Nonnormality and stochastic differential equations

A highly nonnormal Jacobian may give rise to large transients. This behaviour has been shown to have implications for (a) the relevance of linearising a nonlinear system and (b) the timestep restrictions required to keep a numerical method stable. Here, we show that nonnormality also manifests itself for stochastic differential equations. We give an example of a family of systems that is stable without noise, but can be made exponentially unstable in mean-square by a noise perturbation that shrinks to zero as the nonnormality increases. We then show via finite-time convergence theory that an Euler approximation shares the same property, giving a discrete analogue of the result. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 65C30, 34F05