Stochastic differential inclusions and diffusion processes

Connections between weak solutions of stochastic differential inclusions and solutions of partial differential inclusions, generated by given set-valued mappings are considered. The main results are based on some continuous approximation selection theorem and weak compactness of the set of all weak solutions to a given stochastic differential inclusion.     

Limit theorems for a class of diffusion processes

We consider a diffusion process {x(t)} on a compact Riemannian manifold with generator δ/2 + b. A current‐valued continuous stochastic process {X _t} in the sense of Itô [8] corresponds to {x(t)} by considering the stochastic line integral X _t(a) along {x(t)} for every smooth 1-form a. Furthermore {X _t} is decomposed into the martingale part and the bounded variation part as a current-valued continuous process. We show the central limit theorems for {X _t} and the martingale part of {X _t}. Occupation time laws for recurrent diffusions and homogenization problems of periodic diffusions are closely related to these theorems     

Computing conditional expectations of multidimensional diffusion processes

We study multidimensional diffusion processes and give an explicit representation for their conditional expectation. Starting from the solution formula for one dimensional stochastic differential equations found in Lanconelli and Proske [8], we compute the conditional expectation of a certain class of multidimensional diffusions without resorting to the Markov property of the process and therefore without requiring an explicit expression for the semi group associated to it.     

几类三阶非线性常微分方程组边值问题正解的存在性

研究几类二阶非线性常微分方程组的边值问题,在合适的条件下,利用锥拉伸不动点定理获得了其正解的存在性.     

New initial-value method for singularly perturbed boundary-value problems

An initial-value method is given for second-order singularly perturbed boundary-value problems with a boundary layer at one endpoint. The idea is to replace the original two-point boundary value problem by two suitable initial-value problems. The method is very easy to use and to implement. Nontrivial text problems are used to show the feasibility of the given method, its versatility, and its performance in solving linear and nonlinear singularly perturbed problems.This work was supported in part by the Consiglio Nazionale delle Ricerche, Contract No. 86.02108.01, and in part by the Ministero della Pubblica Istruzione.     

Limit theorems for number of diffusion processes, which did not absorb by boundaries

We have random number of independent diffusion processes with absorption on boundaries in some region at initial time t = 0. The initial numbers and positions of processes in region is defined by the Poisson random measure. It is required to estimate the number of the unabsorbed processes for the fixed time τ > 0. The Poisson random measure depends on τ and τ → ∞. This research was partially supported by the Ministry of Education and Science of Ukraine, project 01.07/103 and University of Salerno, Italy.     

Initial-value technique for a class of nonlinear singular perturbation problems

An initial-value technique, which is simple to use and easy to implement, is presented for a class of nonlinear, singularly perturbed two-point boundary-value problems with a boundary layer on the left end of the underlying interval. It is distinguished by the following fact: The original second-order problem is replaced by an asymptotically equivalent first-order problem and is solved as an initial-value problem. Numerical experience with several examples is described.     

Initial-value methods for second-order singularly perturbed boundary-value problems

In this paper, we present a numerical method for solving linear and nonlinear second-order singularly perturbed boundary-value-problems. For linear problems, the method comes from the well-known WKB method. The required approximate solution is obtained by solving the reduced problem and one or two suitable initial-value problems, directly deduced from the given problem. For nonlinear problems, the quasilinearization method is applied. Numerical results are given showing the accuracy and feasibility of the proposed method.This work was supported in part by the Consiglio Nazionale delle Ricerche (Contract No. 86.02108.01 and Progetto Finalizzatto Sistemi Informatia e Calcolo Paralello, Sottoprogetto 1), and in part by the Ministero della Pubblica Istruzione, Rome, Italy.     

The coding complexity of diffusion processes under -norm distortion

We investigate the high resolution quantization and entropy coding problem for solutions of stochastic differential equations under L^p[0,1]-norm distortion. We find explicit high resolution formulas in terms of the average diffusion coefficient seen by the process. The proof is based on a decoupling method introduced in a former article by the author. Given that link it remains to analyze the coding problem for a concatenation of Wiener processes and to solve the corresponding rate allocation problem.     

二阶退化双曲型方程的Darboux型问题

本文讨论二阶退化双曲型方程的一些边值问题.文中先给出第一Darboux问题和一般斜微商边值问题的提法和解的表示式,然后使用复分析方法证明了上述问题解的存在唯一性.     

Quasi-likelihood analysis for nonsynchronously observed diffusion processes

We consider nonsynchronous sampling of parameterized stochastic regression models, which contain stochastic differential equations. Constructing a quasi-likelihood function, we prove that the quasi-maximum likelihood estimator and the Bayes type estimator are consistent and asymptotically mixed normal when the sampling frequency of the nonsynchronous data becomes large.     

Asymptotic Initial-Value Method for Singularly-Perturbed Boundary-Value Problems for Second-Order Ordinary Differential Equations

A computational method is presented to solve a class of nonturning-point singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations with a small parameter multiplying the highest derivative, subject to Dirichlet-type boundary conditions. In this method, first we construct a zeroth order asymptotic expansion for the solution of the given boundary-value problem. Then, this problem is integrated to get an equivalent initial-value problem for first-order ordinary differential equations. This initial-value problem is solved by either a classical method or a fitted operator method after approximating some of the terms in the differential equations by using the zeroth order asymptotic expansion. This method is effective and easy to implement. An error estimate is derived for the numerical solution. Examples are given to illustrate the method.     

The coding complexity of diffusion processes under supremum norm distortion

We investigate the high resolution quantization and entropy coding problem for solutions of stochastic differential equations under supremum norm distortion. Tight asymptotic formulas are found under mild regularity assumptions. The main technical tool is a decoupling method which allows us to relate the complexity of the diffusion process to that of the Wiener process. The technique is also applicable when considering the L^p[0,1]$L^p[0,1]$-norm distortion.     

Almost self-optimizing strategies for the adaptive control of diffusion processes

The ergodic control of a multidimensional diffusion process described by a stochastic differential equation that has some unknown parameters appearing in the drift is investigated. The invariant measure of the diffusion process is shown to be a continuous function of the unknown parameters. For the optimal ergodic cost for the known system, an almost optimal adaptive control is constructed for the unknown system.This research was partially supported by NSF Grants ECS-87-18026, ECS-91-02714, and ECS-91-13029.     

二阶扩散模型的经验似然推断

本文利用经验似然方法得到了二阶扩散模型的漂移系数和扩散系数的经验似然估计量, 并研究这些估计量的相合性和渐近正态性. 进一步在经验似然方法的基础上给出了漂移系数和扩散系数的非对称的置信区间, 并且在一定的条件下证明了调整的对数似然比是渐近卡方分布的.     

Occupation time theorems for a class of one-dimensional diffusion processes

Summary The long time asymptotic behavior of the occupation times on a half line is studied for a class of one-dimensional diffusion processes whose excursion intervals have very heavy tail probability     

Optimal control of diffusion processes with reflection

In this paper, we consider the problem of optimally controlling a diffusion process on a closed bounded region ofR ⁿ with reflection at the boundary. Employing methods similar to Fleming (Ref. 1), we present a constructive proof that there exists an optimal Markov control that is measurable or lower semicontinuous. We prove further that the expected cost function corresponding to the optimal control is the unique solution of the quasilinear parabolic differential equation of dynamic programming with Neumann boundary conditions and that there exists a diffusion process (in the sense of Stroock and Varadhan) corresponding to the optimal control.This work was partially supported by the National Science Foundation, Grant No. GK-18339, by the Office of Naval Research, Grant No. NR-042-264, and by the National Research Council of Canada, Grant No. A3609.The author would like to thank S. R. Pliska, J. Pisa, and N. Trudinger for helpful suggestions. He is especially grateful to Professor A. F. Veinott, Jr., for help and advice in the preparation of the doctoral dissertation, on which part of this paper is based. Finally, he wishes to thank one of the referees for the careful reading and constructive comments on an earlier version of this paper.     

The uniqueness of solutions to discrete, vector, two-point boundary value problems

Difference equations which may arise as discrete approximations to two-point boundary value problems for systems of second-order, ordinary differential equations are investigated and conditions are formulated under which solutions to the discrete problem are unique. Some existence, uniqueness implies existence, and convergence theorems for solutions to the discrete problem are also presented.