Research On KL Expansions For Gaussian Processes And The Optimal Stopping For The Stochastic Logistic Equations

Karhunen-Loeve(KL for short) for some Gaussian processes and the optimal stop-ping problem for the stochastic Logistic models are studied in this dissertation．The firstpart is about some research on the KL expansion for Gaussian processes. KL expansionis one of the important tool for the study of Gaussian processes. There are a lot of appli-cations, such as reproducing kernel Hilbert space, small deviation(small ball probability),statistics and so on. If the KL expansion for the Gaussian process is known, an orthogo-nal basis are given for the associated reproducing kernel Hilbert space. Furthermore, thereproducing kernel Hilbert space is known immediately. In statistics, the eigenvalues ofthe covariance function of the Gaussian processes associated with statistics are used in thecalculation. KL expansion provides the pretty good way to solve the problem, while eventhe Mercer theorem guarantees the existence of the eigenvalues of the Gaussian processes,KL expansions of many Gaussian processes are unknown. Therefore, KL expansion inthis dissertation not only extends the family, but also plays an important role in theory andapplication. The dissertation concentrates on KL expansion for the Detrended Brownianmotion, the mth Detrended Brownian motion, bivariate Brownian bridge and so on. Theresults are as follows：1. KL expansion for the Detrended Brownian motion is studied. Distribution identi-ties are established in connection with the second order Brownian bridge. In addition, asapplication, by using the eigenvalues of the covariance of the Detrended Brownian mo-tion, Laplace transform, large and small deviation asymptotic behaviors for the L2normof the Detrended Brownian motion are given.2. KL expansion for the mth order Detrended Brownian motion is studied. Distri-bution identities are established in connection with the mth order Brownian bridge by KLexpansion and stochastic Fubini approach. In addition, as application, by using the eigen-values of the covariance of the mth order Detrended Brownian motion, Laplace transform,large and small deviation asymptotic behaviors for the L2norm of the mth order DetrendedBrownian motion are given.3. KL expansion for the bivariate Brownian bridge defined by a Brownian sheet,which is a non-tensored Gaussian Field, is studied. By using Dirichlet series and prime decomposition, we claim the conclusion, that is, the eigenvalues and the associated eigen-functions are given for the covariance function of the Gaussian processes partly. KL ex-pansion for Slepian process are discussed. Although the exact expansions are not given,the key troubles are found and will be helpful for further research.The optimal harvesting problems for stochastic Logistic model are discussed in thesecond part of the dissertation. In the biological system, ecological balance and sustain-able development are always the important aspect, among which, the optimal harvestingproblems for biological populations are always permanent research problem. Therefore,Starting from the easiest single population model, stochastic Logistic model, the paperprovides two aspects, results are as follows：1. An optimal stopping problem is formulated for the optimal harvesting problemfor Gilpin-Ayala(a generated Logistic) population model. By applying the smooth pastingtechnique, the optimal stopping boundary and region are given.2. An optimal stopping problem is formulated for the optimal harvesting problemfor Logistic population model driven by Levy process. By applying the smooth pastingtechnique, the optimal stopping boundary and region are given.