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In this paper we study model reduction of linear and bilinear quadraticstochastic control problems with parameter uncertainties. Specifically, weconsider slow-fast systems with unknown diffusion coefficient and study theconvergence of the slow process in the limit of infinite scale separation. Theaim of our work is two-fold: Firstly, we want to propose a general frameworkfor averaging and homogenisation of multiscale systems with parametricuncertainties in the drift or in the diffusion coefficient. Secondly, we wantto use this framework to quantify the uncertainty in the reduced system byderiving a limit equation that represents a worst-case scenario for any given(possibly path-dependent) quantity of interest. We do so by reformulating theslow-fast system as an optimal control problem in which the unknown parameterplays the role of a control variable that can take values in a closed boundedset. For systems with unknown diffusion coefficient, the underlying stochasticcontrol problem admits an interpretation in terms of a stochastic differentialequation driven by a G-Brownian motion. We prove convergence of the slowprocess with respect to the nonlinear expectation on the probability spaceinduced by the G-Brownian motion. The idea here is to formulate the nonlineardynamic programming equation of the underlying control problem as aforward-backward stochastic differential equation in the G-Brownian motionframework (in brief: G-FBSDE), for which convergence can be proved by standardmeans. We illustrate the theoretical findings with two simple numericalexamples, exploiting the connection between fully nonlinear dynamic programmingequations and second-order BSDE (2BSDE): a linear quadratic Gaussian regulatorproblem and a bilinear multiplicative triad that is a standard benchmark systemin turbulence and climate modelling.

Model reduction and uncertainty quantification of multiscale diffusions with parameter uncertainties using nonlinear expectations