  This paper considers importance sampling for estimation of rare-event
probabilities in a specific collection of Markovian jump processes used for
e.g. modelling of credit risk. Previous attempts at designing importance
sampling algorithms have resulted in poor performance and the main contribution
of the paper is the design of efficient importance sampling algorithms using
subsolutions. The dynamics of the jump processes causes the corresponding
Hamilton-Jacobi equations to have an intricate state-dependence, which makes
the design of efficient algorithms difficult. We provide theoretical results
that quantify the performance of importance sampling algorithms in general and
construct asymptotically optimal algorithms for some examples. The
computational gain compared to standard Monte Carlo is illustrated by numerical
examples.
