We design suboptimal filters for a class of continuous-time nonlinear stochastic systems when the measurements are assumed to arrive randomly at discrete times under a Poisson distribution. The proposed filter is a dynamical system with a differential equation and a reset map which updates the estimate whenever a new measurement is received. We analyze the performance of the proposed filter by computing the expected value of the error covariance which is described by a differential equation. We study throughly the conditions under which the error covariance remains bounded, which depend on the system data and the mean sampling rate associated with the measurement process. We also study the particular cases when the error covariance is seen to decrease with the increase in the sampling rate. For the particular case of linear filters, we can also compare the error covariance bounds with the case when the measurements are continuously available.