Bayesian multitarget tracking is an inherently nonlinear problem. Even
when the state models and sensor noise associated with individual targets and observations is Gaussian, the "true" data likelihood, as formulated within the framework of finite-set statistics, is non-Gaussian. Missed detections and false alarms, combined with the fact that targets may enter and leave the scene at random times, complicate matters further. The resulting Bayesian posterior is analytically foreboding, and many conventional estimators are not even defined. We propose an algorithm for generating samples from the posterior based on jump-diffusion processes. When discretized for computer implementation, the jump-diffusion method falls into the general class of Markov chain Monte Carlo methods. The diffusions refine estimates of continuous parameters, such as positions and velocities, whereas the jumps are responsible for major discrete changes, such as adding and removing targets. Jump-diffusion processes have been previously applied to performing automatic target recognition in infrared images and tracking multiple targets using raw narrowband sensor array and high-resolution range profile data. Here, we apply jump-diffusion to the more traditional class of target tracking problems where raw sensor data is preprocessed into reports, but the report-to-target association is unknown. Our formulation maintains the flavor of other recent work employing finite-set statistics, in that no attempts to explicitly associate specific reports with specific targets are needed.