Heart rate is a vital sign, whereas heart rate variability is an important quantitative measure of cardiovascular regulation by the autonomic nervous system. While the design of algorithms to compute heart rate and assess heart rate variability is an active area of research, none of the current approaches considers the natural point process structure of human heart beats and none gives instantaneous estimates of heart rate variability. We model the stochastic structure of heart beat intervals as a history-dependent inverse Gaussian process and derive from it an explicit probability density that gives a new definition of heart rate and a new definition of heart rate variability as the instantaneous R-R interval and heart rate standard deviations. We estimate the time-varying parameters of the inverse Gaussian model by local maximum likelihood and assess model goodness-of-fit by Kolmogorov-Smirnov tests based on the time-rescaling theorem. We illustrate our new definitions in an analysis of human heart beat intervals from ten healthy subjects undergoing a tilt table experiment. While several studies have identified deterministic, non-linear dynamical features in human heart beat intervals, our analysis shows that a highly accurate description of these series at rest and in extreme physiological conditions may be given by an elementary, physiologically-based, stochastic model.