Stochastic models, especially Markov process models, find numerous applications across various fields such as mathematical finance, control theory and systems biology, for example. While in many of these applications stochastic models offer crucial fidelity over their deterministic counterparts, their analysis is generally more involved. In particular, even the simplest stochastic models rarely admit an analytical solution such that numerical approximation techniques dominate the analysis of stochastic models in practice. Many such techniques, however, rely on unverifiable assumptions and only few provide mechanisms to rigorously quantify the induced approximation error. Recently developed moment bounding schemes [1-7] seek to address this shortcoming. By means of convex optimization, such bounding schemes enable the computation of hard, theoretically guaranteed bounds on the stationary as well as transient moments associated with jump-diffusion processes with polynomial data. While the Julia ecosystem has everything in store to set up and solve the respective optimization problems through JuMP with the help of extensions like MomentOpt.jl and SumOfSquares.jl, it still requires expert knowledge to do so. To close this gap and make moment bounding more accessible to practitioners with a diverse background, we present the meta-package MarkovBounds.jl which integrates the functionalities of the aforementioned packages to automatically generate and solve the respective moment bounding problems from high-level input data.

In this brief talk, we sketch the idea of moment bounding schemes and showcase with toy examples drawn from stochastic chemical kinetics and mathematical finance how MarkovBounds.jl enables non-expert users to readily apply them.

[1] Garrett R. Dowdy and Paul I. Barton. Bounds on stochastic chemical kinetic systems at steady state. The Journal of chemical physics, 148(8):84106,2018.

[2] Garrett R. Dowdy and Paul I. Barton. Dynamic bounds on stochastic chemical kinetic systems using semidefinite programming. The Journal of Chemical Physics, 149(7):74103, 2018.

[3] Khem Raj Ghusinga, Cesar A. Vargas-Garcia, Andrew Lamperski, and Abhyudai Singh. Exact lower and upper bounds on stationary moments instochastic biochemical systems. Physical Biology, 14(4):04LT01, 2017.

[4] Juan Kuntz, Michela Ottobre, Guy-Bart Stan, and Mauricio Barahona. Bounding stationary averages of polynomial diffusions via semidefinite programming. SIAM Journal on Scientific Computing, 38(6):A3891–A3920,2016.

[5] Juan Kuntz, Philipp Thomas, Guy-Bart Stan, and Mauricio Barahona. Bounding the stationary distributions of the chemical master equation via mathematical programming. The Journal of Chemical Physics,151(3):34109, 2019.

[6] Yuta Sakurai and Yutaka Hori. A convex approach to steady state moment analysis for stochastic chemical reactions. In2017 IEEE 56th Annual Conference on Decision and Control (CDC), pages 1206–1211. IEEE, 2017

[7] Holtorf, Flemming, and Paul I. Barton. "Tighter bounds on transient moments of stochastic chemical systems." arXiv preprint arXiv:2104.01309 (2021).