  In this paper we study model reduction of linear and bilinear quadratic
stochastic control problems with parameter uncertainties. Specifically, we
consider slow-fast systems with unknown diffusion coefficient and study the
convergence of the slow process in the limit of infinite scale separation. The
aim of our work is two-fold: Firstly, we want to propose a general framework
for averaging and homogenisation of multiscale systems with parametric
uncertainties in the drift or in the diffusion coefficient. Secondly, we want
to use this framework to quantify the uncertainty in the reduced system by
deriving a limit equation that represents a worst-case scenario for any given
(possibly path-dependent) quantity of interest. We do so by reformulating the
slow-fast system as an optimal control problem in which the unknown parameter
plays the role of a control variable that can take values in a closed bounded
set. For systems with unknown diffusion coefficient, the underlying stochastic
control problem admits an interpretation in terms of a stochastic differential
equation driven by a G-Brownian motion. We prove convergence of the slow
process with respect to the nonlinear expectation on the probability space
induced by the G-Brownian motion. The idea here is to formulate the nonlinear
dynamic programming equation of the underlying control problem as a
forward-backward stochastic differential equation in the G-Brownian motion
framework (in brief: G-FBSDE), for which convergence can be proved by standard
means. We illustrate the theoretical findings with two simple numerical
examples, exploiting the connection between fully nonlinear dynamic programming
equations and second-order BSDE (2BSDE): a linear quadratic Gaussian regulator
problem and a bilinear multiplicative triad that is a standard benchmark system
in turbulence and climate modelling.
