Dynamic approximation of a random information functional

Using the probabilistic evaluation of the Marcovian diffusion process by a functional of action, the paper introduces a dynamic approximation of a random information functional defined on the process trajectories and determined by the parameters of a controlled stochastic equation. The developed mathematical formalism is aimed toward a dynamic modeling of a random object.     

Solution to the variation problem for information path functional of a controlled random process

The paper introduces a new approach to dynamic modeling, using the variation principle, applied to a functional on trajectories of a controlled random process, and its connection to the process' information functional. In [V.S. Lerner, Dynamic approximation of a random information functional, J. Math. Anal. Appl. 327 (1) (2007) 494-514, available online 5-24-06], we presented the information path functional with the Lagrangian, determined by the parameters of a controlled stochastic equation. In this paper, the solution to the path functional's variation problem provides both a dynamic model of a random process and the model's optimal control, which allows us to build a two-level information model with a random process at the microlevel and a dynamic process at the macrolevel. A wide class of random objects, modeled by the Markov diffusion process and a common structure of the process' information functional, leads to a universal information structure of the dynamic model, which is specified and identified on a particular object with the applied optimal control functions. The developed mathematical formalism, based on classical methods, is aimed toward the solution of problems identification, combined with an optimal control synthesis, which is practically implemented and also demonstrated in the paper's example.     

Conditional log-Laplace functional for a class of branching processes in random environments

A conditional log-Laplace functional(CLLF) for a class of branching processes in random environments is derived. The basic idea is the decomposition of a dependent branching dynamic into a no-interacting branching and an interacting dynamic generated by the random environments.CLLF will play an important role in the investigation of branching processes and superprocesses with interaction.     

Moderate deviations for time-varying dynamic systems driven by non-homogeneous Markov chains with Two-time Scales

Motivated by problems arising in time-dependent queues and dynamic systems with random environment, this work develops moderate deviations principles for dynamic systems driven by a fast-varying non-homogeneous Markov chain in continuous time. A distinct feature is that the Markov chain is time dependent or inhomogeneous, so are the dynamic systems. Under irreducibility of the non-homogeneous Markov chain, moderate deviations of a non-homogeneous functional are established first. With the help of a martingale problem formulation and a functional central limit theorem for the two timescale system, both upper and lower bounds of moderate deviations are obtained for the rapidly fluctuating Markovian systems. Then applications to queueing systems and dynamic systems modulated by a fast-varying Markov chain are examined.     

On option pricing under a completely random measure via a generalized Esscher transform

In this paper, we develop an option valuation model when the price dynamics of the underlying risky asset is governed by the exponential of a pure jump process specified by a shifted kernel-biased completely random measure. The class of kernel-biased completely random measures is a rich class of jump-type processes introduced in [James, L.F., 2005. Bayesian Poisson process partition calculus with an application to Bayesian Lévy moving averages. Ann. Statist. 33, 1771–1799; James, L.F., 2006. Poisson calculus for spatial neutral to the right processes. Ann. Statist. 34, 416–440] and it provides a great deal of flexibility to incorporate both finite and infinite jump activities. It includes a general class of processes, namely, the generalized Gamma process, which in its turn includes the stable process, the Gamma process and the inverse Gaussian process as particular cases. The kernel-biased representation is a nice representation form and can describe different types of finite and infinite jump activities by choosing different mixing kernel functions. We employ a dynamic version of the Esscher transform, which resembles an exponential change of measures or a disintegration formula based on the Laplace functional used by James, to determine an equivalent martingale measure in the incomplete market. Closed-form option pricing formulae are obtained in some parametric cases, which provide practitioners with a convenient way to evaluate option prices.     

Strong approximations of the quantile process of the product-limit estimator

The quantile process of the product-limit estimator (PL-quantile process) in the random censorship model from the right is studied via strong approximation methods. Some almost sure fluctuation properties of the said process are studied. Sections 3 and 4 contain strong approximations of the PL-quantile process by a generalized Kiefer process. The PL and PL-quantile processes by the same appropriate Kiefer process are approximated and it is demonstrated that this simultaneous approximation cannot be improved in general. Section 5 contains functional LIL for the PL-quantile process and also three methods of constructing confidence bands for theoretical quantiles in the random censorship model from the right.     

Multivariate sensitivity analysis for dynamic models with both random and random process inputs

Dynamic models with both random and random process inputs are frequently used in engineering. However, sensitivity analysis (SA) for such models is still a challenging problem. This paper, therefore, proposes a new multivariate SA technique to aid the safety design of these models. The new method can decompose the SA of dynamic models into a series of SA of their principle components based on singular value decomposition, which will make the SA of dynamic models much more efficient. It is shown that the effect of both random and random process inputs on the uncertainty of dynamic output can be measured from their effects on both the distributions and directions of the principle components, based on which the individual sensitivities are defined. The generalized sensitivities are then proposed to synthesize the information that is spread between the principal components to assess the influence of each input on the entire uncertainty of dynamic output. The properties of the new sensitivities are derived and an efficient estimation algorithm is proposed based on unscented transformation. Numerical results are discussed with application to a hydrokinetic turbine blade model, where the new method is compared with the existing variance-based method.     

Approximation for the Normal Inverse Gaussian Process Using Random Sums

Abstract

We approximate the normal inverse Gaussian (NIG) process with random summations. The random sum we introduce is a random walk subordinated to the first passage time of another independent random walk; the model is interpreted as an internal mechanism at small scale that generates the NIG process. The main result is a functional limit theorem of weak convergence in the Skorohod topology.     

Limit Theorems for Weakly Subcritical Branching Processes in Random Environment

For a branching process in random environment, it is assumed that the offspring distribution of the individuals varies in a random fashion, independently from one generation to the other. Interestingly there is the possibility that the process may at the same time be subcritical and, conditioned on nonextinction, ??supercritical.?? This so-called weakly subcritical case is considered in this paper. We study the asymptotic survival probability and the size of the population conditioned on nonextinction. Also a functional limit theorem is proved, which makes the conditional supercriticality manifest. A main tool is a new type of functional limit theorems for conditional random walks.     

On some functional of the hybrid process in random scenery

In this work we wish to investigate an example based on the so-called Kesten–Spitzer random walk in random scenery. Namely, replacing the one-dimensional random walk in a general i.i.d. scenery by the hybrids of empirical and partial sums process (see, for instance, [L. Horváth, Approximations for hybrids of empirical and partial sums process, J. Statist. Plann. Inference 88 (2000) 1–18]), we establish an upper bound in the strong approximation for the corresponding functional.     

Weak convergence of random <Emphasis Type="Bold">p</Emphasis>-mappings and the exploration process of inhomogeneous continuum random trees

We study the asymptotics of the p-mapping model of random mappings on [n] as n gets large, under a large class of asymptotic regimes for the underlying distribution p. We encode these random mappings in random walks which are shown to converge to a functional of the exploration process of inhomogeneous random trees, this exploration process being derived (Aldous-Miermont-Pitman 2004) from a bridge with exchangeable increments. Our setting generalizes previous results by allowing a finite number of “attracting points” to emerge.Research supported by NSF Grant DMS-0203062.Research supported by NSF Grant DMS-0071468.     

The dynamic multilevel assignment problem as a stochastic extremal process

This paper considers the multilevel assignment problem (i.e. the assignment problem where the supply alternatives are ranked in hierarchical levels) under the assumption that the utility components for each pairwise matching are stochastic. A dynamic version of the multilevel stochastic assignment model is developed, where both demand and supply evaluate alternatives according to a stochastic extremal process, i.e. a process where the maximum of a sequence of random variables is taken into account. The probability distributions of the random variables which describe the joint dynamic behaviour of demand and supply are found. It is also shown that the assignment probabilities assume the structure of a nested-logit model.     

Lower bound for the mean project completion time in dynamic PERT networks

We apply the stochastic dynamic programming to obtain a lower bound for the mean project completion time in a PERT network, where the activity durations are exponentially distributed random variables. Moreover, these random variables are non-static in that the distributions themselves vary according to some randomness in society like strike or inflation. This social randomness is modelled as a function of a separate continuous-time Markov process over the time horizon. The results are verified by simulation.     

考虑调整随机误差的设置调整问题研究

针对存在设置偏差的小批量离散制造过程,研究了每次调整成本固定且调整存在随机误差情形下的质量控制问题。在建立过程状态空间方程模型的基础上,通过贝叶斯方法估计过程的未知参数,利用动态规划的方法得到了边界形式的过程最优调整策略。通过算例验证了所提出调整策略的有效性,并利用仿真对本文提出的调整策略与其他调整策略进行了比较分析,结果表明,本文提出的方法能够更好地减少过程总体质量损失。     

A method for determining the equilibrium states of dynamic systems

A heuristic method is presented for determining the equilibrium states of motion of dynamic systems, in particular, spacecraft. The method can also be applied to the solution of sets of linear or nonlinear algebraic equations. A positive-semidefinite functional is formed to convert the problem to that of finding those minimum points where the functional vanishes. The process is initiated within a selecteddomain of interest by random search; convergence to a minimum is obtained by a modified Davidon's deflected gradient technique. To render this approach feasible in the presence of constraints, the functional is modified to include penalty terms which cause the functional to approach infinity at the constraint boundaries. Close approximations to solutions near the constraint boundaries are found by applying Carroll's approach in successively reducing the weighting factors of the penalty terms. After finding a minimum, the local domain around this point is eliminated by adding to the functional an interior constraint term, representing the surface under a hypersphere centered at the minimum point. The domain of consideration now becomes the subdomain formed by subtracting the space contained within this hypersphere from the previous domain of interest. Minima are now sought within the remaining space, as before.This paper is derived from research performed by the author while employed by TRW Systems Group, Redondo Beach, California.The author acknowledges the helpful suggestions of Dr. G. Bekey, University of Southern California, and those of Mr. E. A. Quast, Dr. M. P. Scher, and Dr. R. J. Wiley, Dynamics Department, TRW Systems Group, Redondo Beach, California.     

The small-time Chung-Wichura law for L��vy processes with non-vanishing Brownian component

We give a ??small time?? functional version of Chung??s ??other?? law of the iterated logarithm for Lévy processes with non-vanishing Brownian component. This is an analogue of the ??other?? law of the iterated logarithm at ??large times?? for Lévy processes and random walks with finite variance, as extended to a functional version by Wichura. As one of many possible applications, we mention a functional law for a two-sided passage time process.     

Fluctuations of shapes of large areas under paths of random walks

Summary We discuss statistical properties of random walks conditioned by fixing a large area under their paths. We prove the functional central limit theorem (invariance principle) for these conditional distributions. The limiting Gaussian measure coincides with the conditional probability distribution of certain timenonhomogeneous Gaussian random process obtained by an integral transformation of the white noise. From the point of view of statistical mechanics the studied problem is the problem of describing the fluctuations of the phase boundary in the one-dimensional SOS-model.     

A note on a simplified approach to the valuation of risky streams

This paper develops an approach to the valuation of risky streams defined by a class of stochastic processes. The valuation is given by a linear functional, which is an extension of the market valuations of the marketed assets. Provided that there exists an interest rate process, it is possible to represent the valuation functional as an appropriate expectation defined on the space of integrable random variables. The paper concludes with the relationship of our approach to the existing literature.     

检修事件的概率分析与结构的维修可靠性

定期检修结构的可靠性分析,需要定量处理大量复杂的随机事件。综合考虑各种随机因素,较全面地分析了检修过程中影响可靠性的诸多复杂随机事件及其发生概率。系统地建立了检修结构的动态可靠性分析模型,整个模型程序化程度高,易于实施。     

A central limit theorem for Gibbs measures relative to Brownian motion

We study a Gibbs measure over Brownian motion with a pair potential which depends only on the increments. Assuming a particular form of this pair potential, we establish that in the infinite volume limit the Gibbs measure can be viewed as Brownian motion moving in a dynamic random environment. Thereby we are in a position to use the technique of Kipnis and Varadhan and to prove a functional central limit theorem.