Date of Award


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Degree Name



Mathematics and Statistics


Sévérien Nkurunziza



Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.


This dissertation proposes three types of processes that are suitable for modeling positive datasets with periodic behavior and mean-reverting level phenomenon. A class of generalized exponential Ornstein–Uhlenbeck process (GEOU) is consid- ered in Chapter 2. This chapter’s key characteristics include the following: first, the classical exponential Ornstein–Uhlenbeck process is generalized to the case where the drift coefficient is driven by a period function of time; second, as opposed to the results in recent literature, the dimension of the drift parameter is considered unknown. This chapter serves to weaken some assumptions, in recent literature, underlying the asymp- totic optimality of some estimators of the drift parameter. Three types of estimators are proposed: unrestricted maximum likelihood estimator (UMLE), restricted maximum likelihood estimator (RMLE) and shrinkage estimators (SEs). Asymptotic distributional risk (ADR) of the proposed estimators is also derived, as well as their relative efficiency. Further, it is proven that the proposed methods improve the goodness-of-fit. Finally, this chapter outlines an analysis of a financial market data set and presents the simulation results, which corroborate the theoretical findings. Chapter 3 proposes a generalized Cox–Ingersoll–Ross (GCIR) process that is suit- able for modeling some periodic financial data. An inference problem, about the drift parameters of the introduced GCIR process is also considered when the target param- eters may satisfy some restrictions. Like in the case of GEOU process, three kinds of estimators are derived: UMLE, RMLE, and SEs. Their joint asymptotic normality is studied. Based on the established asymptotic result, a test is constructed for testing the restriction. The asymptotic power of the proposed test is also derived from this, and it is proven that the proposed test is consistent. This chapter also outlines the ADR of the proposed estimators and their relative efficiency. Finally, simulation results are presented that corroborate the study’s theoretical findings. In Chapter 4, a generalized Chan, Karolyi, Longstaff and Sanders (GCKLS) process is proposed for modeling some financial data that are cyclical in nature. The ergodicity of the solution to the GCKLS model is proven by using the transition probability; the normality and strong consistency of the UMLE are proven by using the ergodicity. Sim- ilarly, UMLE, RMLE, and SEs are derived. A test is performed to assess the restriction. The asymptotic power of the proposed test is consistent. Further, the relative efficiency of the proposed estimators is compared, and simulation results are presented that agree with our theoretical findings.

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