The saddlepoint approximation factors over sample paths of recursively compounded processes

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.