Ruin Theory Problems in Simple SDE Models with Large Deviation Asymptotics

Abstract

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.