The Averaging Principle for Stochastic Slow-Fast Systems Driven by G-Brownian Motion

Introduction: This paper establishes a novel averaging principle for stochastic slow-fast systems where the driving noise is subject to Knightian volatility uncertainty, modeled by a d-dimensional G-Brownian motion B(t). Classical averaging theory fails under volatility ambiguity, and we provide a framework to average the fast dynamics in the worst-case sense.

Methode: We introduce a concept of G-invariant measure, which generalizes the classical invariant measure to the sublinear expectation space. This measure encapsulates a set of possible invariant laws for each admissible volatility scenario. Using this, we define the averaged coefficients for the slow component in a worst-case sense. Under appropriate assumptions, we employ Khasminskii's time-discretization technique and tools from G-stochastic calculus to prove the convergence.

Results: Our main result demonstrates that as the timescale separation parameter tends to zero, the slow component of the original multiscale system converges to the solution of a simplified, averaged equation. The convergence is established in two strong senses: in capacity (quasi-surely) and in the L2-norm under the G-expectation.

Conclusions: This work extends the classical stochastic averaging principle to environments with distributional ambiguity. The results offer a robust mathematical tool for simplifying complex multiscale systems—such as those in financial risk, climate economics, or epidemic modeling—where fast variables are subject to uncertain volatility.