Bounding the Moments of Polynomial Jump-Diffusion Processes

We present MarkovBounds.jl -- A meta-package composing several existing packages from the Julia ecosystem to enable the computation of hard, theoretically guaranteed bounds on the stationary and transient moments of jump-diffusion Markov processes with polynomial data from high-level, practitioner-friendly user inputs.

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.

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