Spurious oscillation in a uniform Euler discretisation of linear stochastic differential equations with vanishing delay

We investigate the oscillatory behaviour of a random Euler-type difference equation, intended to serve as a discrete model of a linear Itô stochastic differential equation with vanishing delay. The oscillatory behaviour of the continuous process satisfying this differential equation was partially described in Appleby and Kelly [Asymptotic and oscillatory properties of linear stochastic delay differential equations with vanishing delay, Funct. Differential Equation 11(3–4) (2004) 235–265.] The construction of a discrete model that successfully mimics some of the properties of the continuous process would simplify the analysis, allowing the partial description to be completed. However, care must be taken; a uniform Euler discretisation yields spurious oscillatory behaviour. We present a complete analysis of the uniform scheme.