Bounding stationary averages of polynomial diffusions via semidefinite programming

Juan Kuntz, Michela Ottobre, Guy Bart Stan, Mauricio Barahona

Research output: Contribution to journalArticlepeer-review

21 Citations (Scopus)
42 Downloads (Pure)


We introduce an algorithm based on semidefinite programming that yields increasing (resp., decreasing) sequences of lower (resp., upper) bounds on polynomial stationary averages of diffusions with polynomial drift vector and diffusion coefficients. The bounds are obtained by optimizing an objective, determined by the stationary average of interest, over the set of real vectors defined by certain linear equalities and semidefinite inequalities which are satisfied by the moments of any stationary measure of the diffusion. We exemplify the use of the approach through several applications: a Bayesian inference problem; the computation of Lyapunov exponents of linear ordinary differential equations perturbed by multiplicative white noise; and a reliability problem from structural mechanics. Additionally, we prove that the bounds converge to the infimum and supremum of the set of stationary averages for certain SDEs associated with the computation of the Lyapunov exponents, and we provide numerical evidence of convergence in more general settings.

Original languageEnglish
Pages (from-to)A3891-A3920
Number of pages30
JournalSIAM Journal on Scientific Computing
Issue number6
Publication statusPublished - 20 Dec 2016


  • Lyapunov exponents
  • Moment problems
  • Semidefinite programming
  • Stationary measures
  • Stochastic differential equations

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


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