  The present paper introduces a jump-diffusion extension of the classical
diffusion default intensity model by means of subordination in the sense of
Bochner. We start from the bi-variate process $(X,D)$ of a diffusion state
variable $X$ driving default intensity and a default indicator process $D$ and
time change it with a L\'{e}vy subordinator ${\mathcal{T}}$. We characterize
the time-changed process
$(X^{\phi}_t,D^{\phi}_t)=(X({\mathcal{T}}_t),D({\mathcal{T}}_t))$ as a
Markovian--It\^{o} semimartingale and show from the Doob--Meyer decomposition
of $D^{\phi}$ that the default time in the time-changed model has a
jump-diffusion or a pure jump intensity. When $X$ is a CIR diffusion with
mean-reverting drift, the default intensity of the subordinate model (SubCIR)
is a jump-diffusion or a pure jump process with mean-reverting jumps in both
directions that stays nonnegative. The SubCIR default intensity model is
analytically tractable by means of explicitly computed eigenfunction expansions
of relevant semigroups, yielding closed-form pricing of credit-sensitive
securities.
