Mathematical modeling, estimation and application in finance

Abstract

Parameter estimation has wide applications in such fields as finance, oil deposit detection,etc. In this dissertation, we discuss the parameter estimation problems in a stochastic differential equation and a partial differential equation. In chapter one, we provide a general moment estimator for the Ornstein-Uhlenbeck Process driven by a-stable Lévy motion. When the noise is an a-stable Lévy motion, the process does not have the second moment which makes the parametric estimation problem more difficult. In this case, there are limited papers dealing with the parametric estimation problem. In previous work, one can only estimate the drift parameter q assuming the other parameters (a, s and b) are known and under discrete observations. In most literature, one also needs to assume that the time step h depends on n and converges to 0 as n goes to infinity. This means that a high frequency data must be available for the estimators to be effective. The main mathematical tool that we use is ergodic theory and sample characteristic functions so that we can estimate all the parameters simultaneously. We also obtain the strong consistency and asymptotic normality of the proposed joint estimators when the time step h remains constant. In chapter three, we describe how to use implicit sampling in parameter estimation problems where the goal is to find parameters of a partial differential equation, such that the output of the numerical model is compatible with data. We could generate independent samples, so that some of the practical difficulties one encounters with Markov Chain Monte Carlo methods, e.g. burn-in time or correlations among dependent samples, are avoided. We describe a new implementation of implicit sampling for parameter estimation problems that makes use of a class of overlapping Newton Krylov-Schwarz algorithms to solve it. With a reasonably large overlap, the Newton Krylov-Schwarz method is scalable and capable of finding the solution with noise. The comparison with BFGS method demonstrates the superiority of our method. We also use the local Karhunan-Loève expansion to reduce the dimension of the parameter which enables the parallel and efficient computation of a possibly large number of dominant KL modes. Another important topic considered in this dissertation is in chapter two, a novel approach for solving optimal price adjustment problems, when the underlying process is geometric Brownian motion process. Several countries use the administratively-set fuel prices close to their international free market counterparts. However, chasing a global market price of energy has the disadvantage that the domestic prices need to fluctuate daily. This creates uncertainties for households and firms and expose them to global price shocks. In chapter two, we offers a model of adjustment rule which is based on optimal lower and upper price barriers. Once the ratio of the domestic and global price hit the bounds, the domestic price will then be readjusted to the original desired level. We offer a procedure to use expected hitting time approach to solve the model, which does not require solving a PDE or running Monte-Carlo simulations. We characterize the optimal policy behavior as a function of underlying parameters and also compare the gains from adopting an optimal policy versus a mechanical policy.