Measure-valued Processes Research Articles

On the use of measure-valued strategies in bond markets

We propose here a theory of cylindrical stochastic integration, recently developed by Mikulevicius and Rozovskii, as mathematical background to the theory of bond markets. In this theory, since there is a continuum of securities, it seems natural to define a portfolio as a measure on maturities. However, it turns out that this set of strategies is not complete, and the theory of cylindrical integration allows one to overcome this difficulty. Our approach generalizes the measure-valued strategies: this explains some known results, such as approximate completeness, but at the same time it also shows that either the optimal strategy is based on a finite number of bonds or it is not necessarily a measure-valued process.

Comparisons for measure valued processes with interactions

This paper considers some measure-valued processes $\{X_t\dvtx t \in [0,T]\}$ based on an underlying critical branching particle structure with random branching rates. In the case of constant branching these processes are Dawson--Watanabe processes. Sufficient conditions on functionals $\Phi$ of the process are given that imply that the expectations $E(\Phi(X_T))$ are comparable to the constant branching case. Applications to hitting estimates and regularity of solutions are discussed. The result is established via the martingale optimality principle of stochastic control theory. Key steps, which are of independent interest, are the proof of a version of Itô's lemma for $\Phi(X_t)$, suitable for a large class of functions of measures (Theorem 3) and the proof of various smoothing properties of the Dawson--Watanabe transition semigroup (Section 3).

Stochastic flows associated to coalescent processes

We study a class of stochastic flows connected to the coalescent processes that have been studied recently by Mohle, Pitman, Sagitov and Schweinsberg in connection with asymptotic models for the genealogy of populations with a large fixed size. We define a bridge to be a right-continuous process (B(r),r[0,1]) with nondecreasing paths and exchangeable increments, such that B(0)=0 and B(1)=1. We show that flows of bridges are in one-to-one correspondence with the so-called exchangeable coalescents. This yields an infinite-dimensional version of the classical Kingman representation for exchangeable partitions of ℕ. We then propose a Poissonian construction of a general class of flows of bridges and identify the associated coalescents. We also discuss an important auxiliary measure-valued process, which is closely related to the genealogical structure coded by the coalescent and can be viewed as a generalized Fleming-Viot process.

On Measure-Valued Processes Generated by Differential Equations

We study the problem of representation of a homogeneous semigroup {Θ t }t ≥ 0 of transformations of probability measures on \(\mathbb{R}^d \) in the form \(\Theta _t (\mu) = \mu \circ u_{\mu}^{-1} (\cdot ,t),\) where \(u_{\mu} :\mathbb{R}^d \times [0, T] \to \mathbb{R}^d\) satisfies a differential equation of a special form dependent on the measure μ. We give necessary and sufficient conditions for this representation.

Stochastic flows with interaction and measure‐valued processes

We consider the new class of the Markov measure‐valued stochastic processes with constant mass. We give the construction of such processes with the family of the probabilities which describe the motion of single particles. We also consider examples related to stochastic flows with the interactions and the local times for such processes.

A Criterion for Relative Compactness of a Sequence of Measure-Valued Random Processes

A criterion for relative compactness of a sequence of measure-valued processes is proved and exemplified.

Competing Species Superprocesses with Infinite Variance

We study pairs of interacting measure-valued branching processes (superprocesses) with alpha-stable migration and $(1+\beta)$-branching mechanism. The interaction is realized via some killing procedure. The collision local time for such processes is constructed as a limit of approximating collision local times. For certain dimensions this convergence holds uniformly over all pairs of such interacting superprocesses. We use this uniformity to prove existence of a solution to a competing species martingale problem under a natural dimension restriction. The competing species model describes the evolution of two populations where individuals of different types may kill each other if they collide. In the case of Brownian migration and finite variance branching, the model was introduced by Evans and Perkins (1994). The fact that now the branching mechanism does not have finite variance requires the development of new methods for handling the collision local time which we believe are of some independent interest.

Multiple-input heavy-traffic real-time queues

A single queueing station that serves K input streams is considered. Each stream is an independent renewal process, with customers having random lead times. Customers are served by processor sharing across streams. Within each stream, two disciplines are considered--earliest deadline first and first-in, first-out. The set of current lead times of the K streams is modeled as a K-dimensional vector of random counting measures on $\mathbb{R}$, and the limit of this vector of measure-valued processes is obtained under heavy traffic conditions.

Strong law of large numbers and mixing for the invariant distributions of measure-valued diffusions

Let M(R d) denote the space of locally finite measures on R d and let M 1( M(R d)) denote the space of probability measures on M(R d) . Define the mean measure π ν of ν∈ M 1( M(R d)) by π ν(B)= ∫ M(R d) η(B) dν(η), for B⊂R d. For such a measure ν with locally finite mean measure π ν , let f be a nonnegative, locally bounded test function satisfying 〈 f, π ν 〉=∞. ν is said to satisfy the strong law of large numbers with respect to f if 〈 f n , η〉/〈 f n , π ν 〉 converges almost surely to 1 with respect to ν as n→∞, for any increasing sequence { f n } of compactly supported functions which converges to f. ν is said to be mixing with respect to two sequences of sets { A n } and { B n } if ∫ M(R d) f(η(A n))g(η(B n)) dν(η)− ∫ M(R d) f(η(A n)) dν(η) ∫ M(R d) g(η(B n)) dν(η) converges to 0 as n→∞ for every pair of functions f, g∈ C b 1([0,∞)). It is known that certain classes of measure-valued diffusion processes possess a family of invariant distributions. These distributions belong to M 1( M(R d)) and have locally finite mean measures. We prove the strong law of large numbers and mixing for many such distributions.

Catalytic branching and the Brownian snake

We construct a catalytic super process X (measure-valued spatial branching process) where the local branching rate is governed by an additive functional A of the motion process. These processes have been investigated before but under restrictive assumptions on A. Here we do not even need continuity of A. The key is to introduce a new time scale in which motion and branching occur at a varying speed but are continuous. Another aspect is to consider X in the generic time scale of the branching—and not of the motion process. This allows to give an explicit construction of X using the Brownian snake. As a by-product this yields an almost sure approximation by the corresponding branching particle systems.

Mutually catalytic branching in the plane: Finite measure states

We study a pair of populations in $\mathbb{R}^{2}$ which undergo diffusion and branching. The system is interactive in that the branching rate of each type is proportional to the local density of the other type. For a diffusion rate sufficiently large compared with the branching rate, the model is constructed as the unique pair of finite measure-valued processes which satisfy a martingale problem involving the collision local time of the solutions. The processes are shown to have densities at fixed times which live on disjoint sets and explode as they approach the interface of the two populations. In the long-term limit, global extinction of one type is shown. The process constructed is a rescaled limit of the corresponding $\mathbb{Z}^{2}$-lattice model studied by D. A. Dawson and E. A. Perkins and resolves the large scale mass-time-space behavior of that model.

Self-Intersection Local Time for ′(ℝd)-Ornstein-Uhlenbeck Processes Arising from Immigration Systems

Fluctuation limits of an immigration branching particle system and an immigration branching measure-valued process yield different types of 𝒮′(ℝd)-valued Ornstein-Uhlenbeck processes whose covariances are given in terms of an excessive measure for the underlying motion in Rd, which is taken to be a symmetric α-stable process. In this paper we prove existence and path continuity results for the self-intersection local time of these Ornstein-Uhlenbeck processes. The results depend on relationships between the dimension d and the parameter α.

On Central Limit Theorems for Vector Random Measures and Measure-Valued Processes

Let B be a separable Banach space. Suppose that ($F,F_i, i\ge 1$) is a sequence of independent identically distributed (i.i.d.) and symmetrical independently scattered (s.i.s.) B-valued random measures. We first establish the central limit theorem for $Y_n=\frac 1{\sqrt n} \sum_{i=1}^nF_i$ by taking the viewpoint of random linear functionals on Schwartz distribution spaces. Then, let ($X,X_i, i\ge 1$) be a sequence of i.i.d. symmetric B-valued random vectors and ($B,B_i, i\ge 1$) a sequence of independent standard Brownian motions on [0,1] independent of ($X,X_i, i\ge 1$). The central limit theorem for measure-valued processes $Z_n(t)=\frac 1{\sqrt n} \sum_{i=1}^nX_i\delta_{B_i(t)}$, $t\in [0,1]$, will be investigated in the same frame. Our main results concerning $Y_n$ differ from D. H. Thang's [Probab. Theory Related Fields, 88 (1991), pp. 1-16] in that we take into account~F as a whole; while the results related to $Z_n$ are extensions of I.~Mitoma [Ann. Probab., 11 (1983), pp.~989--999] to random ...

Skew Convolution Semigroups and Related Immigration Processes

A special type of immigration associated with measure-valued branching processes is formulated by using skew convolution semigroups. We give a characterization for a general inhomogeneous skew convolution semigroup in terms of probability entrance laws. The related immigration process is constructed by summing up measure-valued paths in the Kuznetsov process determined by an entrance rule. The behavior of the Kuznetsov process is then studied, which provides insight into trajectory structures of the immigration process. Some well-known results on excessive measures are formulated in terms of stationary immigration processes.

Multiple Scale Analysis of Spatial Branching Processes under the Palm Distribution

We consider two types of measure-valued branching processes on the lattice $Z^d$. These are on the one hand side a particle system, called branching random walk, and on the other hand its continuous mass analogue, a system of interacting diffusions also called super random walk. It is known that the long-term behavior differs sharply in low and high dimensions: if $d\leq 2$ one gets local extinction, while, for $d\geq 3$, the systems tend to a non-trivial equilibrium. Due to Kallenberg's criterion, local extinction goes along with clumping around a 'typical surviving particle.' This phenomenon is called clustering. A detailed description of the clusters has been given for the corresponding processes on $R^2$ in Klenke (1997). Klenke proved that with the right scaling the mean number of particles over certain blocks are asymptotically jointly distributed like marginals of a system of coupled Feller diffusions, called system of tree indexed Feller diffusions, provided that the initial intensity is appropriately increased to counteract the local extinction. The present paper takes different remedy against the local extinction allowing also for state-dependent branching mechanisms. Instead of increasing the initial intensity, the systems are described under the Palm distribution. It will turn out together with the results in Klenke (1997) that the change to the Palm measure and the multiple scale analysis commute, as $t\to\infty$. The method of proof is based on the fact that the tree indexed systems of the branching processes and of the diffusions in the limit are completely characterized by all their moments. We develop a machinery to describe the space-time moments of the superprocess effectively and explicitly.

On the Log-Laplace Equation for Nonlocal Operators Generating Sub-Markovian Semigroups

We consider the semilinear Cauchy problem for a class of pseudo-differential operators generating sub-Markovian semigroups. Solutions of such problems with negative definite nonlinearity play an important role in constructing branching measure-valued processes. We establish local existence and uniqueness of solutions in the context of the Dirichlet space associated to the problem. Comparison and global properties of solutions are also studied.

Asymptotic study of measure-valued processes generated by randomly moving particles

Asymptotic study of measure-valued processes generated by randomly moving particles

Invariant Probability Distributions for Measure-Valued Diffusions

We investigate the set of invariant probability distributions for measure-valued diffusion processes corresponding to semilinear operators of the form $u_t = L_0 u + \beta u - \alpha u^2$, where $L_0 = 1/2 \sum_{i,j=1}^d a_{i,j} \frac{\partial^2}{\partial x_i \partial x_j}+ \sum_{i=1}^d b_1\frac{\partial}{\partial x_i}$

A LIFO queue in heavy traffic

This paper describes the heavy-traffic behavior of an M/G/1 last-in-first-out preemptive resume queue. An appropriate framework for the analysis is provided by measure-valued processes. In particular, the paper exploits the setting of recent works by Le Gall and Le Jan.Their finite-measure-valued exploration process corresponds to our RES-measure (residual services measure) process, that captures all the relevant information about the evolution of the queue, while their height process corresponds to the queue-length process. The heavy-traffic “diffusion” approximations for the RES-measure and the queue-length processes are derived under the usual second moment assumptions on the service distributions. The tightness of queue lengths argument uses estimates for the total size and height of large Galton–Watson trees.

CONSTRUCTION OF SUPER-BROWNIAN MOTIONS

Suppose A is a continuous additive functional of a Brownian motion. A d-dimensional super-Brownian motion with general branching rate functional A and general mechanism ψ is constructed under a condition on A, which is weaker than the conditions imposed by Dynkin (Ann. Prob. 1991, 19, 1157–1194; An Introduction to Branching Measure-Valued Processes; Amer. Math. Soc.: Providence, RI., 1994).