Document Type




Degree Granting Department

Mathematics and Statistics

Major Professor

Gangaram S. Ladde


Delayed Volatility, Extended Kalman Filter, Local Lagged adapted Generalized Method of Moments, Risk-Neutral Dynamics, Stochastic Hybrid System


Supply and demand in the World oil market are balanced through responses to price movement with considerable complexity in the evolution of underlying supply-demand

expectation process. In order to be able to understand the price balancing process, it is important to know the economic forces and the behavior of energy commodity spot price processes. The relationship between the different energy sources and its utility together with uncertainty also play a role in many important energy issues.

The qualitative and quantitative behavior of energy commodities in which the trend in price of one commodity coincides with the trend in price of other commodities, have always raised the questions regarding their interactions.

Moreover, if there is any interaction, then one would like to know the extent of influence on each other.

In this work, we undertake the study to shed a light on the above highlighted processes and issues. The presented study systematically deals with the development of stochastic dynamic models and mathematical, statistical and computational analysis of energy commodity spot price and interaction processes.

Below we list the main components of the research carried out in this dissertation.

(1) Employing basic economic principles, interconnected deterministic and stochastic models of linear log-spot and expected log-spot price processes coupled with non-linear volatility process are initiated. (2) Closed form solutions of the models are analyzed.

(3) Introducing a change of probability measure, a risk-neutral interconnected stochastic model is derived.

(4) Furthermore, under the risk-neutral measure, expectation of the square of volatility is reduced to a continuous-time deterministic delay differential equation. (5) The by-product of this exhibits the hereditary effects on the mean-square volatility process.

(6) Using a numerical scheme, a time-series model is developed and utilized to estimate the state and parameters of the dynamic model.

In fact, the developed time-series model includes the extended GARCH model as special case.

(7) Using the Henry Hub natural gas data set, the usefulness of the linear interconnected stochastic models is outlined.

(8) Using natural and basic economic ideas, interconnected deterministic and stochastic models in (1) are extended to non-linear log-spot price, expected log-spot price and volatility processes. (9) The presented extended models are validated. (10) Closed form solution and risk-neutral models of (8) are outlined.

(11) To exhibit the usefulness of the non-linear interconnected stochastic model, to increase the efficiency and to reduce the magnitude of error, it was essential to develop a modified version of extended Kalman filtering approach.

The modified approach exhibits the reduction of magnitude of error.

Furthermore, Henry Hub natural gas data set is used to show the advantages of the non-linear interconnected stochastic model.

(12) Parameter and state estimation problems of continuous time non-linear stochastic dynamic process is motivated to initiate an alternative innovative approach. This led to introduce the concept of statistic processes, namely, local sample mean and sample variance. (13) Then it led to the development of an interconnected discrete-time dynamic system of local statistic processes and (14) its mathematical model. (15) This paved the way for developing an innovative approach referred as Local Lagged adapted Generalized Method of Moments (LLGMM). This approach exhibits the balance between model specification and model prescription of continuous time dynamic processes. (16) In addition, it motivated to initiate conceptual computational state and parameter estimation and simulation schemes that generates a mean square sub-optimal procedure. (17) The usefulness of this approach is illustrated by applying this technique to four energy commodity data sets, the U. S. Treasury Bill Yield Interest Rate and the U.S. Eurocurrency Exchange Rate data sets for state and parameter estimation problems. (18) Moreover, the forecasting and confidence-interval problems are also investigated.

(19) The non-linear interconnected stochastic model (8) was further extended to multivariate interconnected energy commodities and sources with and without external random intervention processes. (20) Moreover, it was essential to extend the interconnected discrete-time dynamic system of local sample mean and variance processes to multivariate discrete-time dynamic system. (21) Extending the LLGMM approach in (15) to a multivariate interconnected stochastic dynamic model under intervention process, the parameters in the multivariate interconnected stochastic model are estimated. These estimated parameters help in analyzing the short term and long term relationship between the energy commodities. These developed results are applied to the Henry Hub natural gas, crude oil and coal data sets.