An optimization-based method for bounding state functionals of nonlinear stochastic systems

We propose a method for bounding state functionals of a class of nonlinear stochastic differential equations. Given a class of state functionals of a stochastic system, the Feynman-Kac Lemma is a backward in time partial differential equation that describes the evolution of the state functional. We bound these state functionals based on a method which uses barrier functionals. We show that, under the assumption of polynomial data, the bounds can be obtained by using semi-definite programming. The proposed method is then applied to the case study of noise in genetic negative autoregulation to bound a functional of the second moment, which is of specific interest to experimental assays. The bound obtained is found to be in good agreement with experimental results in the literature.