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Observed oscillations of the Antarctic stratospheric polar vortex often resemble those in Kida's model of an elliptical vortex in a linear background flow. Here, Kida's model is used to investigate the dynamics of "vortex-splitting" stratospheric sudden warmings (SSWs), such as the Antarctic event of 2002. SSWs are identified with a bifurcation in the periodic orbits of the model. The influence of "tropospheric macroturbulence" on the vortex is modelled by allowing the linear background forcing flow to be driven by a random process, with a finite decorrelation time (an Ornstein-Uhlenbeck process). It is shown that this stochasticity generates a random walk across the state-space of periodic orbits, which will eventually lead to a bifurcation point after which an SSW will occur. In certain asymptotic limits, the expected time before an SSW occurs can be found by solving a "first passage time" problem for a stochastic differential equation, allowing the dependence of the expected time to an SSW on the model parameters to be elucidated. Results are verified using both Kida's model and single-layer quasi-geostrophic simulations. The results point towards a "noise-memory" paradigm of the winter stratosphere, according to which the forcing history determines whether the vortex is quiescent, whether it undergoes large amplitude nonlinear oscillations or, in extreme cases, whether the vortex will eventually split.