  We consider the problem of estimating parameter sensitivities for stochastic
models of multiscale reaction networks. These sensitivity values are important
for model analysis, and, the methods that currently exist for sensitivity
estimation mostly rely on simulations of the stochastic dynamics. This is
problematic because these simulations become computationally infeasible for
multiscale networks due to reactions firing at several different timescales.
However it is often possible to exploit the multiscale property to derive a
"model reduction" and approximate the dynamics as a Piecewise Deterministic
Markov process (PDMP), which is a hybrid process consisting of both discrete
and continuous components. The aim of this paper is to show that such PDMP
approximations can be used to accurately and efficiently estimate the parameter
sensitivity for the original multiscale stochastic model. We prove the
convergence of the original sensitivity to the corresponding PDMP sensitivity,
in the limit where the PDMP approximation becomes exact. Moreover we establish
a representation of the PDMP parameter sensitivity that separates the
contributions of discrete and continuous components in the dynamics, and allows
one to efficiently estimate both contributions.
