A class of stochastic processes characterized by rapid transitions in their structure is considered and the quickest detection of such transitions is studied in a Bayesian framework. The emphasis is on stochastic processes consisting of a randomly arrived causal pulse (possibly with a set of random parameters such as amplitude and duration) and an additive white Gaussian noise. In this model, the pulse shape and the prior joint density of the arrival time and other random parameters are assumed known. The task of quickest detection in this paper is described mathematically by minimizing the expected detection error. The detection error is represented by a nonlinear function of the distance between the actual transition time and its associated detection time. The assumptions on this function are fairly mild and allow to flexibly design its shape for a desired trade-off between the detection delay and the false alarm rate. Two special cases of such design are well known error measures: mean squared and mean absolute error. The quickest detection problem-a subclass of optimal stopping time problems-is formulated as a stochastic optimal control problem and is resolved using dynamic programming. The optimal detection rule is determined in terms of the solution of an integral equation that cannot be directly solved due to its complexity. This equation is later used to develop a class of suboptimal detection rules and a lower bound on the minimum error. Using this lower bound, it is shown for a numerical example that the suboptimal detector is nearly optimal.