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.

[1]  Chase L. Beisel,et al.  Understanding and exploiting feedback in synthetic biology , 2013 .

[2]  Antonis Papachristodoulou,et al.  Introducing INTSOSTOOLS: A SOSTOOLS plug-in for integral inequalities , 2015, 2015 European Control Conference (ECC).

[3]  Mohamadreza Ahmadi,et al.  Barrier functionals for output functional estimation of PDEs , 2015, 2015 American Control Conference (ACC).

[4]  Antonis Papachristodoulou,et al.  Designing Genetic Feedback Controllers , 2015, IEEE Transactions on Biomedical Circuits and Systems.

[5]  X. Zhou,et al.  Stochastic Controls: Hamiltonian Systems and HJB Equations , 1999 .

[6]  Mohamadreza Ahmadi,et al.  Stability Analysis for a Class of Partial Differential Equations via Semidefinite Programming , 2016, IEEE Transactions on Automatic Control.

[7]  L. Serrano,et al.  Engineering stability in gene networks by autoregulation , 2000, Nature.

[8]  Abhyudai Singh,et al.  Conditional Moment Closure Schemes for Studying Stochastic Dynamics of Genetic Circuits , 2015, IEEE Transactions on Biomedical Circuits and Systems.

[9]  Andreas Milias-Argeitis,et al.  Optimization-based Lyapunov function construction for continuous-time Markov chains with affine transition rates , 2014, 53rd IEEE Conference on Decision and Control.

[10]  D. Gillespie The chemical Langevin equation , 2000 .

[11]  Konstantinos Michalodimitrakis,et al.  Noise in transcription negative feedback loops: simulation and experimental analysis , 2006, Molecular systems biology.

[12]  João Pedro Hespanha,et al.  Approximate Moment Dynamics for Chemically Reacting Systems , 2011, IEEE Transactions on Automatic Control.

[13]  S. Prajna Barrier certificates for nonlinear model validation , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[14]  R. Khasminskii Stability of Stochastic Differential Equations , 2012 .

[15]  S. Meyn,et al.  Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes , 1993, Advances in Applied Probability.

[16]  Kevin Burrage,et al.  Numerical solutions of stochastic differential equations – implementation and stability issues , 2000 .

[17]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[18]  R. Khasminskii Stochastic Stability of Differential Equations , 1980 .

[19]  Antonis Papachristodoulou,et al.  Advanced Methods and Algorithms for Biological Networks Analysis , 2006, Proceedings of the IEEE.

[20]  Mohamadreza Ahmadi,et al.  Dissipation inequalities for the analysis of a class of PDEs , 2016, Autom..

[21]  C. Gillespie Moment-closure approximations for mass-action models. , 2009, IET systems biology.

[22]  D. Gillespie A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions , 1976 .

[23]  R. C. Merton,et al.  Theory of Rational Option Pricing , 2015, World Scientific Reference on Contingent Claims Analysis in Corporate Finance.