  This paper presents an identity between the multivariate and univariate
saddlepoint approximations applied to sample path probabilities for a certain
class of stochastic processes. This class, which we term the recursively
compounded processes, includes branching processes and other models featuring
sums of a random number of i.i.d. terms; and compound Poisson processes and
other L\'evy processes in which the additive parameter is itself chosen
randomly. For such processes, $\hat{f}_{X_1,\dotsc,X_N |
X_0=x_0}(x_1,\dots,x_N) = \prod_{n=1}^N \hat{f}_{X_n |
X_0=x_0,\dots,X_{n-1}=x_{n-1}}(x_n),$ where the left-hand side is a
multivariate saddlepoint approximation applied to the random vector
$(X_1,\dots,X_N)$ and the right-hand side is a product of univariate
saddlepoint approximations applied to the conditional one-step distributions
given the past. Two proofs are given. The first proof is analytic, based on a
change-of-variables identity linking the functions that arise in the respective
saddlepoint approximations. The second proof is probabilistic, based on a
representation of the saddlepoint approximation in terms of tilted
distributions, changes of measure, and relative entropies.
