| This dissertation consists of two parts. The first part studies subordinate Ornstein-Uhlenbeck (SubOU) processes, i.e., OU diffusions time changed by Levy subordinators. I construct their sample path decomposition, show that they possess mean-reverting jumps, study their equivalent measure transformations, and the eigenfunction expansion of their transition semigroups in terms of Hermite polynomials. As an application, I propose a new class of commodity models with mean-reverting jumps based on SubOU process. Further time changing by the integral of a CIR process plus a deterministic function of time induces stochastic volatility and time inhomogeneity, such as seasonality, into the models. I 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.;The second part proposes a new approach to solve finite and infinite horizon optimal stopping problems for symmetric Hunt processes, a large class of Markov processes that includes one-dimensional diffusions, birth-death processes, jump-diffusions and pure jumps and continuous-time Markov chains obtained by time changing diffusions and BD processes with Levy subordinators. When the expectation operator has a purely discrete spectrum, the value function of a discrete time optimal stopping problem with square-integrable rewards has the expansion in the eigenfunctions of the expectation operator. The Bellman's backward induction for the value function then reduces to an explicit recursion for the expansion coefficients. The value function of the continuous optimal stopping problem is then obtained by extrapolating the value function of the discrete problem to the limit via Richardson extrapolation. To illustrate this approach, the dissertation develops several applications in evaluation of American-style commodity futures options, Bermudan-style real options and callable consols. |
