The saddlepoint approximation factors over sample paths of recursively compounded processes

TL;DR

This paper establishes an identity linking multivariate and univariate saddlepoint approximations for sample path probabilities in recursively compounded stochastic processes, simplifying their analysis.

Contribution

It introduces a novel identity connecting multivariate and univariate saddlepoint approximations for a broad class of processes, with two rigorous proofs provided.

Findings

The identity applies to processes like branching, compound Poisson, and Lévy processes.

It simplifies the computation of sample path probabilities in these processes.

Two proofs—analytic and probabilistic—validate the identity.

Abstract

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.