Continuous Time Random Matching

We show the existence of independent random matching of a large population in a continuous-time dynamical system, where the matching intensities could be general non-negative jointly continuous functions on the space of type distributions and the time line. In particular, we construct a continuum of independent continuous-time Markov processes that is derived from random mutation, random matching, random type changing and random break up with time-dependent and distribution dependent parameters. It follows from the exact law of large numbers that the deterministic evolution of the agents’ realized type distribution for such a continuous-time dynamical system can be determined by a system of differential equations. The results provide the first mathematical foundation for a large literature on continuous-time search-based models of labor markets, money, and over-the-counter markets for financial products.