Ornstein-Uhlenbeck Process for Horse Race Betting: A Micro-Macro Analysis of Herding and Informed Bettors

We model the time evolution of single win odds in Japanese horse racing as a stochastic process, deriving an Ornstein--Uhlenbeck process by analyzing the probability dynamics of vote shares and the empirical time series of odds movements. Our framework incorporates two types of bettors: herders, who adjust their bets based on current odds, and fundamentalists, who wager based on a horse's true winning probability. Using data from 3450 Japan Racing Association races in 2008, we identify a microscopic probability rule governing individual bets and a mean-reverting macroscopic pattern in odds convergence. This structure parallels financial markets, where traders' decisions are influenced by market fluctuations, and the interplay between herding and fundamentalist strategies shapes price dynamics. These results highlight the broader applicability of our approach to non-equilibrium financial and betting markets, where mean-reverting dynamics emerge from simple behavioral interactions.