Cramér--Lundberg model
with finite exponential moments. The additional
nonclassical feature is that
the company is also allowed to invest in some stock market,
modeled by geometric Brownian motion.
We obtain an exact analogue
of the classical estimate for the ruin probability without
investment, that is, an exponential
inequality. The exponent is larger than
the one obtained without investment, the classical Lundberg
adjustment coefficient, and thus one gets a sharper bound on the ruin probability.
¶ A surprising result is that the trading strategy yielding the
optimal asymptotic decay of the ruin probability simply
consists in holding a fixed quantity (which can be explicitly
calculated) in the risky asset, independent of the current reserve. This
result is in apparent contradiction to the common believe that
"rich" companies should invest more in risky assets than "poor"
ones. The reason for this seemingly paradoxical result is that
the minimization of the ruin probability is an extremely
conservative optimization criterion, especially for "rich"
companies.