高斯过程的KL展开及随机Logistic方程最优停时的研究
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摘要
本文主要研究一些Gaussian过程的Karhunen-Loeve展开(以下简称KL展开)及随机Logistic模型的最优停时。本文第一部分研究了一些Gaussian过程的KL展开问题。KL展开是研究Gaussian过程的重要工具之一，最早由Karhunen和Loeve分别于1946和1955年提出。KL展开理论在研究Gaussian过程的再生核Hilbert空间，小偏差估计(又称小球概率)以及在统计等方面都有重要的应用。如果已知Gaussian过程的KL展开，就相当于给出了对应的再生核Hilbert空间的一组正交基，相应的再生核Hilbert空间自然就找到了。Gaussian过程的KL展开方法提供的简洁的特征值能够帮助给出精确的小概率估计。在统计上，许多统计量的计算往往要用到与之相关的Gaussian过程协方差函数的特征值，KL展开方法很好的解决了这个问题，然而即使有Mercer定理保证了Gaussian过程特征值的存在性，大多数Gaussian过程的KL展开仍然是未知的，因此，本文研究的几类Gaussian过程的KL展开不仅扩大了已知Gaussian过程KL展开的集合，而且具有一定的理论价值和应用价值。本文研究了Detrended Brownian运动，m阶Detrended Brownian运动以及双变量Brownian桥的KL展开等问题，所得结果如下：
1.研究了Detrended Brownian运动的KL展开，并推得此Gaussian过程和二阶Brownian桥在分布意义上相等的结论。此外，作为应用，利用得到的Detrended Brownian运动协方差的特征值及Fredholm行列式，给出了DetrendedBrownian运动L2范数意义下精确的Laplace变换以及大，小偏差的精确估计。
2.研究了m阶Detrended Brownian运动的KL展开，利用KL展开的结论和随机Fubini定理两种途径，推得此Gaussian过程和广义Brownian桥在分布意义上相等的结论。此外，作为应用部分，利用得到的m阶Detrended Brownian运动协方差的特征值及Fredholm行列式，给出了Detrended Brownian运动L2范数意义下精确的Laplace变换以及大小偏差的精确估计。
3.研究了一个非张量的Gaussian过程，即由Brownian单定义的双变量Brownian桥的KL展开问题。利用Dirichlet级数和质数分解讨论了KL展开完备性，并得出结论，即部分的得到了此Gaussian过程的协方差函数的精确的特征值和相应的特征向量。讨论了Slepian过程的KL展开，虽然没有给出其精确的展开形式，但是还是发现了困难所在，对于将来的研究会有很多的帮助。
本文的第二部分讨论了随机Logistic模型的最优捕获问题。在生物系统中，生态平衡和保持可持续发展一直是研究的重点，其中对于生物种群的最优捕获问题一直是一个恒久的研究问题，所以本文从最简单的单种群模型，随机Logistic模型出发研究问题，做了以下两方面工作，所得结果如下：
1.把Gilpin-Ayala模型(广义Logistic模型)的最优捕获问题形成为一个最优停时去研究，利用光滑相切技术，得到了精确的最优停止边界和最优停止区域。
2.把Levy过程驱动的Logistic模型的最优捕获问题形成为一个最优停时去研究，利用光滑相切技术，得到了精确的最优停止边界和最优停止区域。
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.
1.研究了Detrended Brownian运动的KL展开，并推得此Gaussian过程和二阶Brownian桥在分布意义上相等的结论。此外，作为应用，利用得到的Detrended Brownian运动协方差的特征值及Fredholm行列式，给出了DetrendedBrownian运动L2范数意义下精确的Laplace变换以及大，小偏差的精确估计。
2.研究了m阶Detrended Brownian运动的KL展开，利用KL展开的结论和随机Fubini定理两种途径，推得此Gaussian过程和广义Brownian桥在分布意义上相等的结论。此外，作为应用部分，利用得到的m阶Detrended Brownian运动协方差的特征值及Fredholm行列式，给出了Detrended Brownian运动L2范数意义下精确的Laplace变换以及大小偏差的精确估计。
3.研究了一个非张量的Gaussian过程，即由Brownian单定义的双变量Brownian桥的KL展开问题。利用Dirichlet级数和质数分解讨论了KL展开完备性，并得出结论，即部分的得到了此Gaussian过程的协方差函数的精确的特征值和相应的特征向量。讨论了Slepian过程的KL展开，虽然没有给出其精确的展开形式，但是还是发现了困难所在，对于将来的研究会有很多的帮助。
本文的第二部分讨论了随机Logistic模型的最优捕获问题。在生物系统中，生态平衡和保持可持续发展一直是研究的重点，其中对于生物种群的最优捕获问题一直是一个恒久的研究问题，所以本文从最简单的单种群模型，随机Logistic模型出发研究问题，做了以下两方面工作，所得结果如下：
1.把Gilpin-Ayala模型(广义Logistic模型)的最优捕获问题形成为一个最优停时去研究，利用光滑相切技术，得到了精确的最优停止边界和最优停止区域。
2.把Levy过程驱动的Logistic模型的最优捕获问题形成为一个最优停时去研究，利用光滑相切技术，得到了精确的最优停止边界和最优停止区域。
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.
引文
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