  We examine hitting probability problems for Ornstein-Uhlenbeck (OU) processes
and Geometric Brownian motions (GBM) with respect to exponential boundaries
related to problems arising in risk theory and asset and liability models in
pension funds. In Section 2 we consider the OU process described by the
Stochastic Differential Equation (SDE) $dX_t = \mu X_t dt + \sigma dW_t$ with
$X_0=x_0$ evolving between a lower and an upper deterministic exponential
boundary. Both the finite horizon ``ruin probability'' problem and the
corresponding infinite horizon problem is examined in the low noise case, using
the Wentzell-Freidlin approach in order to obtain logarithmic asymptotics for
the probability of hitting either the lower or the upper boundary. The
resulting variational problems are studied in detail. The exponential rate
characterizing the ruin probability and the ``path to ruin'' are obtained by
their solution. Logarithmic asymptotics for the meeting probability in a pair
of OU processes with different positive drift coefficients, driven by
independent Brownian motions is also obtained using Wentzell-Freidlin
techniques. The optimal paths followed by the two processes and the meeting
time $T$ are determined by solving a variational problem with transversality
conditions. In Section 3 a corresponding problem involving a Geometric Brownian
motion is considered. Since in this case, an exact, closed form solution is
also available and we take advantage of this situation in order to explore
numerically the quality of the Large Deviations results obtained using the
Wentzell-Freidlin approach.
