The output of an ideal photodetector that is modeled as a doubly stochastic Poisson process is used to excite a phase-locked loop. The rate of this Poisson process contains an angle-modulated subcarrier process that is modeled as a Gauss-Markov process. A measure transformation involving a martingale translation is applied to the suboptimal filter used to estimate the phase. Cycle slipping is then discussed with respect to the evolution of the phase error process under the new measure. The approach is in the context of a first-passage problem and uses a Kramers-Moyal expansion of the equation satisfied by the mean slip time. Various approximations to the jump rate and size are assumed, and a perturbation analysis results in asymptotic representations of the corresponding solutions. Comparisons with solutions obtained from so-called diffusion approximations are also included, as well as a specific example to illustrate the procedure.

An extension of the well-known Cameron-Martin formula can be interpreted as the expectation of a stochastic reliability function applicable in those situations where nondecreasing failure rates are desired. This follows ff the failure rate is modeled as the square of a Gauss-Markov process. We describe the methodology for the general vector case, and then specialize the results to the one-dimensional case so as to obtain an exact closed-form expression for the reliability function. Using the theory of recurrent and transient processes, we then show how the choice of a model parameter and the initial state influence reliability.

The evaluation of error-probability bounds for binary detection problems involving continuons-time stochastic processes as signals is considered. These bounds are of interest because, in even the simplest detection problems, the computation of the exact probabilities of error is usually mathematically intractable. The method used consists of applying some results from martingale theory to detection and estimation problems. Only discontinuous observations that contain the rate process associated with a counting process are considered. The problem addressed is to evaluate Chernoff bounds on error probabilities for the likelihood-ratio test. The solution procedure consists of a measure transformation technique that makes it possible to obtain an expression for the Chernoff bound in terms of an expectation of a multiplicative functional of the conditional mean signal (rate process) estimates. If the processes involved are Markov, it is then possible to represent the above expression as a solution to a partial differential equation that is derived from the backward equation of Kolmogorov. The above procedure is repeated when the optimal estimates are replaced by suboptimal estimates. Examples are given to illustrate the technique.