Unlike traditional graph-based linear dynamics, where agents exchange opinions with their neighbors in a static social graph regardless of their differences in opinions, the bounded confidence opinion dynamics models exchange between agents with similar opinions. We generalize the bounded confidence opinion dynamics model by incorporating pairwise stochastic interactions, probabilistic influencing based on opinion differences and the self or endogenous evolutions of the agent opinions, which are represented by random processes. The opinion exchanges resulting from influencing have pairwise contraction effects, whereas endogenous motions have an expansive effect, for instance, of a diffusive nature. We analytically characterize the conditions under which this stochastic dynamics is stable in an appropriate sense. In the diffusive case, the presence of heavy tailed influence functions with a Pareto exponent of 2 is critical for stability (for a pair of agents an influence function maps opinion differences to probabilities of influence). Moreover, this model sheds light on dynamics that combine aspects of graph-based and bounded confidence dynamics.