Stochastic systems with small noise, analysis and simulation; a phase locked loop example

Systems with wide bandwidth noise inputs are a common occurrence in stochastic control and communication theory and elsewhere, e.g., tracking or synchronization systems such as phase locked loops (PLL). One is often interested in calculating such quantities as the probability of escape from a desired “error” set, in some time interval, or the mean time for such escape. Diffusion approximations (the system obtained in the limit bandwidth as the $ \to \infty $) are often used for this since they are easier to analyze. When the noise effects in the physical system are small, one is tempted to do an asymptotic analysis (noise intensity $ \to 0$) on the diffusion approximation, and use this for the desired estimates on the original system. Such a procedure does not work in general: the double limit bandwidth $ \to \infty $, intensity $ \to 0$ is not always justified. Under quite broad conditions on the noise processes, it is justified for the systems studied here. We study a particular form of the PLL owing to...