  We establish a recursive representation that fully decouples jumps from a
large class of multivariate inhomogeneous stochastic differential equations
with jumps of general time-state dependent unbounded intensity, not of
L\'evy-driven type that essentially benefits a lot from independent and
stationary increments. The recursive representation, along with a few related
ones, are derived by making use of a jump time of the underlying dynamics as an
information relay point in passing the past on to a previous iteration step to
fill in the missing information on the unobserved trajectory ahead. We prove
that the proposed recursive representations are convergent exponentially fast
in the limit, and can be represented in a similar form to Picard iterates under
the probability measure with its jump component suppressed. On the basis of
each iterate, we construct upper and lower bounding functions that are also
convergent towards the true solution as the iterations proceed. We provide
numerical results to justify our theoretical findings.
