  This work provides a semi-analytic approximation method for decoupled
forwardbackward SDEs (FBSDEs) with jumps. In particular, we construct an
asymptotic expansion method for FBSDEs driven by the random Poisson measures
with {\sigma}-finite compensators as well as the standard Brownian motions
around the small-variance limit of the forward SDE. We provide a semi-analytic
solution technique as well as its error estimate for which we only need to
solve essentially a system of linear ODEs. In the case of a finite jump measure
with a bounded intensity, the method can also handle state-dependent and hence
non-Poissonian jumps, which are quite relevant for many practical applications.
