Let M be a finite set, P be a stochastic matrix and U={(Z_n)} be the family of all finite Markov chains (MC) (Z_n) defined by M, P, and all possible initial distributions. The behavior of a MC (Z_n) is a classical result of probability theory derived in the 1930s by A. N. Kolmogorov and W. Doeblin. If a stochastic matrix P is replaced by a sequence of stochastic matrices (P_n) and transitions at moment n are defined by P_n, then U becomes a family of nonhomogeneous MCs. There are numerous results concerning the behavior of such MCs given some specific properties of the sequence (P_n). But what if there are no assumptions about sequence (P_n)? Is it possible to say something about the behavior of the family U? The surprising answer to this question is Yes. Such behavior is described by a theorem which we call a decomposition- separation (DS) theorem, and which was initiated by a small paper of A. N. Kolmogorov (1936) and formulated and proved in a few stages in a series of papers including D. Blackwell (1945), H. Cohn (1971, 1989), and I. Sonin (1987, 1991, 1996).

Let {Z_n:n≥0} denote a single type supercritical branching process initiated by a single ancestor. This paper studies the asymptotic behavior of the history of generation sizes conditioned on different notions of information about the “current” population size. A “suppression property” under the large deviation conditioning, namely that R_n≡Z_n+1/Z_n>a, is observed. Furthermore, under a more refined conditioning, the asymptotic aposteriori distribution of the original offspring distribution is developed. Implications of our results to conditional consistency property of age is discussed.

Consider the random walk on the set of nonnegative integers that takes two steps to the left (just one step from state 1) with probability p∈[1/3,1) and one step to the right with probability 1−p. State 0 is absorbing and the initial state is a fixed positive integer j₀. Here we find the distribution of the absorption time. The absorption time is the duration of (or the number of coups in) the well-known Labouchere betting system. As a consequence of this, we obtain in the fair case (p=1/2) the asymptotic behavior of the Labouchere bettor’s conditional expected deficit after n coups, given that the system has not yet been completed.

We consider the systems of random differential equations. The coefficients of the equations depend on a small parameter. The first equation, “slow” component, Ordinary Differential Equation (ODE), has unbounded highly oscillating in space variable coefficients and random perturbations, which are described by the second equation, “fast” component, Stochastic Differential Equation (SDE) with periodic coefficients. Sufficient conditions for weak convergence as small parameter goes to zero of the solutions of the “slow” components to the certain stochastic process are given.

We consider large time and infinite particle limit for a system of particles living in random potentials. The randomness enters the potential through an external ergodic Markov process, modeling oscillating environment with good statistical averaging properties. From each individual particle’s point of view, both law of large number and central limit theorem type of averaging are possible. Problems of this type have been well studied and are known as random evolutions. Instead of one particle, we focus on the collective behavior of infinite particles. We separately rescale potential functions (type one) which annihilates the equilibrium measure of the ergodic environment process, and the potential functions which may not annihilate such measure (type two). Appropriately rescaled to the macroscopic limit, type two potentials give a transport term while type one potentials give a nonlinear diffusion term. The resulting equation is a version of nonlinear Vlasov–Fokker–Planck equation. We will also prove the uniqueness of solution for such equation.

This is an informal introduction to stochastic analysis on both Riemannian and Lorentzian manifolds. We review the basics underlying the construction of diffusions on manifolds, highlighting the important differences between the Riemannian and Lorentzian cases. We also discuss a few recent applications which range from biophysics to cosmology.

Using the method of Krylov’s estimates, we prove the existence of (weak) solutions of the one-dimensional stochastic equation dX_t=b(X_t−)dZ_t+a(X_t)dt with arbitrary initial value x₀∈ℝ and the driven symmetric Cauchy process Z. The bounded coefficient b is assumed to be of non-degenerate form and the drift a to satisfy the condition |a(x)|≤(1/2)|b(x)| for all x∈ℝ.

Classification of data as true or fabricated has applications in fraud detection and verification of data samples. In this paper, we apply nonlinear filtering to a simplified fraud-detection problem: classifying coin flip sequences as either real or faked. On the way, we propose a method for generating Bernoulli variables with given marginal probabilities and pair-wise covariances. Finally, we present the empirical performance of the classification algorithm.

A filtering model with counting process observations has been recently developed as a reasonable framework for the micromovement of asset price. In this paper, we first highlight such an extension to multivariate case for modeling multi-stocks and related results on Bayes estimation via filtering. For this rich class of multivariate models, we develop the Bayesian model selection using Bayes factor. Based on the unnormalized, Duncan-Mortensen-Zakai-like filtering equation, we derive a system of SPDE characterizing the evolution of the Bayes factors and prove their uniqueness. Furthermore, applying Kushner’s Markov chain approximation method, we propose a numerical scheme to derive recursive algorithms, and we prove the consistency (or robustness) of the recursive algorithms.

This paper examines the numerical implementation of a linear programming (LP) formulation of stochastic control problems involving singular stochastic processes. The decision maker has the ability to influence a diffusion process through the selection of its drift rate (a control that acts absolutely continuously in time) and may also decide to instantaneously move the process to some other level (a singular control). The first goal of the paper is to show that linear programming provides a viable approach to solving singular control problems. A second goal is the determination of the absolutely continuous control from the LP results and is intimately tied to the particular numerical implementation. The original stochastic control problem is equivalent to an infinite-dimensional linear program in which the variables are measures on appropriate bounded regions. The implementation method replaces the LP formulation involving measures by one involving the moments of the measures. This moment approach does not directly provide the optimal control in feedback form of the current state. The second goal of this paper is to show that the feedback form of the optimal control can be obtained using sensitivity analysis.

We consider an infinite horizon stochastic control problem with discretionary stopping. The state process is given by a one dimensional stochastic differential equation. The diffusion coefficient is chosen by an adaptive choice of the controller and it is allowed to take the value zero. The controller also chooses the quitting time to stop the system. Here we develop a martingale characterization of the value function and use it and the principle of smooth fit to derive an explicit optimal strategy when the drift coefficient of the state process is of the form b(x)=−θx where θ>0 is a constant.

An explicit formula for the Skorokhod map Γ_0,a on [0,a] for a>0 is provided and related to similar formulas in the literature. Specically, it is shown that on the space $\mathcal{D}$[0,∞) of right-continuous functions with left limits taking values in ℝ,

Γ_0,a(ψ)(t)=ψ(t)−[(ψ(0)−a)⁺∧inf_u∈[0,t]ψ(u)]∨sup_s∈[0,t][(ψ(s)−a)∧inf_u∈[s,t]ψ(u)]

is the unique function taking values in [0,a] that is obtained from by minimal “pushing” at the endpoints 0 and a. An application of this result to real-time queues with reneging is outlined.

This paper develops a simple and systematic approach for obtaining bounds on stationary expectations of Markov processes. Given a function f which one is interested in evaluating, the main idea is to find a function g that satisfies a certain “mean drift” inequality with respect to f, which in turn leads to bounds on the stationary expectation of the latter. The approach developed in the paper is broadly applicable and can be used to bound steady-state expectations in general state space Markov chains, continuous time chains, and diffusion processes (with, or without, reflecting boundaries).

Traditional Internet traffic studies have primarily focused on the temporal characteristics of packet traces as observed on a single link within an ISP’s network. They have contributed to advances in the areas of self-similar stochastic processes, long-range dependence, and heavy-tailed distributions and have demonstrated the benefits of applying a wavelet-based multireso- lution analysis (MRA) approach when analyzing these traces. However, an ISP’s physical infrastructure typically consists of 100s or 1000s of such links which are connected by routers or switches, and the Internet as a whole is made up of about 20,000 such ISPs. When viewed within this bigger context, the importance of the traffic’s spatial characteristics becomes evident, and traffic matrices—compact and succinct descriptions of the traffic exchanges between nodes in a given network structure—are used in practice to capture and explore critical aspects of this spatial component of Internet traffic. In this paper, we first review some of the known results about the observed multi- faceted scaling behavior of Internet traffic as seen on a single link. Next, we give a detailed account of how the architectural design of the Internet gives rise to natural representation of traffic matrices at different scales or levels of resolution. Moreover, we discuss the development of a MRA-like framework of traffic matrices that respects the different physically or logically meaningful Internet connectivity structures and provides new insights into Internet traffic as a spatio-temporal object.

A fractional Brownian queueing model, that is, a fluid queue with an input of a fractional Brownian motion, has been applied in network model- ing since the self-similarity and long-range dependence were observed in Inter- net traffic. In this paper, a fluid queue with an aggregated fractional Brownian input, which is a generalization of a fractional Brownian queueing model, is considered and the maximum queue length over a time interval [0,t] is studied. The impact of an aggregated fractional Brownian input on the queue length process is analyzed and the main results on the maximum queue length are compared with some related known results in the literature.

The design and analysis of congestion control mechanisms for modern data networks such as the Internet is a challenging problem. Mathematical models at various levels have been introduced in an effort to provide insight to some aspects of this problem. A model introduced and studied by Roberts and Massoulié [13] aims to capture the dynamics of document arrivals and departures in a network where bandwidth is shared fairly amongst flows that correspond to continuous transfers of individual elastic documents. Here we consider this model under a family of bandwidth sharing policies introduced by Mo and Walrand [14]. With generally distributed interarrival times and document sizes, except for a few special cases, it is an open problem to establish stability of this stochastic flow level model under the nominal condition that the average load on each resource is less than its capacity. As a step towards the study of this model, in a separate work [8], we introduced a measure valued process to describe the dynamic evolution of the residual document sizes and proved a fluid limit result: under mild assumptions, rescaled measure valued processes corresponding to a sequence of flow level models (with fixed network structure) are tight, and any weak limit point of the sequence is almost surely a solution of a certain fluid model. The invariant states for the fluid model were also characterized in [8]. In this paper, we review the structure of the stochastic flow level model, describe our fluid model approximation and then give two interesting examples of network topologies for which stability of the fluid model can be established under a nominal condition. The two types of networks are linear networks and tree networks.

No free lunch with vanishing risk (NFLVR) is known to be equivalent to the existence of an equivalent martingale measure for the price process semimartingale. We give necessary conditions for such a semimartingale to have the property NFLVR. We also extend Novikov’s criterion for the stochastic exponential of a local martingale to be a martingale to the general case; that is, the case where the paths need not be continuous.

This work presents a novel concept in stochastic optimization, namely, the notion of forward performance. As an application, we analyze a portfolio management problem with exponential criteria. Under minimal model assumptions we explicitly construct the forward performance process and the associated optimal wealth and asset allocations. For various model parameters, we recover a range of investment policies that correspond to distinct financial applications.

Let X_t be a stochastic process driven by a differential equation of the form dX_t=σ(t,X_t)dW_t+b(t,X_t)dt, t>0, and let X^⋆_s,t=sup_s≤u≤tX_u, be the maximum of the diffusion. In this work we obtain bounds for the tail distribution of X^*_s,t, define several dynamic VaR type quantiles for this process and give upper and lower bounds for both, the VaR quantile and the conditioned mean loss associated to it. The results we obtain are based in the change of time property of the Brownian Motion, and can be applied to a a large class of examples used in Finance, in particular where σ(t,X_t)=σ_tX^γ_t , where 0≤γ<1. The estimates we obtain are sharp. We discuss carefully the Geometric Brownian Motion, the Cox-Ingersoll-Ross and the Vasicek type models, and give an application to Russian options.

In this article, we establish, using a duality argument, an identity stating that the Laplace transform of the length of a contiguous bacterial recombination region equals the probability of choosing a given allele in a stationary population evolving according to the one-dimensional Wright–Fisher diffusion model. Beyond giving us an improved inferential strategy for parameter estimation in bacterial recombination, the matching of the selection and recombination parameters in the identity also suggests the existence of an intriguing formal relationship between gene conversion and the ancestral selection graph.