The Annals of Applied Probability Articles (Project Euclid)

On optimal arbitrage

Thu, 05 Aug 2010 15:41 EDT

Daniel Fernholz, Ioannis Karatzas

Source: Ann. Appl. Probab., Volume 20, Number 4, 1179--1204.

Abstract:
In a Markovian model for a financial market, we characterize the best arbitrage with respect to the market portfolio that can be achieved using nonanticipative investment strategies, in terms of the smallest positive solution to a parabolic partial differential inequality; this is determined entirely on the basis of the covariance structure of the model. The solution is intimately related to properties of strict local martingales and is used to generate the investment strategy which realizes the best possible arbitrage. Some extensions to non-Markovian situations are also presented.

The total path length of split trees

Fri, 12 Oct 2012 14:53 EDT

Nicolas Broutin, Cecilia Holmgren

Source: Ann. Appl. Probab., Volume 22, Number 5, 1745--1777.

Abstract:
We consider the model of random trees introduced by Devroye [ SIAM J. Comput. 28 (1999) 409–432]. The model encompasses many important randomized algorithms and data structures. The pieces of data (items) are stored in a randomized fashion in the nodes of a tree. The total path length (sum of depths of the items) is a natural measure of the efficiency of the algorithm/data structure. Using renewal theory, we prove convergence in distribution of the total path length toward a distribution characterized uniquely by a fixed point equation. Our result covers, using a unified approach, many data structures such as binary search trees, $m$-ary search trees, quad trees, median-of-$(2k+1)$ trees, and simplex trees.

Spreading speeds in reducible multitype branching random walk

Fri, 12 Oct 2012 14:53 EDT

J. D. Biggins

Source: Ann. Appl. Probab., Volume 22, Number 5, 1778--1821.

Abstract:
This paper gives conditions for the rightmost particle in the $n$th generation of a multitype branching random walk to have a speed, in the sense that its location divided by $n$ converges to a constant as $n$ goes to infinity. Furthermore, a formula for the speed is obtained in terms of the reproduction laws. The case where the collection of types is irreducible was treated long ago. In addition, the asymptotic behavior of the number in the $n$th generation to the right of $na$ is obtained. The initial motive for considering the reducible case was results for a deterministic spatial population model with several types of individual discussed by Weinberger, Lewis and Li [ J. Math. Biol. 55 (2007) 207–222]: the speed identified here for the branching random walk corresponds to an upper bound for the speed identified there for the deterministic model.

Convergence of stochastic gene networks to hybrid piecewise deterministic processes

Fri, 12 Oct 2012 14:53 EDT

A. Crudu, A. Debussche, A. Muller, O. Radulescu

Source: Ann. Appl. Probab., Volume 22, Number 5, 1822--1859.

Abstract:
We study the asymptotic behavior of multiscale stochastic gene networks using weak limits of Markov jump processes. Depending on the time and concentration scales of the system, we distinguish four types of limits: continuous piecewise deterministic processes (PDP) with switching, PDP with jumps in the continuous variables, averaged PDP, and PDP with singular switching. We justify rigorously the convergence for the four types of limits. The convergence results can be used to simplify the stochastic dynamics of gene network models arising in molecular biology.

Tracking a random walk first-passage time through noisy observations

Fri, 12 Oct 2012 14:53 EDT

Marat V. Burnashev, Aslan Tchamkerten

Source: Ann. Appl. Probab., Volume 22, Number 5, 1860--1879.

Abstract:
Given a Gaussian random walk (or a Wiener process), possibly with drift, observed through noise, we consider the problem of estimating its first-passage time $\tau_{\ell}$ of a given level $\ell$ with a stopping time $\eta$ defined over the noisy observation process. Main results are upper and lower bounds on the minimum mean absolute deviation $\inf_{\eta}{\mathbb{E} }|\eta-\tau_{\ell}|$ which become tight as $\ell\to\infty$. Interestingly, in this regime the estimation error does not get smaller if we allow $\eta$ to be an arbitrary function of the entire observation process, not necessarily a stopping time. In the particular case where there is no drift, we show that it is impossible to track $\tau_{\ell}$: $\inf_{\eta}{\mathbb{E} }|\eta-\tau_{\ell}|^{p}=\infty$ for any $\ell>0$ and $p\geq1/2$.

Optimal scaling of random walk Metropolis algorithms with discontinuous target densities

Fri, 12 Oct 2012 14:53 EDT

Peter Neal, Gareth Roberts, Wai Kong Yuen

Source: Ann. Appl. Probab., Volume 22, Number 5, 1880--1927.

Abstract:
We consider the optimal scaling problem for high-dimensional random walk Metropolis (RWM) algorithms where the target distribution has a discontinuous probability density function. Almost all previous analysis has focused upon continuous target densities. The main result is a weak convergence result as the dimensionality $d$ of the target densities converges to $\infty$. In particular, when the proposal variance is scaled by $d^{-2}$, the sequence of stochastic processes formed by the first component of each Markov chain converges to an appropriate Langevin diffusion process. Therefore optimizing the efficiency of the RWM algorithm is equivalent to maximizing the speed of the limiting diffusion. This leads to an asymptotic optimal acceptance rate of $e^{-2}$ $(=0.1353)$ under quite general conditions. The results have major practical implications for the implementation of RWM algorithms by highlighting the detrimental effect of choosing RWM algorithms over Metropolis-within-Gibbs algorithms.

Self-similar solutions in one-dimensional kinetic models: A probabilistic view

Fri, 12 Oct 2012 14:53 EDT

Federico Bassetti, Lucia Ladelli

Source: Ann. Appl. Probab., Volume 22, Number 5, 1928--1961.

Abstract:
This paper deals with a class of Boltzmann equations on the real line, extensions of the well-known Kac caricature. A distinguishing feature of the corresponding equations is that therein, the collision gain operators are defined by $N$-linear smoothing transformations. These kind of problems have been studied, from an essentially analytic viewpoint, in a recent paper by Bobylev, Cercignani and Gamba [ Comm. Math. Phys. 291 (2009) 599–644]. Instead, the present work rests exclusively on probabilistic methods, based on techniques pertaining to the classical central limit problem and to the so-called fixed-point equations for probability distributions. An advantage of resorting to methods from the probability theory is that the same results—relative to self-similar solutions—as those obtained by Bobylev, Cercignani and Gamba, are here deduced under weaker conditions. In particular, it is shown how convergence to a self-similar solution depends on the belonging of the initial datum to the domain of attraction of a specific stable distribution. Moreover, some results on the speed of convergence are given in terms of Kantorovich–Wasserstein and Zolotarev distances between probability measures.

Fast approach to the Tracy–Widom law at the edge of GOE and GUE

Fri, 12 Oct 2012 14:53 EDT

Iain M. Johnstone, Zongming Ma

Source: Ann. Appl. Probab., Volume 22, Number 5, 1962--1988.

Abstract:
We study the rate of convergence for the largest eigenvalue distributions in the Gaussian unitary and orthogonal ensembles to their Tracy–Widom limits. We show that one can achieve an $O(N^{-2/3})$ rate with particular choices of the centering and scaling constants. The arguments here also shed light on more complicated cases of Laguerre and Jacobi ensembles, in both unitary and orthogonal versions. Numerical work shows that the suggested constants yield reasonable approximations, even for surprisingly small values of $N$.

Bootstrap percolation on the random graph $G_{n,p}$

Fri, 12 Oct 2012 14:53 EDT

Svante Janson, Tomasz Łuczak, Tatyana Turova, Thomas Vallier

Source: Ann. Appl. Probab., Volume 22, Number 5, 1989--2047.

Abstract:
Bootstrap percolation on the random graph $G_{n,p}$ is a process of spread of “activation” on a given realization of the graph with a given number of initially active nodes. At each step those vertices which have not been active but have at least $r\geq2$ active neighbors become active as well. We study the size $A^{\ast}$ of the final active set. The parameters of the model are, besides $r$ (fixed) and $n$ (tending to $\infty$), the size $a=a(n)$ of the initially active set and the probability $p=p(n)$ of the edges in the graph. We show that the model exhibits a sharp phase transition: depending on the parameters of the model, the final size of activation with a high probability is either $n-o(n)$ or it is $o(n)$. We provide a complete description of the phase diagram on the space of the parameters of the model. In particular, we find the phase transition and compute the asymptotics (in probability) for $A^{\ast}$; we also prove a central limit theorem for $A^{\ast}$ in some ranges. Furthermore, we provide the asymptotics for the number of steps until the process stops.

Nonuniform random geometric graphs with location-dependent radii

Fri, 12 Oct 2012 14:53 EDT

Srikanth K. Iyer, Debleena Thacker

Source: Ann. Appl. Probab., Volume 22, Number 5, 2048--2066.

Abstract:
We propose a distribution-free approach to the study of random geometric graphs. The distribution of vertices follows a Poisson point process with intensity function $nf(\cdot)$, where $n\in \mathbb{N}$, and $f$ is a probability density function on $\mathbb{R}^{d}$. A vertex located at $x$ connects via directed edges to other vertices that are within a cut-off distance $r_{n}(x)$. We prove strong law results for (i) the critical cut-off function so that almost surely, the graph does not contain any node with out-degree zero for sufficiently large $n$ and (ii) the maximum and minimum vertex degrees. We also provide a characterization of the cut-off function for which the number of nodes with out-degree zero converges in distribution to a Poisson random variable. We illustrate this result for a class of densities with compact support that have at most polynomial rates of decay to zero. Finally, we state a sufficient condition for an enhanced version of the above graph to be almost surely connected eventually.

Simple arbitrage

Fri, 12 Oct 2012 14:53 EDT

Christian Bender

Source: Ann. Appl. Probab., Volume 22, Number 5, 2067--2085.

Abstract:
We characterize absence of arbitrage with simple trading strategies in a discounted market with a constant bond and several risky assets. We show that if there is a simple arbitrage, then there is a 0-admissible one or an obvious one, that is, a simple arbitrage which promises a minimal riskless gain of $\varepsilon$, if the investor trades at all. For continuous stock models, we provide an equivalent condition for absence of 0-admissible simple arbitrage in terms of a property of the fine structure of the paths, which we call “two-way crossing.” This property can be verified for many models by the law of the iterated logarithm. As an application we show that the mixed fractional Black–Scholes model, with Hurst parameter bigger than a half, is free of simple arbitrage on a compact time horizon. More generally, we discuss the absence of simple arbitrage for stochastic volatility models and local volatility models which are perturbed by an independent 1$/$2-Hölder continuous process.

The asymptotic distribution of the length of Beta-coalescent trees

Fri, 12 Oct 2012 14:53 EDT

Götz Kersting

Source: Ann. Appl. Probab., Volume 22, Number 5, 2086--2107.

Abstract:
We derive the asymptotic distribution of the total length $L_{n}$ of a $\operatorname{Beta}(2-\alpha,\alpha)$-coalescent tree for $1<\alpha<2$, starting from $n$ individuals. There are two regimes: If $\alpha\le\frac{1}{2}(1+\sqrt{5})$, then $L_{n}$ suitably rescaled has a stable limit distribution of index $\alpha$. Otherwise $L_{n}$ just has to be shifted by a constant (depending on $n$) to get convergence to a nondegenerate limit distribution. As a consequence, we obtain the limit distribution of the number $S_{n}$ of segregation sites. These are points (mutations), which are placed on the tree’s branches according to a Poisson point process with constant rate.

Muller’s ratchet with compensatory mutations

Fri, 12 Oct 2012 14:53 EDT

P. Pfaffelhuber, P. R. Staab, A. Wakolbinger

Source: Ann. Appl. Probab., Volume 22, Number 5, 2108--2132.

Abstract:
We consider an infinite-dimensional system of stochastic differential equations describing the evolution of type frequencies in a large population. The type of an individual is the number of deleterious mutations it carries, where fitness of individuals carrying $k$ mutations is decreased by $\alpha k$ for some $\alpha>0$. Along the individual lines of descent, new mutations accumulate at rate $\lambda$ per generation, and each of these mutations has a probability $\gamma$ per generation to disappear. While the case $\gamma=0$ is known as (the Fleming–Viot version of) Muller’s ratchet , the case $\gamma>0$ is associated with compensatory mutations in the biological literature. We show that the system has a unique weak solution. In the absence of random fluctuations in type frequencies (i.e., for the so-called infinite population limit) we obtain the solution in a closed form by analyzing a probabilistic particle system and show that for $\gamma>0$, the unique equilibrium state is the Poisson distribution with parameter $\lambda/(\gamma+\alpha)$.

Stochastic approximation, cooperative dynamics and supermodular games

Fri, 12 Oct 2012 14:53 EDT

Michel Benaïm, Mathieu Faure

Source: Ann. Appl. Probab., Volume 22, Number 5, 2133--2164.

Abstract:
This paper considers a stochastic approximation algorithm, with decreasing step size and martingale difference noise. Under very mild assumptions, we prove the nonconvergence of this process toward a certain class of repulsive sets for the associated ordinary differential equation (ODE). We then use this result to derive the convergence of the process when the ODE is cooperative in the sense of Hirsch [ SIAM J. Math. Anal. 16 (1985) 423–439]. In particular, this allows us to extend significantly the main result of Hofbauer and Sandholm [ Econometrica 70 (2002) 2265–2294] on the convergence of stochastic fictitious play in supermodular games .

The spatial $\Lambda$-Fleming–Viot process on a large torus: Genealogies in the presence of recombination

Fri, 23 Nov 2012 13:39 EST

A. M. Etheridge, A. Véber

Source: Ann. Appl. Probab., Volume 22, Number 6, 2165--2209.

Abstract:
We extend the spatial $\Lambda$-Fleming–Viot process introduced in [ Electron. J. Probab. 15 (2010) 162–216] to incorporate recombination. The process models allele frequencies in a population which is distributed over the two-dimensional torus $\mathbb{T} (L)$ of sidelength $L$ and is subject to two kinds of reproduction events: small events of radius $\mathcal{O} (1)$ and much rarer large events of radius $\mathcal{O} (L^{\alpha})$ for some $\alpha\in(0,1]$. We investigate the correlation between the times to the most recent common ancestor of alleles at two linked loci for a sample of size two from the population. These individuals are initially sampled from “far apart” on the torus. As $L$ tends to infinity, depending on the frequency of the large events, the recombination rate and the initial distance between the two individuals sampled, we obtain either a complete decorrelation of the coalescence times at the two loci, or a sharp transition between a first period of complete correlation and a subsequent period during which the remaining times needed to reach the most recent common ancestor at each locus are independent. We use our computations to derive approximate probabilities of identity by descent as a function of the separation at which the two individuals are sampled.

Phase transition for the mixing time of the Glauber dynamics for coloring regular trees

Fri, 23 Nov 2012 13:39 EST

Prasad Tetali, Juan C. Vera, Eric Vigoda, Linji Yang

Source: Ann. Appl. Probab., Volume 22, Number 6, 2210--2239.

Abstract:
We prove that the mixing time of the Glauber dynamics for random $k$-colorings of the complete tree with branching factor $b$ undergoes a phase transition at $k=b(1+o_{b}(1))/\ln{b}$. Our main result shows nearly sharp bounds on the mixing time of the dynamics on the complete tree with $n$ vertices for $k=Cb/\ln{b}$ colors with constant $C$. For $C\geq1$ we prove the mixing time is $O(n^{1+o_{b}(1)}\ln{n})$. On the other side, for $C<1$ the mixing time experiences a slowing down; in particular, we prove it is $O(n^{1/C+o_{b}(1)}\ln{n})$ and $\Omega(n^{1/C-o_{b}(1)})$. The critical point $C=1$ is interesting since it coincides (at least up to first order) with the so-called reconstruction threshold which was recently established by Sly. The reconstruction threshold has been of considerable interest recently since it appears to have close connections to the efficiency of certain local algorithms, and this work was inspired by our attempt to understand these connections in this particular setting.

Crossings of smooth shot noise processes

Fri, 23 Nov 2012 13:39 EST

Hermine Biermé, Agnès Desolneux

Source: Ann. Appl. Probab., Volume 22, Number 6, 2240--2281.

Abstract:
In this paper, we consider smooth shot noise processes and their expected number of level crossings. When the kernel response function is sufficiently smooth, the mean number of crossings function is obtained through an integral formula. Moreover, as the intensity increases, or equivalently, as the number of shots becomes larger, a normal convergence to the classical Rice’s formula for Gaussian processes is obtained. The Gaussian kernel function, that corresponds to many applications in physics, is studied in detail and two different regimes are exhibited.

Large deviations of the empirical currents for a boundary-driven reaction diffusion model

Fri, 23 Nov 2012 13:39 EST

Thierry Bodineau, Maxime Lagouge

Source: Ann. Appl. Probab., Volume 22, Number 6, 2282--2319.

Abstract:
We derive a large deviation principle for the empirical currents of lattice gas dynamics which combine a fast stirring mechanism (Symmetric Simple Exclusion Process) and creation/annihilation mechanisms (Glauber dynamics). Previous results on the density large deviations can be recovered from this general large deviation principle. The contribution of external driving forces due to reservoirs at the boundary of the system is also taken into account.

Optimal scaling and diffusion limits for the Langevin algorithm in high dimensions

Fri, 23 Nov 2012 13:39 EST

Natesh S. Pillai, Andrew M. Stuart, Alexandre H. Thiéry

Source: Ann. Appl. Probab., Volume 22, Number 6, 2320--2356.

Abstract:
The Metropolis-adjusted Langevin (MALA) algorithm is a sampling algorithm which makes local moves by incorporating information about the gradient of the logarithm of the target density. In this paper we study the efficiency of MALA on a natural class of target measures supported on an infinite dimensional Hilbert space. These natural measures have density with respect to a Gaussian random field measure and arise in many applications such as Bayesian nonparametric statistics and the theory of conditioned diffusions. We prove that, started in stationarity, a suitably interpolated and scaled version of the Markov chain corresponding to MALA converges to an infinite dimensional diffusion process. Our results imply that, in stationarity, the MALA algorithm applied to an $N$-dimensional approximation of the target will take $\mathcal{O}(N^{1/3})$ steps to explore the invariant measure, comparing favorably with the Random Walk Metropolis which was recently shown to require $\mathcal{O}(N)$ steps when applied to the same class of problems. As a by-product of the diffusion limit, it also follows that the MALA algorithm is optimized at an average acceptance probability of $0.574$. Previous results were proved only for targets which are products of one-dimensional distributions, or for variants of this situation, limiting their applicability. The correlation in our target means that the rescaled MALA algorithm converges weakly to an infinite dimensional Hilbert space valued diffusion, and the limit cannot be described through analysis of scalar diffusions. The limit theorem is proved by showing that a drift-martingale decomposition of the Markov chain, suitably scaled, closely resembles a weak Euler–Maruyama discretization of the putative limit. An invariance principle is proved for the martingale, and a continuous mapping argument is used to complete the proof.

A class of multifractal processes constructed using an embedded branching process

Fri, 23 Nov 2012 13:39 EST

Geoffrey Decrouez, Owen Dafydd Jones

Source: Ann. Appl. Probab., Volume 22, Number 6, 2357--2387.

Abstract:
We present a new class of multifractal process on $\mathbb{R}$, constructed using an embedded branching process. The construction makes use of known results on multitype branching random walks, and along the way constructs cascade measures on the boundaries of multitype Galton–Watson trees. Our class of processes includes Brownian motion subjected to a continuous multifractal time-change. In addition, if we observe our process at a fixed spatial resolution, then we can obtain a finite Markov representation of it, which we can use for on-line simulation. That is, given only the Markov representation at step $n$, we can generate step $n+1$ in $O(\log n)$ operations. Detailed pseudo-code for this algorithm is provided.

Mean-variance hedging via stochastic control and BSDEs for general semimartingales

Fri, 23 Nov 2012 13:39 EST

Monique Jeanblanc, Michael Mania, Marina Santacroce, Martin Schweizer

Source: Ann. Appl. Probab., Volume 22, Number 6, 2388--2428.

Abstract:
We solve the problem of mean-variance hedging for general semimartingale models via stochastic control methods. After proving that the value process of the associated stochastic control problem has a quadratic structure, we characterize its three coefficient processes as solutions of semimartingale backward stochastic differential equations and show how they can be used to describe the optimal trading strategy for each conditional mean-variance hedging problem. For comparison with the existing literature, we provide alternative equivalent versions of the BSDEs and present a number of simple examples.

Phylogenetic mixtures: Concentration of measure in the large-tree limit

Fri, 23 Nov 2012 13:39 EST

Elchanan Mossel, Sebastien Roch

Source: Ann. Appl. Probab., Volume 22, Number 6, 2429--2459.

Abstract:
The reconstruction of phylogenies from DNA or protein sequences is a major task of computational evolutionary biology. Common phenomena, notably variations in mutation rates across genomes and incongruences between gene lineage histories, often make it necessary to model molecular data as originating from a mixture of phylogenies. Such mixed models play an increasingly important role in practice. Using concentration of measure techniques, we show that mixtures of large trees are typically identifiable. We also derive sequence-length requirements for high-probability reconstruction.

A diffusion approximation theorem for a nonlinear PDE with application to random birefringent optical fibers

Fri, 23 Nov 2012 13:39 EST

A. de Bouard, M. Gazeau

Source: Ann. Appl. Probab., Volume 22, Number 6, 2460--2504.

Abstract:
In this article we propose a generalization of the theory of diffusion approximation for random ODE to a nonlinear system of random Schrödinger equations. This system arises in the study of pulse propagation in randomly birefringent optical fibers. We first show existence and uniqueness of solutions for the random PDE and the limiting equation. We follow the work of Garnier and Marty [ Wave Motion 43 (2006) 544–560], Marty [Problèmes d’évolution en milieux aléatoires: Théorèmes limites, schémas numériques et applications en optique (2005) Univ. Paul Sabatier], where a linear electric field is considered, and we get an asymptotic dynamic for the nonlinear electric field.

Stochastic functional differential equations driven by Lévy processes and quasi-linear partial integro-differential equations

Fri, 23 Nov 2012 13:39 EST

Xicheng Zhang

Source: Ann. Appl. Probab., Volume 22, Number 6, 2505--2538.

Abstract:
In this article we study a class of stochastic functional differential equations driven by Lévy processes (in particular, $\alpha$-stable processes), and obtain the existence and uniqueness of Markov solutions in small time intervals. This corresponds to the local solvability to a class of quasi-linear partial integro-differential equations. Moreover, in the constant diffusion coefficient case, without any assumptions on the Lévy generator, we also show the existence of a unique maximal weak solution for a class of semi-linear partial integro-differential equation systems under bounded Lipschitz assumptions on the coefficients. Meanwhile, in the nondegenerate case (corresponding to $\Delta^{\alpha/2}$ with $\alpha\in(1,2]$), based upon some gradient estimates, the existence of global solutions is established too. In particular, this provides a probabilistic treatment for the nonlinear partial integro-differential equations, such as the multi-dimensional fractal Burgers equations and the fractal scalar conservation law equations.

On the rate of approximation in finite-alphabet longest increasing subsequence problems

Fri, 23 Nov 2012 13:39 EST

Christian Houdré, Zsolt Talata

Source: Ann. Appl. Probab., Volume 22, Number 6, 2539--2559.

Abstract:
The rate of convergence of the distribution of the length of the longest increasing subsequence, toward the maximal eigenvalue of certain matrix ensembles, is investigated. For finite-alphabet uniform and nonuniform i.i.d. sources, a rate of $\log n/\sqrt{n}$ is obtained. The uniform binary case is further explored, and an improved $1/\sqrt{n}$ rate obtained.

Tree-valued Fleming–Viot dynamics with mutation and selection

Fri, 23 Nov 2012 13:39 EST

Andrej Depperschmidt, Andreas Greven, Peter Pfaffelhuber

Source: Ann. Appl. Probab., Volume 22, Number 6, 2560--2615.

Abstract:
The Fleming–Viot measure-valued diffusion is a Markov process describing the evolution of (allelic) types under mutation, selection and random reproduction. We enrich this process by genealogical relations of individuals so that the random type distribution as well as the genealogical distances in the population evolve stochastically. The state space of this tree-valued enrichment of the Fleming–Viot dynamics with mutation and selection (TFVMS) consists of marked ultrametric measure spaces, equipped with the marked Gromov-weak topology and a suitable notion of polynomials as a separating algebra of test functions. The construction and study of the TFVMS is based on a well-posed martingale problem. For existence, we use approximating finite population models, the tree-valued Moran models, while uniqueness follows from duality to a function-valued process. Path properties of the resulting process carry over from the neutral case due to absolute continuity, given by a new Girsanov-type theorem on marked metric measure spaces. To study the long-time behavior of the process, we use a duality based on ideas from Dawson and Greven [On the effects of migration in spatial Fleming–Viot models with selection and mutation (2011c) Unpublished manuscript] and prove ergodicity of the TFVMS if the Fleming–Viot measure-valued diffusion is ergodic. As a further application, we consider the case of two allelic types and additive selection. For small selection strength, we give an expansion of the Laplace transform of genealogical distances in equilibrium, which is a first step in showing that distances are shorter in the selective case.

Large deviation principles for nongradient weakly asymmetric stochastic lattice gases

Fri, 25 Jan 2013 09:33 EST

Lorenzo Bertini, Alessandra Faggionato, Davide Gabrielli

Source: Ann. Appl. Probab., Volume 23, Number 1, 1--65.

Abstract:
We consider a lattice gas on the discrete $d$-dimensional torus $(\mathbb{Z}/N\mathbb{Z})^d$ with a generic translation invariant, finite range interaction satisfying a uniform strong mixing condition. The lattice gas performs a Kawasaki dynamics in the presence of a weak external field $E/N$. We show that, under diffusive rescaling, the hydrodynamic behavior of the lattice gas is described by a nonlinear driven diffusion equation. We then prove the associated dynamical large deviation principle. Under suitable assumptions on the external field (e.g., $E$ constant), we finally analyze the variational problem defining the quasi-potential and characterize the optimal exit trajectory. From these results we deduce the asymptotic behavior of the stationary measures of the stochastic lattice gas, which are not explicitly known. In particular, when the external field $E$ is constant, we prove a stationary large deviation principle for the empirical density and show that the rate function does not depend on $E$.

Adaptive Gibbs samplers and related MCMC methods

Fri, 25 Jan 2013 09:33 EST

Krzysztof Łatuszyński, Gareth O. Roberts, Jeffrey S. Rosenthal

Source: Ann. Appl. Probab., Volume 23, Number 1, 66--98.

Abstract:
We consider various versions of adaptive Gibbs and Metropolis-within-Gibbs samplers, which update their selection probabilities (and perhaps also their proposal distributions) on the fly during a run by learning as they go in an attempt to optimize the algorithm. We present a cautionary example of how even a simple-seeming adaptive Gibbs sampler may fail to converge. We then present various positive results guaranteeing convergence of adaptive Gibbs samplers under certain conditions.

The coalescent point process of branching trees

Fri, 25 Jan 2013 09:33 EST

Amaury Lambert, Lea Popovic

Source: Ann. Appl. Probab., Volume 23, Number 1, 99--144.

Abstract:
We define a doubly infinite, monotone labeling of Bienaymé–Galton–Watson (BGW) genealogies. The genealogy of the current generation backwards in time is uniquely determined by the coalescent point process $(A_{i};i\ge1)$, where $A_{i}$ is the coalescence time between individuals $i$ and $i+1$. There is a Markov process of point measures $(B_{i};i\ge1)$ keeping track of more ancestral relationships, such that $A_{i}$ is also the first point mass of $B_{i}$. This process of point measures is also closely related to an inhomogeneous spine decomposition of the lineage of the first surviving particle in generation $h$ in a planar BGW tree conditioned to survive $h$ generations. The decomposition involves a point measure $\rho$ storing the number of subtrees on the right-hand side of the spine. Under appropriate conditions, we prove convergence of this point measure to a point measure on $\mathbb{R}_{+}$ associated with the limiting continuous-state branching (CSB) process. We prove the associated invariance principle for the coalescent point process, after we discretize the limiting CSB population by considering only points with coalescence times greater than $\varepsilon $. The limiting coalescent point process $(B^{\varepsilon}_{i};i\ge1)$ is the sequence of depths greater than $\varepsilon$ of the excursions of the height process below some fixed level. In the diffusion case, there are no multiple ancestries and (it is known that) the coalescent point process is a Poisson point process with an explicit intensity measure. We prove that in the general case the coalescent process with multiplicities $(B^{\varepsilon}_{i};i\ge1)$ is a Markov chain of point masses and we give an explicit formula for its transition function. The paper ends with two applications in the discrete case. Our results show that the sequence of $A_{i}$’s are i.i.d. when the offspring distribution is linear fractional. Also, the law of Yaglom’s quasi-stationary population size for subcritical BGW processes is disintegrated with respect to the time to most recent common ancestor of the whole population.

SPDE limits of many-server queues

Fri, 25 Jan 2013 09:33 EST

Haya Kaspi, Kavita Ramanan

Source: Ann. Appl. Probab., Volume 23, Number 1, 145--229.

Abstract:
This paper studies a queueing system in which customers with independent and identically distributed service times arrive to a queue with many servers and enter service in the order of arrival. The state of the system is represented by a process that describes the total number of customers in the system, and a measure-valued process that keeps track of the ages of customers in service, leading to a Markovian description of the dynamics. Under suitable assumptions, a functional central limit theorem is established for the sequence of (centered and scaled) state processes as the number of servers goes to infinity. The limit process describing the total number in system is shown to be an Itô diffusion with a constant diffusion coefficient that is insensitive to the service distribution beyond its mean. In addition, the limit of the sequence of (centered and scaled) age processes is shown to be a diffusion taking values in a Hilbert space and is characterized as the unique solution of a stochastic partial differential equation that is coupled with the Itô diffusion describing the limiting number in system. Furthermore, the limit processes are shown to be semimartingales and to possess a strong Markov property.

Population genetics of neutral mutations in exponentially growing cancer cell populations

Fri, 25 Jan 2013 09:33 EST

Rick Durrett

Source: Ann. Appl. Probab., Volume 23, Number 1, 230--250.

Abstract:
In order to analyze data from cancer genome sequencing projects, we need to be able to distinguish causative, or “driver,” mutations from “passenger” mutations that have no selective effect. Toward this end, we prove results concerning the frequency of neutural mutations in exponentially growing multitype branching processes that have been widely used in cancer modeling. Our results yield a simple new population genetics result for the site frequency spectrum of a sample from an exponentially growing population.

Optimal stopping under probability distortion

Fri, 25 Jan 2013 09:33 EST

Zuo Quan Xu, Xun Yu Zhou

Source: Ann. Appl. Probab., Volume 23, Number 1, 251--282.

Abstract:
We formulate an optimal stopping problem for a geometric Brownian motion where the probability scale is distorted by a general nonlinear function. The problem is inherently time inconsistent due to the Choquet integration involved. We develop a new approach, based on a reformulation of the problem where one optimally chooses the probability distribution or quantile function of the stopped state. An optimal stopping time can then be recovered from the obtained distribution/quantile function, either in a straightforward way for several important cases or in general via the Skorokhod embedding. This approach enables us to solve the problem in a fairly general manner with different shapes of the payoff and probability distortion functions. We also discuss economical interpretations of the results. In particular, we justify several liquidation strategies widely adopted in stock trading, including those of “buy and hold,” “cut loss or take profit,” “cut loss and let profit run” and “sell on a percentage of historical high.”

Poisson–Dirichlet branching random walks

Fri, 25 Jan 2013 09:33 EST

Louigi Addario-Berry, Kevin Ford

Source: Ann. Appl. Probab., Volume 23, Number 1, 283--307.

Abstract:
We determine, to within $O(1)$, the expected minimal position at level $n$ in certain branching random walks. The walks under consideration have displacement vector $(v_{1},v_{2},\ldots)$, where each $v_{j}$ is the sum of $j$ independent $\operatorname{Exponential}(1)$ random variables and the different $v_{i}$ need not be independent. In particular, our analysis applies to the Poisson–Dirichlet branching random walk and to the Poisson-weighted infinite tree. As a corollary, we also determine the expected height of a random recursive tree to within $O(1)$.

Dual formulation of second order target problems

Fri, 25 Jan 2013 09:33 EST

H. Mete Soner, Nizar Touzi, Jianfeng Zhang

Source: Ann. Appl. Probab., Volume 23, Number 1, 308--347.

Abstract:
This paper provides a new formulation of second order stochastic target problems introduced in [ SIAM J. Control Optim. 48 (2009) 2344–2365] by modifying the reference probability so as to allow for different scales. This new ingredient enables us to prove a dual formulation of the target problem as the supremum of the solutions of standard backward stochastic differential equations. In particular, in the Markov case, the dual problem is known to be connected to a fully nonlinear, parabolic partial differential equation and this connection can be viewed as a stochastic representation for all nonlinear, scalar, second order, parabolic equations with a convex Hessian dependence.

Default clustering in large portfolios: Typical events

Fri, 25 Jan 2013 09:33 EST

Kay Giesecke, Konstantinos Spiliopoulos, Richard B. Sowers

Source: Ann. Appl. Probab., Volume 23, Number 1, 348--385.

Abstract:
We develop a dynamic point process model of correlated default timing in a portfolio of firms, and analyze typical default profiles in the limit as the size of the pool grows. In our model, a firm defaults at a stochastic intensity that is influenced by an idiosyncratic risk process, a systematic risk process common to all firms, and past defaults. We prove a law of large numbers for the default rate in the pool, which describes the “typical” behavior of defaults.

Alpha-diversity processes and normalized inverse-Gaussian diffusions

Fri, 25 Jan 2013 09:33 EST

Matteo Ruggiero, Stephen G. Walker, Stefano Favaro

Source: Ann. Appl. Probab., Volume 23, Number 1, 386--425.

Abstract:
The infinitely-many-neutral-alleles model has recently been extended to a class of diffusion processes associated with Gibbs partitions of two-parameter Poisson–Dirichlet type. This paper introduces a family of infinite-dimensional diffusions associated with a different subclass of Gibbs partitions, induced by normalized inverse-Gaussian random probability measures. Such diffusions describe the evolution of the frequencies of infinitely-many types together with the dynamics of the time-varying mutation rate, which is driven by an $\alpha$-diversity diffusion. Constructed as a dynamic version, relative to this framework, of the corresponding notion for Gibbs partitions, the latter is explicitly derived from an underlying population model and shown to coincide, in a special case, with the diffusion approximation of a critical Galton–Watson branching process. The class of infinite-dimensional processes is characterized in terms of its infinitesimal generator on an appropriate domain, and shown to be the limit in distribution of a certain sequence of Feller diffusions with finitely-many types. Moreover, a discrete representation is provided by means of appropriately transformed Moran-type particle processes, where the particles are samples from a normalized inverse-Gaussian random probability measure. The relationship between the limit diffusion and the two-parameter model is also discussed.

No-arbitrage of second kind in countable markets with proportional transaction costs

Tue, 12 Feb 2013 10:14 EST

Bruno Bouchard, Erik Taflin

Source: Ann. Appl. Probab., Volume 23, Number 2, 427--454.

Abstract:
Motivated by applications to bond markets, we propose a multivariate framework for discrete time financial markets with proportional transaction costs and a countable infinite number of tradable assets. We show that the no-arbitrage of second kind property (NA2 in short), recently introduced by Rásonyi for finite-dimensional markets, allows us to provide a closure property for the set of attainable claims in a very natural way, under a suitable efficient friction condition. We also extend to this context the equivalence between NA2 and the existence of many (strictly) consistent price systems.

Optimal investment under multiple defaults risk: A BSDE-decomposition approach

Tue, 12 Feb 2013 10:14 EST

Ying Jiao, Idris Kharroubi, Huyên Pham

Source: Ann. Appl. Probab., Volume 23, Number 2, 455--491.

Abstract:
We study an optimal investment problem under contagion risk in a financial model subject to multiple jumps and defaults. The global market information is formulated as a progressive enlargement of a default-free Brownian filtration, and the dependence of default times is modeled by a conditional density hypothesis. In this Itô-jump process model, we give a decomposition of the corresponding stochastic control problem into stochastic control problems in the default-free filtration, which are determined in a backward induction. The dynamic programming method leads to a backward recursive system of quadratic backward stochastic differential equations (BSDEs) in Brownian filtration, and our main result proves, under fairly general conditions, the existence and uniqueness of a solution to this system, which characterizes explicitly the value function and optimal strategies to the optimal investment problem. We illustrate our solutions approach with some numerical tests emphasizing the impact of default intensities, loss or gain at defaults and correlation between assets. Beyond the financial problem, our decomposition approach provides a new perspective for solving quadratic BSDEs with a finite number of jumps.

Stochastic coalescence in logarithmic time

Tue, 12 Feb 2013 10:14 EST

Po-Shen Loh, Eyal Lubetzky

Source: Ann. Appl. Probab., Volume 23, Number 2, 492--528.

Abstract:
The following distributed coalescence protocol was introduced by Dahlia Malkhi in 2006 motivated by applications in social networking. Initially there are $n$ agents wishing to coalesce into one cluster via a decentralized stochastic process, where each round is as follows: every cluster flips a fair coin to dictate whether it is to issue or accept requests in this round. Issuing a request amounts to contacting a cluster randomly chosen proportionally to its size . A cluster accepting requests is to select an incoming one uniformly (if there are such) and merge with that cluster. Empirical results by Fernandess and Malkhi suggested the protocol concludes in $O(\log n)$ rounds with high probability, whereas numerical estimates by Oded Schramm, based on an ingenious analytic approximation, suggested that the coalescence time should be super-logarithmic. Our contribution is a rigorous study of the stochastic coalescence process with two consequences. First, we confirm that the above process indeed requires super-logarithmic time w.h.p., where the inefficient rounds are due to oversized clusters that occasionally develop. Second, we remedy this by showing that a simple modification produces an essentially optimal distributed protocol; if clusters favor their smallest incoming merge request then the process does terminate in $O(\log n)$ rounds w.h.p., and simulations show that the new protocol readily outperforms the original one. Our upper bound hinges on a potential function involving the logarithm of the number of clusters and the cluster-susceptibility, carefully chosen to form a supermartingale. The analysis of the lower bound builds upon the novel approach of Schramm which may find additional applications: rather than seeking a single parameter that controls the system behavior, instead one approximates the system by the Laplace transform of the entire cluster-size distribution.

Separation of time-scales and model reduction for stochastic reaction networks

Tue, 12 Feb 2013 10:14 EST

Hye-Won Kang, Thomas G. Kurtz

Source: Ann. Appl. Probab., Volume 23, Number 2, 529--583.

Abstract:
A stochastic model for a chemical reaction network is embedded in a one-parameter family of models with species numbers and rate constants scaled by powers of the parameter. A systematic approach is developed for determining appropriate choices of the exponents that can be applied to large complex networks. When the scaling implies subnetworks have different time-scales, the subnetworks can be approximated separately, providing insight into the behavior of the full network through the analysis of these lower-dimensional approximations.

Bounds on the suprema of Gaussian processes, and omega results for the sum of a random multiplicative function

Tue, 12 Feb 2013 10:14 EST

Adam J. Harper

Source: Ann. Appl. Probab., Volume 23, Number 2, 584--616.

Abstract:
We prove new lower bounds for the upper tail probabilities of suprema of Gaussian processes. Unlike many existing bounds, our results are not asymptotic, but supply strong information when one is only a little into the upper tail. We present an extended application to a Gaussian version of a random process studied by Halász. This leads to much improved lower bound results for the sum of a random multiplicative function. We further illustrate our methods by improving lower bounds for some classical constants from extreme value theory, the Pickands constants $H_{\alpha}$, as $\alpha\rightarrow0$.

A Berry–Esseen bound with applications to vertex degree counts in the Erdős–Rényi random graph

Tue, 12 Feb 2013 10:14 EST

Larry Goldstein

Source: Ann. Appl. Probab., Volume 23, Number 2, 617--636.

Abstract:
Applying Stein’s method, an inductive technique and size bias coupling yields a Berry–Esseen theorem for normal approximation without the usual restriction that the coupling be bounded. The theorem is applied to counting the number of vertices in the Erdős–Rényi random graph of a given degree.

Sharp benefit-to-cost rules for the evolution of cooperation on regular graphs

Tue, 12 Feb 2013 10:14 EST

Yu-Ting Chen

Source: Ann. Appl. Probab., Volume 23, Number 2, 637--664.

Abstract:
We study two of the simple rules on finite graphs under the death–birth updating and the imitation updating discovered by Ohtsuki, Hauert, Lieberman and Nowak [ Nature 441 (2006) 502–505]. Each rule specifies a payoff-ratio cutoff point for the magnitude of fixation probabilities of the underlying evolutionary game between cooperators and defectors. We view the Markov chains associated with the two updating mechanisms as voter model perturbations. Then we present a first-order approximation for fixation probabilities of general voter model perturbations on finite graphs subject to small perturbation in terms of the voter model fixation probabilities. In the context of regular graphs, we obtain algebraically explicit first-order approximations for the fixation probabilities of cooperators distributed as certain uniform distributions. These approximations lead to a rigorous proof that both of the rules of Ohtsuki et al. are valid and are sharp.

On utility maximization under convex portfolio constraints

Tue, 12 Feb 2013 10:14 EST

Kasper Larsen, Gordan Žitković

Source: Ann. Appl. Probab., Volume 23, Number 2, 665--692.

Abstract:
We consider a utility-maximization problem in a general semimartingale financial model, subject to constraints on the number of shares held in each risky asset. These constraints are modeled by predictable convex-set-valued processes whose values do not necessarily contain the origin; that is, it may be inadmissible for an investor to hold no risky investment at all. Such a setup subsumes the classical constrained utility-maximization problem, as well as the problem where illiquid assets or a random endowment are present. Our main result establishes the existence of optimal trading strategies in such models under no smoothness requirements on the utility function. The result also shows that, up to attainment, the dual optimization problem can be posed over a set of countably-additive probability measures, thus eschewing the need for the usual finitely-additive enlargement.

Alignment-free phylogenetic reconstruction: Sample complexity via a branching process analysis

Tue, 12 Feb 2013 10:14 EST

Constantinos Daskalakis, Sebastien Roch

Source: Ann. Appl. Probab., Volume 23, Number 2, 693--721.

Abstract:
We present an efficient phylogenetic reconstruction algorithm allowing insertions and deletions which provably achieves a sequence-length requirement (or sample complexity) growing polynomially in the number of taxa. Our algorithm is distance-based, that is, it relies on pairwise sequence comparisons. More importantly, our approach largely bypasses the difficult problem of multiple sequence alignment.

Large deviations for the degree structure in preferential attachment schemes

Tue, 12 Feb 2013 10:14 EST

Jihyeok Choi, Sunder Sethuraman

Source: Ann. Appl. Probab., Volume 23, Number 2, 722--763.

Abstract:
Preferential attachment schemes, where the selection mechanism is linear and possibly time-dependent, are considered, and an infinite-dimensional large deviation principle for the sample path evolution of the empirical degree distribution is found by Dupuis–Ellis-type methods. Interestingly, the rate function, which can be evaluated, contains a term which accounts for the cost of assigning a fraction of the total degree to an “infinite” degree component, that is, when an atypical “condensation” effect occurs with respect to the degree structure. As a consequence of the large deviation results, a sample path a.s. law of large numbers for the degree distribution is deduced in terms of a coupled system of ODEs from which power law bounds for the limiting degree distribution are given. However, by analyzing the rate function, one can see that the process can deviate to a variety of atypical nonpower law distributions with finite cost, including distributions typically associated with sub and superlinear selection models.

Cone-constrained continuous-time Markowitz problems

Tue, 12 Feb 2013 10:14 EST

Christoph Czichowsky, Martin Schweizer

Source: Ann. Appl. Probab., Volume 23, Number 2, 764--810.

Abstract:
The Markowitz problem consists of finding, in a financial market, a self-financing trading strategy whose final wealth has maximal mean and minimal variance. We study this in continuous time in a general semimartingale model and under cone constraints: trading strategies must take values in a (possibly random and time-dependent) closed cone. We first prove existence of a solution for convex constraints by showing that the space of constrained terminal gains, which is a space of stochastic integrals, is closed in $L^{2}$. Then we use stochastic control methods to describe the local structure of the optimal strategy, as follows. The value process of a naturally associated constrained linear-quadratic optimal control problem is decomposed into a sum with two opportunity processes $L^{\pm}$ appearing as coefficients. The martingale optimality principle translates into a drift condition for the semimartingale characteristics of $L^{\pm}$ or equivalently into a coupled system of backward stochastic differential equations for $L^{\pm}$. We show how this can be used to both characterize and construct optimal strategies. Our results explain and generalize all the results available in the literature so far. Moreover, we even obtain new sharp results in the unconstrained case.

Convergence analysis of some multivariate Markov chains using stochastic monotonicity

Tue, 12 Feb 2013 10:14 EST

Kshitij Khare, Nabanita Mukherjee

Source: Ann. Appl. Probab., Volume 23, Number 2, 811--833.

Abstract:
We provide a nonasymptotic analysis of convergence to stationarity for a collection of Markov chains on multivariate state spaces, from arbitrary starting points, thereby generalizing results in [Khare and Zhou Ann. Appl. Probab. 19 (2009) 737–777]. Our examples include the multi-allele Moran model in population genetics and its variants in community ecology, a generalized Ehrenfest urn model and variants of the Pólya urn model. It is shown that all these Markov chains are stochastically monotone with respect to an appropriate partial ordering. Then, using a generalization of the results in [Diaconis, Khare and Saloff-Coste Sankhya 72 (2010) 45–76] and [Wilson Ann. Appl. Probab. 14 (2004) 274–325] (for univariate totally ordered spaces) to multivariate partially ordered spaces, we obtain explicit nonasymptotic bounds for the distance to stationarity from arbitrary starting points. In previous literature, bounds, if any, were available only from special starting points. The analysis also works for nonreversible Markov chains, and allows us to analyze cases of the multi-allele Moran model not considered in [Khare and Zhou Ann. Appl. Probab. 19 (2009) 737–777].

Error distributions for random grid approximations of multidimensional stochastic integrals

Tue, 12 Feb 2013 10:14 EST

Carl Lindberg, Holger Rootzén

Source: Ann. Appl. Probab., Volume 23, Number 2, 834--857.

Abstract:
This paper proves joint convergence of the approximation error for several stochastic integrals with respect to local Brownian semimartingales, for nonequidistant and random grids. The conditions needed for convergence are that the Lebesgue integrals of the integrands tend uniformly to zero and that the squared variation and covariation processes converge. The paper also provides tools which simplify checking these conditions and which extend the range for the results. These results are used to prove an explicit limit theorem for random grid approximations of integrals based on solutions of multidimensional SDEs, and to find ways to “design” and optimize the distribution of the approximation error. As examples we briefly discuss strategies for discrete option hedging.

Root’s barrier: Construction, optimality and applications to variance options

Thu, 07 Mar 2013 14:35 EST

Alexander M. G. Cox, Jiajie Wang

Source: Ann. Appl. Probab., Volume 23, Number 3, 859--894.

Abstract:
Recent work of Dupire and Carr and Lee has highlighted the importance of understanding the Skorokhod embedding originally proposed by Root for the model-independent hedging of variance options. Root’s work shows that there exists a barrier from which one may define a stopping time which solves the Skorokhod embedding problem. This construction has the remarkable property, proved by Rost, that it minimizes the variance of the stopping time among all solutions. In this work, we prove a characterization of Root’s barrier in terms of the solution to a variational inequality, and we give an alternative proof of the optimality property which has an important consequence for the construction of subhedging strategies in the financial context.

Three-dimensional Brownian motion and the golden ratio rule

Thu, 07 Mar 2013 14:35 EST

Kristoffer Glover, Hardy Hulley, Goran Peskir

Source: Ann. Appl. Probab., Volume 23, Number 3, 895--922.

Abstract:
Let $X=(X_{t})_{t\ge0}$ be a transient diffusion process in $(0,\infty)$ with the diffusion coefficient $\sigma>0$ and the scale function $L$ such that $X_{t}\rightarrow\infty$ as $t\rightarrow\infty$, let $I_{t}$ denote its running minimum for $t\ge0$, and let $\theta$ denote the time of its ultimate minimum $I_{\infty}$. Setting $c(i,x)=1-2L(x)/L(i)$ we show that the stopping time \[\tau_{*}=\inf\{t\ge0\vert X_{t}\ge f_{*}(I_{t})\}\] minimizes $\mathsf{E}(\vert\theta-\tau\vert-\theta)$ over all stopping times $\tau$ of $X$ (with finite mean) where the optimal boundary $f_{*}$ can be characterized as the minimal solution to \[f'(i)=-\frac{\sigma^{2}(f(i))L'(f(i))}{c(i,f(i))[L(f(i))-L(i)]}\int_{i}^{f(i)}\frac{c_{i}'(i,y)[L(y)-L(i)]}{\sigma^{2}(y)L'(y)}\,dy\] staying strictly above the curve $h(i)=L^{-1}(L(i)/2)$ for $i>0$. In particular, when $X$ is the radial part of three-dimensional Brownian motion, we find that \[\tau_{*}=\inf\biggl\{t\ge0\Big\vert\frac{X_{t}-I_{t}}{I_{t}}\ge\varphi\biggr\},\] where $\varphi=(1+\sqrt{5})/2=1.61\ldots$ is the golden ratio. The derived results are applied to problems of optimal trading in the presence of bubbles where we show that the golden ratio rule offers a rigorous optimality argument for the choice of the well-known golden retracement in technical analysis of asset prices.

Random interlacements and amenability

Thu, 07 Mar 2013 14:35 EST

Augusto Teixeira, Johan Tykesson

Source: Ann. Appl. Probab., Volume 23, Number 3, 923--956.

Abstract:
We consider the model of random interlacements on transient graphs, which was first introduced by Sznitman [ Ann. of Math. (2) (2010) 171 2039–2087] for the special case of ${\mathbb{Z}}^{d}$ (with $d\geq3$). In Sznitman [ Ann. of Math. (2) (2010) 171 2039–2087], it was shown that on ${\mathbb{Z}}^{d}$: for any intensity $u>0$, the interlacement set is almost surely connected. The main result of this paper says that for transient, transitive graphs, the above property holds if and only if the graph is amenable. In particular, we show that in nonamenable transitive graphs, for small values of the intensity $u$ the interlacement set has infinitely many infinite clusters. We also provide examples of nonamenable transitive graphs, for which the interlacement set becomes connected for large values of $u$. Finally, we establish the monotonicity of the transition between the “disconnected” and the “connected” phases, providing the uniqueness of the critical value $u_{c}$ where this transition occurs.

Averaging over fast variables in the fluid limit for Markov chains: Application to the supermarket model with memory

Thu, 07 Mar 2013 14:35 EST

M. J. Luczak, J. R. Norris

Source: Ann. Appl. Probab., Volume 23, Number 3, 957--986.

Abstract:
We set out a general procedure which allows the approximation of certain Markov chains by the solutions of differential equations. The chains considered have some components which oscillate rapidly and randomly, while others are close to deterministic. The limiting dynamics are obtained by averaging the drift of the latter with respect to a local equilibrium distribution of the former. Some general estimates are proved under a uniform mixing condition on the fast variable which give explicit error probabilities for the fluid approximation. Mitzenmacher, Prabhakar and Shah [In Proc. 43rd Ann. Symp. Found. Comp. Sci. (2002) 799–808, IEEE] introduced a variant with memory of the “join the shortest queue” or “supermarket” model, and obtained a limit picture for the case of a stable system in which the number of queues and the total arrival rate are large. In this limit, the empirical distribution of queue sizes satisfies a differential equation, while the memory of the system oscillates rapidly and randomly. We illustrate our general fluid limit estimate by giving a proof of this limit picture.

Random permutation matrices under the generalized Ewens measure

Thu, 07 Mar 2013 14:35 EST

Christopher Hughes, Joseph Najnudel, Ashkan Nikeghbali, Dirk Zeindler

Source: Ann. Appl. Probab., Volume 23, Number 3, 987--1024.

Abstract:
We consider a generalization of the Ewens measure for the symmetric group, calculating moments of the characteristic polynomial and similar multiplicative statistics. In addition, we study the asymptotic behavior of linear statistics (such as the trace of a permutation matrix or of a wreath product) under this new measure.

Exact and high-order discretization schemes for Wishart processes and their affine extensions

Thu, 07 Mar 2013 14:35 EST

Abdelkoddousse Ahdida, Aurélien Alfonsi

Source: Ann. Appl. Probab., Volume 23, Number 3, 1025--1073.

Abstract:
This work deals with the simulation of Wishart processes and affine diffusions on positive semidefinite matrices. To do so, we focus on the splitting of the infinitesimal generator in order to use composition techniques as did Ninomiya and Victoir [ Appl. Math. Finance 15 (2008) 107–121] or Alfonsi [ Math. Comp. 79 (2010) 209–237]. Doing so, we have found a remarkable splitting for Wishart processes that enables us to sample exactly Wishart distributions without any restriction on the parameters. It is related but extends existing exact simulation methods based on Bartlett’s decomposition. Moreover, we can construct high-order discretization schemes for Wishart processes and second-order schemes for general affine diffusions. These schemes are, in practice, faster than the exact simulation to sample entire paths. Numerical results on their convergence are given.

Examples of nonpolygonal limit shapes in i.i.d. first-passage percolation and infinite coexistence in spatial growth models

Thu, 07 Mar 2013 14:35 EST

Michael Damron, Michael Hochman

Source: Ann. Appl. Probab., Volume 23, Number 3, 1074--1085.

Abstract:
We construct an edge-weight distribution for i.i.d. first-passage percolation on $\mathbb{Z}^{2}$ whose limit shape is not a polygon and whose extreme points are arbitrarily dense in the boundary. Consequently, the associated Richardson-type growth model can support coexistence of a countably infinite number of distinct species, and the graph of infection has infinitely many ends.

Singular forward–backward stochastic differential equations and emissions derivatives

Thu, 07 Mar 2013 14:35 EST

René Carmona, François Delarue, Gilles-Edouard Espinosa, Nizar Touzi

Source: Ann. Appl. Probab., Volume 23, Number 3, 1086--1128.

Abstract:
We introduce two simple models of forward–backward stochastic differential equations with a singular terminal condition and we explain how and why they appear naturally as models for the valuation of CO${}_{2}$ emission allowances. Single phase cap-and-trade schemes lead readily to terminal conditions given by indicator functions of the forward component, and using fine partial differential equations estimates, we show that the existence theory of these equations, as well as the properties of the candidates for solution, depend strongly upon the characteristics of the forward dynamics. Finally, we give a first order Taylor expansion and show how to numerically calibrate some of these models for the purpose of CO${}_{2}$ option pricing.

Distributional convergence for the number of symbol comparisons used by QuickSort

Thu, 07 Mar 2013 14:35 EST

James Allen Fill

Source: Ann. Appl. Probab., Volume 23, Number 3, 1129--1147.

Abstract:
Most previous studies of the sorting algorithm QuickSort have used the number of key comparisons as a measure of the cost of executing the algorithm. Here we suppose that the $n$ independent and identically distributed (i.i.d.) keys are each represented as a sequence of symbols from a probabilistic source and that QuickSort operates on individual symbols, and we measure the execution cost as the number of symbol comparisons. Assuming only a mild “tameness” condition on the source, we show that there is a limiting distribution for the number of symbol comparisons after normalization: first centering by the mean and then dividing by $n$. Additionally, under a condition that grows more restrictive as $p$ increases, we have convergence of moments of orders $p$ and smaller. In particular, we have convergence in distribution and convergence of moments of every order whenever the source is memoryless, that is, whenever each key is generated as an infinite string of i.i.d. symbols. This is somewhat surprising; even for the classical model that each key is an i.i.d. string of unbiased (“fair”) bits, the mean exhibits periodic fluctuations of order $n$.

Quenched limits for the fluctuations of transient random walks in random environment on $\mathbb{Z}$

Thu, 07 Mar 2013 14:35 EST

Nathanaël Enriquez, Christophe Sabot, Laurent Tournier, Olivier Zindy

Source: Ann. Appl. Probab., Volume 23, Number 3, 1148--1187.

Abstract:
We consider transient nearest-neighbor random walks in random environment on $\mathbb{Z}$. For a set of environments whose probability is converging to 1 as time goes to infinity, we describe the fluctuations of the hitting time of a level $n$, around its mean, in terms of an explicit function of the environment. Moreover, their limiting law is described using a Poisson point process whose intensity is computed. This result can be considered as the quenched analog of the classical result of Kesten, Kozlov and Spitzer [ Compositio Math . 30 (1975) 145–168].

Degree asymptotics with rates for preferential attachment random graphs

Thu, 07 Mar 2013 14:35 EST

Erol A. Peköz, Adrian Röllin, Nathan Ross

Source: Ann. Appl. Probab., Volume 23, Number 3, 1188--1218.

Abstract:
We provide optimal rates of convergence to the asymptotic distribution of the (properly scaled) degree of a fixed vertex in two preferential attachment random graph models. Our approach is to show that these distributions are unique fixed points of certain distributional transformations which allows us to obtain rates of convergence using a new variation of Stein’s method. Despite the large literature on these models, there is surprisingly little known about the limiting distributions so we also provide some properties and new representations, including an explicit expression for the densities in terms of the confluent hypergeometric function of the second kind.

On a preferential attachment and generalized Pólya’s urn model

Thu, 07 Mar 2013 14:35 EST

Andrea Collevecchio, Codina Cotar, Marco LiCalzi

Source: Ann. Appl. Probab., Volume 23, Number 3, 1219--1253.

Abstract:
We study a general preferential attachment and Pólya’s urn model. At each step a new vertex is introduced, which can be connected to at most one existing vertex. If it is disconnected, it becomes a pioneer vertex. Given that it is not disconnected, it joins an existing pioneer vertex with probability proportional to a function of the degree of that vertex. This function is allowed to be vertex-dependent, and is called the reinforcement function. We prove that there can be at most three phases in this model, depending on the behavior of the reinforcement function. Consider the set whose elements are the vertices with cardinality tending a.s. to infinity. We prove that this set either is empty, or it has exactly one element, or it contains all the pioneer vertices. Moreover, we describe the phase transition in the case where the reinforcement function is the same for all vertices. Our results are general, and in particular we are not assuming monotonicity of the reinforcement function. Finally, consider the regime where exactly one vertex has a degree diverging to infinity. We give a lower bound for the probability that a given vertex ends up being the leading one, that is, its degree diverges to infinity. Our proofs rely on a generalization of the Rubin construction given for edge-reinforced random walks, and on a Brownian motion embedding.

Degree and clustering coefficient in sparse random intersection graphs

Thu, 07 Mar 2013 14:35 EST

Mindaugas Bloznelis

Source: Ann. Appl. Probab., Volume 23, Number 3, 1254--1289.

Abstract:
We establish asymptotic vertex degree distribution and examine its relation to the clustering coefficient in two popular random intersection graph models of Godehardt and Jaworski [ Electron. Notes Discrete Math. 10 (2001) 129–132]. For sparse graphs with a positive clustering coefficient, we examine statistical dependence between the (local) clustering coefficient and the degree. Our results are mathematically rigorous. They are consistent with the empirical observation of Foudalis et al. [In Algorithms and Models for Web Graph (2011) Springer] that, “clustering correlates negatively with degree.” Moreover, they explain empirical results on $k^{-1}$ scaling of the local clustering coefficient of a vertex of degree $k$ reported in Ravasz and Barabási [ Phys. Rev. E 67 (2003) 026112].