A level crossing predictor is a predictor process $Y(t)$, possibly multivariate, which can be used to predict whether a specified process $X(t)$ will cross a predetermined level or not. A natural criterion on how good a predictor is, can be the probability that a crossing is detected a sufficient time ahead, and the number of times the predictor makes a false alarm. If $X$ is Gaussian and the process $Y$ is designed to detect only level crossings, one is led to consider a multivariate predictor process $Y(t)$ such that a level crossing is predicted for $X(t)$ if $Y(t)$ enters some nonlinear region in $R^p$. In the present paper we develop the probabilistic methods for evaluation of such an alarm system. The basic tool is a model for the behavior of $X(t)$ near the points where $Y(t)$ enters the alarm region. This model includes the joint distribution of location and direction of $Y(t)$ at the crossing points.
"Model Processes in Nonlinear Prediction with Application to Detection and Alarm." Ann. Probab. 8 (4) 775 - 792, August, 1980. https://doi.org/10.1214/aop/1176994665