Recent Development in Stochastic Dynamics and Stochastic Analysis

The following sections are included:

• Editorial Foreword

• Preface

• Contents

The following sections are included:

• Introduction
  • TheWave equation
  • The Transport equation

• Abstract formulation

• The Wave equation – introduction

• Weak solution to the wave equation

• Mild formulations

• Scales of Hilbert spaces
  • Application to the boundary value problem

• Equivalence of weak and mild solutions

• The Fundamental Solution

• Applications

• The Transport equation

• References

Quantum operations (channels) are natural generalizations of transition matrices in stochastic analysis. A quantum operation usually causes decoherence of quantum states. Speaking in a broad sense, decoherence is the loss of some kind of correlations (in particular, entanglement). This is also recognized as the origin of the emergence of classicality. While this decoherence has been widely studied and characterized from various qualitative perspectives, there are relatively fewer quantitative characterizations. In this work, motivated by the notion of coherent information and consideration of correlating capability and transferring of correlations, we study an informational measure of decoherent capability of quantum operations in terms of quantum mutual information. The latter is usually regarded as a suitable measure of total correlations in a bipartite system. This measure possesses a variety of desirable properties required for quantitative characterizations of decoherent information, and is complementary to the coherent information introduced by Schumacher and Nielsen (Phys. Rev. A, 54, 2629, 1996) in a loose sense. Apart from the significance in its own right, the decoherent information also provides an alternative and simple interpretation of, and sheds new light on, the coherent information. Several examples are worked out. A quantum Fano type inequality, an informational no-broadcasting result for bipartite correlations, and a continuity estimate for the decoherent information, are also established.

Some recent results on stabilization by noise in systems modeled by evolution equations is reviewed, mostly for partial differential equations but also for delay differential equations and systems without uniqueness.

Complex systems often involve multiple scales and multiple physical, chemical and biological mechanisms. Due to the lack of scientific understanding, some mechanisms are not represented (i.e., “unresolved”) in mathematical models for these complex systems. The impact of these unresolved processes on the resolved ones may be delicate and needs to be quantified. A stochastic dynamical approach is proposed to parameterize these missing mechanisms or unresolved scales. An example is briefly discussed to demonstrate this stochastic approach.

Let X be a complex Banach space (not necessary to be reflexive). In this chapter, we prove a moment characterization theorem and a convergent theorem for X-valued generalized functionals of white noise, which refine the corresponding results recently obtained by Wang and Huang (Acta Math. Sin. Engl. Ser. 22 (2006), 157-168).

The statistical estimation of the Hurst index is one of the fundamental problems in the literature of long-range dependent and self-similar processes. In this article, the Hurst index estimation problem is addressed for a special class of self-similar processes that exhibit long-memory, the Hermite processes. These processes generalize the fractional Brownian motion, in the sense that they share its covariance function, but are non-Gaussian. Existing estimators such as the R/S statistic, the variogram, the maximum likelihood and the wavelet-based estimators are reviewed and compared with a class of consistent estimators which are constructed based on the discrete variations of the process. Convergence theorems (asymptotic distributions) of the latter are derived using multiple Wiener-Itô integrals and Malliavin calculus techniques. Based on these results, it is shown that the latter are asymptotically more efficient than the former.

Complex systems are usually under the influences of noises. Appropriately modeling these noises requires knowledge about generalized time derivatives and generalized stochastic processes. To this end, a brief introduction to generalized functions theory is provided. Then this theory is applied to fractional Brownian motion and its derivative, both regarded as generalized stochastic processes, and it is demonstrated that the “time derivative of fractional Brownian motion” is correlated and thus is a mathematical model for colored noise. In particular, the “time derivative of the usual Brownian motion” is uncorrelated and hence is an appropriate model for white noise.

To facilitate random dynamical systems approach for stochastic partial differential equations arising as mathematical models for complex systems under fluctuations in science and engineering, global existence and uniqueness of mild solutions for a class of stochastic partial differential equations with local Lipschitz coefficients are considered. A sufficient condition for non-explosion, or global existence and uniqueness, of mild solutions is provided and a few examples are presented to demonstrate the result.

In this chapter, we consider the optimal proportional reinsurance policy for an insurance company and focus on the case that the reinsurer asks for positive transaction costs from the insurance company. That is, the reserve of the insurance company {R_t} is governed by a SDE , where W_t is a standard Brownian motion, π denotes the admissible proportional reinsurance policy, μ, λ and σ are positive constants with μ ≤ λ. The aim of this paper is to find a policy that maximizes the return function , where c > 0, τ_π is the time of bankruptcy and x represents the initial reserve. We find the optimal policy and the optimal return function for the insurance company via stochastic control theory. We also give transparent economic interpretation of the return function J(x, π) and show that it could be a better interpretation than the traditional one. Finally, we make some numerical calculations and give some economic analysis on the influence of the transaction costs.

In this chapter asymptotic properties of the p-variations of stochastic processes of the type X = Y + L are studied, where L is an α-stable Lévy process, and Y a perturbation which satisfies some mild Lipschitz continuity assumptions. Local functional limit theorems for the power variation processes of X are established. In case X is a solution of a stochastic differential equation driven by the process L, these limit theorems provide estimators of the stability index α. They are applied to paleo-climatic temperature time series taken from the Greenland ice core, describing the fine structure of temperature variability during the last ice age, in particular exhibiting the intermediate warming periods known as Dansgaard–Oeschger events. Our results provide the best fitting α in a parameter test in which the time series are modeled by stochastic differential equations with α-stable noise component.

The infinite dimensional rotation group plays important roles in white noise analysis. In particular, the class II subgroup describes significant probabilistic properties. We shall find a local Lie group involving some half whiskers in the class II and discuss its Lie algebra.

It is shown that, in order that the multiplicativity rule , which essentially characterizes Weyl processes, continue to hold when the sure times a < b < c are replaced by stop times, it is sufficient to use left stopping at the lower limit a and right stopping at the upper limit b in .

This chapter aims at firstly establishing a Karhunen-Loéve expansion for stochastic convolution of cylindrical fractional Brownian motion, then studying asymptotic behavior of conditional exponential moments of the stochastic convolution via this expansion.

This is an overview of some recent techniques involving the Malliavin calculus of variations and the so-called ‘Stein's method’ for the Gaussian approximations of probability distributions. Special attention is devoted to establishing explicit connections with the classic method of moments: in particular, interpolation techniques are used in order to deduce some new estimates for the moments of random variables belonging to a fixed Wiener chaos. As an illustration, a class of central limit theorems associated with the quadratic variation of a fractional Brownian motion is studied in detail.

This chapter investigates stochastic integrals with respect to an infinite number of Poisson point processes and its’ corresponding martingale representations. Furthermore, with non-Markovian and non-Lipschitz coefficients, stochastic differential equations driven by sequence of Poisson point processes are discussed, where the results are extensions of the linear continuous cases.

In this article, we start with a briefly survey of the recent development on Euclidean random fields along with constructive, relativistic quantum field theory. We then present a unified account of Lévy white noise and related elliptic SPDEs driven by Lévy white noise. We explicate the link, via analytic continuation, from the Euclidean random fields obtained as solutions of the elliptic SPDEs to local, relativistic quantum fields with indefinite metric. By comparing the derived vector and scalar relativistic quantum field models, we reformulate the elliptic SPDE for the scalar model with a new Gaussian white noise term so that the associated Euclidean random field possesses a feature of avoiding the re-definition of the two point Schwinger functions needed for nontrivial scattering in the relativistic model, which then leads to a scalar model of local, relativistic quantum field theory in indefinite metric with nontrivial scattering behavior. Finally, we demonstrate a lattice approximation for the induced Euclidean random field.

The paper provides a short representation of Choquet integral. The main content is from the book D. Denneberg (1994) and the paper Zhou (1998).