A quadratic BSDE framework for risk-adjusted noise-induced switching in bistable nonlinear oscillators under stochastic volatility

Noise-induced transitions between coexisting attractors are ubiquitous in nonlinear devices and natural systems, yet risk quantification becomes nontrivial when excitation intensity is intermittent and exhibits volatility clustering. We propose a quadratic backward stochastic differential equation (BSDE) approach to compute a dynamic, risk-adjusted switching metric for bistable oscillators driven by stochastic volatility. The forward dynamics couples a double-well drift with a Cox–Ingersoll–Ross-type variance process and correlated Brownian motions, capturing bursts of agitation often observed in ambient vibrations and complex environments. For an overload or switching payoff at a terminal horizon, we adopt the entropic risk measure, leading to a quadratic BSDE whose exponential transform is a martingale. This yields a numerically stable regression Monte Carlo algorithm that estimates the time-resolved conditional risk along trajectories and constructs parametric risk surfaces with respect to initial state and volatility. Numerical experiments demonstrate that volatility and cross-correlation strongly modulate the probability–risk gap: regimes with comparable raw switching probabilities may exhibit markedly different risk profiles under risk aversion. The proposed BSDE-based metric provides an early-warning indicator for impending transitions and a practical tool for robust design and reliability assessment of bistable nonlinear systems under intermittent stochastic forcing.