  This paper studies subordinate Ornstein-Uhlenbeck (OU) processes, i.e., OU
diffusions time changed by L\'{e}vy subordinators. We construct their sample
path decomposition, show that they possess mean-reverting jumps, study their
equivalent measure transformations, and the spectral representation of their
transition semigroups in terms of Hermite expansions. As an application, we
propose a new class of commodity models with mean-reverting jumps based on
subordinate OU process. Further time changing by the integral of a CIR process
plus a deterministic function of time, we induce stochastic volatility and time
inhomogeneity, such as seasonality, in the models. We obtain analytical
solutions for commodity futures options in terms of Hermite expansions. The
models are consistent with the initial futures curve, exhibit Samuelson's
maturity effect, and are flexible enough to capture a variety of implied
volatility smile patterns observed in commodities futures options.
