We model the dynamics $X_t$ of a stock index as a
non--lognormally--distributed generalization of the geometric
Brownian motion. In detail, $X_t$ is supposed to be a
weak solution of a one-dimensional stochastic differential equation
of the form
$$ dX_t = b(X_t) dt + \sigma X_t dW_t, $$
with volatility $\sigma > 0$ and Brownian motion $W_t$.
For a dichotomous Bayesian decision problem concerning the size
of the drift $b$, we investigate
the (average) reduction of decision risk
that can be obtained by observing the path of $X$. We also
show some connections with relative entropy
and with contingent claim pricing.