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On optimal arbitrage
http://projecteuclid.org/euclid.aoap/1279638783
<strong>Daniel Fernholz</strong>, <strong>Ioannis Karatzas</strong><p><strong>Source: </strong>Ann. Appl. Probab., Volume 20, Number 4, 1179--1204.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.aoap/1279638783_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTChange-point detection for Lévy processeshttps://projecteuclid.org/euclid.aoap/1548298928<strong>José E. Figueroa-López</strong>, <strong>Sveinn Ólafsson</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 2, 717--738.</p><p><strong>Abstract:</strong><br/>
Since the work of Page in the 1950s, the problem of detecting an abrupt change in the distribution of stochastic processes has received a great deal of attention. In particular, a deep connection has been established between Lorden’s minimax approach to change-point detection and the widely used CUSUM procedure, first for discrete-time processes, and subsequently for some of their continuous-time counterparts. However, results for processes with jumps are still scarce, while the practical importance of such processes has escalated since the turn of the century. In this work, we consider the problem of detecting a change in the distribution of continuous-time processes with independent and stationary increments, that is, Lévy processes, and our main result shows that CUSUM is indeed optimal in Lorden’s sense. This is the most natural continuous-time analogue of the seminal work of Moustakides [ Ann. Statist. 14 (1986) 1379–1387] for sequentially observed random variables that are assumed to be i.i.d. before and after the change-point. From a practical perspective, the approach we adopt is appealing as it consists in approximating the continuous-time problem by a suitable sequence of change-point problems with equispaced sampling points, and for which a CUSUM procedure is shown to be optimal.
</p>projecteuclid.org/euclid.aoap/1548298928_20190123220226Wed, 23 Jan 2019 22:02 ESTSuper-replication with fixed transaction costshttps://projecteuclid.org/euclid.aoap/1548298929<strong>Peter Bank</strong>, <strong>Yan Dolinsky</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 2, 739--757.</p><p><strong>Abstract:</strong><br/>
We study super-replication of contingent claims in markets with fixed transaction costs. This can be viewed as a stochastic impulse control problem with a terminal state constraint. The first result in this paper reveals that in reasonable continuous time financial market models the super-replication price is prohibitively costly and leads to trivial buy-and-hold strategies. Our second result derives nontrivial scaling limits of super-replication prices for binomial models with small fixed costs.
</p>projecteuclid.org/euclid.aoap/1548298929_20190123220226Wed, 23 Jan 2019 22:02 ESTFirst-order Euler scheme for SDEs driven by fractional Brownian motions: The rough casehttps://projecteuclid.org/euclid.aoap/1548298930<strong>Yanghui Liu</strong>, <strong>Samy Tindel</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 2, 758--826.</p><p><strong>Abstract:</strong><br/>
In this article, we consider the so-called modified Euler scheme for stochastic differential equations (SDEs) driven by fractional Brownian motions (fBm) with Hurst parameter $\frac{1}{3}<H<\frac{1}{2}$. This is a first-order time-discrete numerical approximation scheme, and has been introduced in [ Ann. Appl. Probab. 26 (2016) 1147–1207] recently in order to generalize the classical Euler scheme for Itô SDEs to the case $H>\frac{1}{2}$. The current contribution generalizes the modified Euler scheme to the rough case $\frac{1}{3}<H<\frac{1}{2}$. Namely, we show a convergence rate of order $n^{\frac{1}{2}-2H}$ for the scheme, and we argue that this rate is exact. We also derive a central limit theorem for the renormalized error of the scheme, thanks to some new techniques for asymptotics of weighted random sums. Our main idea is based on the following observation: the triple of processes obtained by considering the fBm, the scheme process and the normalized error process, can be lifted to a new rough path. In addition, the Hölder norm of this new rough path has an estimate which is independent of the step-size of the scheme.
</p>projecteuclid.org/euclid.aoap/1548298930_20190123220226Wed, 23 Jan 2019 22:02 ESTMalliavin calculus approach to long exit times from an unstable equilibriumhttps://projecteuclid.org/euclid.aoap/1548298931<strong>Yuri Bakhtin</strong>, <strong>Zsolt Pajor-Gyulai</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 2, 827--850.</p><p><strong>Abstract:</strong><br/>
For a one-dimensional smooth vector field in a neighborhood of an unstable equilibrium, we consider the associated dynamics perturbed by small noise. Using Malliavin calculus tools, we obtain precise vanishing noise asymptotics for the tail of the exit time and for the exit distribution conditioned on atypically long exits.
</p>projecteuclid.org/euclid.aoap/1548298931_20190123220226Wed, 23 Jan 2019 22:02 ESTOptimal mean-based algorithms for trace reconstructionhttps://projecteuclid.org/euclid.aoap/1548298932<strong>Anindya De</strong>, <strong>Ryan O’Donnell</strong>, <strong>Rocco A. Servedio</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 2, 851--874.</p><p><strong>Abstract:</strong><br/>
In the (deletion-channel) trace reconstruction problem , there is an unknown $n$-bit source string $x$. An algorithm is given access to independent traces of $x$, where a trace is formed by deleting each bit of $x$ independently with probability $\delta$. The goal of the algorithm is to recover $x$ exactly (with high probability), while minimizing samples (number of traces) and running time.
Previously, the best known algorithm for the trace reconstruction problem was due to Holenstein et al. [in Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms 389–398 (2008) ACM]; it uses $\exp(\widetilde{O}(n^{1/2}))$ samples and running time for any fixed $0<\delta<1$. It is also what we call a “mean-based algorithm,” meaning that it only uses the empirical means of the individual bits of the traces. Holenstein et al. also gave a lower bound, showing that any mean-based algorithm must use at least $n^{\widetilde{\Omega}(\log n)}$ samples.
In this paper, we improve both of these results, obtaining matching upper and lower bounds for mean-based trace reconstruction. For any constant deletion rate $0<\delta<1$, we give a mean-based algorithm that uses $\exp(O(n^{1/3}))$ time and traces; we also prove that any mean-based algorithm must use at least $\exp(\Omega(n^{1/3}))$ traces. In fact, we obtain matching upper and lower bounds even for $\delta$ subconstant and $\rho\:=1-\delta$ subconstant: when $(\log^{3}n)/n\ll\delta\leq1/2$ the bound is $\exp(-\Theta(\delta n)^{1/3})$, and when $1/\sqrt{n}\ll\rho\leq1/2$ the bound is $\exp(-\Theta(n/\rho)^{1/3})$.
Our proofs involve estimates for the maxima of Littlewood polynomials on complex disks. We show that these techniques can also be used to perform trace reconstruction with random insertions and bit-flips in addition to deletions. We also find a surprising result: for deletion probabilities $\delta>1/2$, the presence of insertions can actually help with trace reconstruction.
</p>projecteuclid.org/euclid.aoap/1548298932_20190123220226Wed, 23 Jan 2019 22:02 ESTA shape theorem for the scaling limit of the IPDSAW at criticalityhttps://projecteuclid.org/euclid.aoap/1548298933<strong>Philippe Carmona</strong>, <strong>Nicolas Pétrélis</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 2, 875--930.</p><p><strong>Abstract:</strong><br/>
In this paper we give a complete characterization of the scaling limit of the critical Interacting Partially Directed Self-Avoiding Walk (IPDSAW) introduced in Zwanzig and Lauritzen [ J. Chem. Phys. 48 (1968) 3351]. As the system size $L\in\mathbb{N}$ diverges, we prove that the set of occupied sites, rescaled horizontally by $L^{2/3}$ and vertically by $L^{1/3}$ converges in law for the Hausdorff distance toward a nontrivial random set. This limiting set is built with a Brownian motion $B$ conditioned to come back at the origin at $a_{1}$ the time at which its geometric area reaches $1$. The modulus of $B$ up to $a_{1}$ gives the height of the limiting set, while its center of mass process is an independent Brownian motion.
Obtaining the shape theorem requires to derive a functional central limit theorem for the excursion of a random walk with Laplace symmetric increments conditioned on sweeping a prescribed geometric area. This result is proven in a companion paper Carmona and Pétrélis (2017).
</p>projecteuclid.org/euclid.aoap/1548298933_20190123220226Wed, 23 Jan 2019 22:02 ESTNormal approximation for stabilizing functionalshttps://projecteuclid.org/euclid.aoap/1548298934<strong>Raphaël Lachièze-Rey</strong>, <strong>Matthias Schulte</strong>, <strong>J. E. Yukich</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 2, 931--993.</p><p><strong>Abstract:</strong><br/>
We establish presumably optimal rates of normal convergence with respect to the Kolmogorov distance for a large class of geometric functionals of marked Poisson and binomial point processes on general metric spaces. The rates are valid whenever the geometric functional is expressible as a sum of exponentially stabilizing score functions satisfying a moment condition. By incorporating stabilization methods into the Malliavin–Stein theory, we obtain rates of normal approximation for sums of stabilizing score functions which either improve upon existing rates or are the first of their kind.
Our general rates hold for functionals of marked input on spaces more general than full-dimensional subsets of $\mathbb{R}^{d}$, including $m$-dimensional Riemannian manifolds, $m\leq d$. We use the general results to deduce improved and new rates of normal convergence for several functionals in stochastic geometry, including those whose variances re-scale as the volume or the surface area of an underlying set. In particular, we improve upon rates of normal convergence for the $k$-face and $i$th intrinsic volume functionals of the convex hull of Poisson and binomial random samples in a smooth convex body in dimension $d\geq 2$. We also provide improved rates of normal convergence for statistics of nearest neighbors graphs and high-dimensional data sets, the number of maximal points in a random sample, estimators of surface area and volume arising in set approximation via Voronoi tessellations, and clique counts in generalized random geometric graphs.
</p>projecteuclid.org/euclid.aoap/1548298934_20190123220226Wed, 23 Jan 2019 22:02 ESTErgodicity of an SPDE associated with a many-server queuehttps://projecteuclid.org/euclid.aoap/1548298935<strong>Reza Aghajani</strong>, <strong>Kavita Ramanan</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 2, 994--1045.</p><p><strong>Abstract:</strong><br/>
We consider the so-called GI/GI/N queueing network in which a stream of jobs with independent and identically distributed service times arrive according to a renewal process to a common queue served by $N$ identical servers in a first-come-first-serve manner. We introduce a two-component infinite-dimensional Markov process that serves as a diffusion model for this network, in the regime where the number of servers goes to infinity and the load on the network scales as $1-\beta N^{-1/2}+o(N^{-1/2})$ for some $\beta>0$. Under suitable assumptions, we characterize this process as the unique solution to a pair of stochastic evolution equations comprised of a real-valued Itô equation and a stochastic partial differential equation on the positive half line, which are coupled together by a nonlinear boundary condition. We construct an asymptotic (equivalent) coupling to show that this Markov process has a unique invariant distribution. This invariant distribution is shown in a companion paper [Aghajani and Ramanan (2016)] to be the limit of the sequence of suitably scaled and centered stationary distributions of the GI/GI/N network, thus resolving (for a large class service distributions) an open problem raised by Halfin and Whitt in [ Oper. Res. 29 (1981) 567–588]. The methods introduced here are more generally applicable for the analysis of a broader class of networks.
</p>projecteuclid.org/euclid.aoap/1548298935_20190123220226Wed, 23 Jan 2019 22:02 ESTCentral limit theorems in the configuration modelhttps://projecteuclid.org/euclid.aoap/1548298936<strong>A. D. Barbour</strong>, <strong>Adrian Röllin</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 2, 1046--1069.</p><p><strong>Abstract:</strong><br/>
We prove a general normal approximation theorem for local graph statistics in the configuration model, together with an explicit bound on the error in the approximation with respect to the Wasserstein metric. Such statistics take the form $T:=\sum_{v\in V}H_{v}$, where $V$ is the vertex set, and $H_{v}$ depends on a neighbourhood in the graph around $v$ of size at most $\ell$. The error bound is expressed in terms of $\ell$, $|V|$, an almost sure bound on $H_{v}$, the maximum vertex degree $d_{\max}$ and the variance of $T$. Under suitable assumptions on the convergence of the empirical degree distributions to a limiting distribution, we deduce that the size of the giant component in the configuration model has asymptotically Gaussian fluctuations.
</p>projecteuclid.org/euclid.aoap/1548298936_20190123220226Wed, 23 Jan 2019 22:02 ESTErgodicity of a Lévy-driven SDE arising from multiclass many-server queueshttps://projecteuclid.org/euclid.aoap/1548298937<strong>Ari Arapostathis</strong>, <strong>Guodong Pang</strong>, <strong>Nikola Sandrić</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 2, 1070--1126.</p><p><strong>Abstract:</strong><br/>
We study the ergodic properties of a class of multidimensional piecewise Ornstein–Uhlenbeck processes with jumps, which contains the limit of the queueing processes arising in multiclass many-server queues with heavy-tailed arrivals and/or asymptotically negligible service interruptions in the Halfin–Whitt regime as special cases. In these queueing models, the Itô equations have a piecewise linear drift, and are driven by either (1) a Brownian motion and a pure-jump Lévy process, or (2) an anisotropic Lévy process with independent one-dimensional symmetric $\alpha $-stable components or (3) an anisotropic Lévy process as in (2) and a pure-jump Lévy process. We also study the class of models driven by a subordinate Brownian motion, which contains an isotropic (or rotationally invariant) $\alpha $-stable Lévy process as a special case. We identify conditions on the parameters in the drift, the Lévy measure and/or covariance function which result in subexponential and/or exponential ergodicity. We show that these assumptions are sharp, and we identify some key necessary conditions for the process to be ergodic. In addition, we show that for the queueing models described above with no abandonment, the rate of convergence is polynomial, and we provide a sharp quantitative characterization of the rate via matching upper and lower bounds.
</p>projecteuclid.org/euclid.aoap/1548298937_20190123220226Wed, 23 Jan 2019 22:02 ESTOn one-dimensional Riccati diffusionshttps://projecteuclid.org/euclid.aoap/1548298938<strong>A. N. Bishop</strong>, <strong>P. Del Moral</strong>, <strong>K. Kamatani</strong>, <strong>B. Rémillard</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 2, 1127--1187.</p><p><strong>Abstract:</strong><br/>
This article is concerned with the fluctuation analysis and the stability properties of a class of one-dimensional Riccati diffusions. These one-dimensional stochastic differential equations exhibit a quadratic drift function and a non-Lipschitz continuous diffusion function. We present a novel approach, combining tangent process techniques, Feynman–Kac path integration and exponential change of measures, to derive sharp exponential decays to equilibrium. We also provide uniform estimates with respect to the time horizon, quantifying with some precision the fluctuations of these diffusions around a limiting deterministic Riccati differential equation. These results provide a stronger and almost sure version of the conventional central limit theorem. We illustrate these results in the context of ensemble Kalman–Bucy filtering. To the best of our knowledge, the exponential stability and the fluctuation analysis developed in this work are the first results of this kind for this class of nonlinear diffusions.
</p>projecteuclid.org/euclid.aoap/1548298938_20190123220226Wed, 23 Jan 2019 22:02 ESTOn Poisson approximations for the Ewens sampling formula when the mutation parameter grows with the sample sizehttps://projecteuclid.org/euclid.aoap/1548298939<strong>Koji Tsukuda</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 2, 1188--1232.</p><p><strong>Abstract:</strong><br/>
The Ewens sampling formula was first introduced in the context of population genetics by Warren John Ewens in 1972, and has appeared in a lot of other scientific fields. There are abundant approximation results associated with the Ewens sampling formula especially when one of the parameters, the sample size $n$ or the mutation parameter $\theta$ which denotes the scaled mutation rate, tends to infinity while the other is fixed. By contrast, the case that $\theta$ grows with $n$ has been considered in a relatively small number of works, although this asymptotic setup is also natural. In this paper, when $\theta$ grows with $n$, we advance the study concerning the asymptotic properties of the total number of alleles and of the component counts in the allelic partition assuming the Ewens sampling formula, from the viewpoint of Poisson approximations. Specifically, the main contributions of this paper are deriving Poisson approximations of the total number of alleles, an independent process approximation of small component counts, and functional central limit theorems, under the asymptotic regime that both $n$ and $\theta$ tend to infinity.
</p>projecteuclid.org/euclid.aoap/1548298939_20190123220226Wed, 23 Jan 2019 22:02 ESTThe critical greedy server on the integers is recurrenthttps://projecteuclid.org/euclid.aoap/1548298940<strong>James R. Cruise</strong>, <strong>Andrew R. Wade</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 2, 1233--1261.</p><p><strong>Abstract:</strong><br/>
Each site of $\mathbb{Z}$ hosts a queue with arrival rate $\lambda $. A single server, starting at the origin, serves its current queue at rate $\mu $ until that queue is empty, and then moves to the longest neighbouring queue. In the critical case $\lambda =\mu $, we show that the server returns to every site infinitely often. We also give a sharp iterated logarithm result for the server’s position. Important ingredients in the proofs are that the times between successive queues being emptied exhibit doubly exponential growth, and that the probability that the server changes its direction is asymptotically equal to $1/4$.
</p>projecteuclid.org/euclid.aoap/1548298940_20190123220226Wed, 23 Jan 2019 22:02 ESTJoin-the-shortest queue diffusion limit in Halfin–Whitt regime: Tail asymptotics and scaling of extremahttps://projecteuclid.org/euclid.aoap/1548298941<strong>Sayan Banerjee</strong>, <strong>Debankur Mukherjee</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 2, 1262--1309.</p><p><strong>Abstract:</strong><br/>
Consider a system of $N$ parallel single-server queues with unit-ex ponential service time distribution and a single dispatcher where tasks arrive as a Poisson process of rate $\lambda(N)$. When a task arrives, the dispatcher assigns it to one of the servers according to the Join-the-Shortest Queue (JSQ) policy. Eschenfeldt and Gamarnik ( Math. Oper. Res. 43 (2018) 867–886) established that in the Halfin–Whitt regime where $(N-\lambda(N))/\sqrt{N}\to\beta>0$ as $N\to\infty$, appropriately scaled occupancy measure of the system under the JSQ policy converges weakly on any finite time interval to a certain diffusion process as $N\to\infty$. Recently, it was further established by Braverman (2018) that the convergence result extends to the steady state as well, that is, stationary occupancy measure of the system converges weakly to the steady state of the diffusion process as $N\to\infty$, proving the interchange of limits result.
In this paper, we perform a detailed analysis of the steady state of the above diffusion process. Specifically, we establish precise tail-asymptotics of the stationary distribution and scaling of extrema of the process on large time interval. Our results imply that the asymptotic steady-state scaled number of servers with queue length two or larger exhibits an exponential tail, whereas that for the number of idle servers turns out to be Gaussian. From the methodological point of view, the diffusion process under consideration goes beyond the state-of-the-art techniques in the study of the steady state of diffusion processes. Lack of any closed-form expression for the steady state and intricate interdependency of the process dynamics on its local times make the analysis significantly challenging. We develop a technique involving the theory of regenerative processes that provides a tractable form for the stationary measure, and in conjunction with several sharp hitting time estimates, acts as a key vehicle in establishing the results. The technique and the intermediate results might be of independent interest, and can possibly be used in understanding the bulk behavior of the process.
</p>projecteuclid.org/euclid.aoap/1548298941_20190123220226Wed, 23 Jan 2019 22:02 ESTThe length of the longest common subsequence of two independent mallows permutationshttps://projecteuclid.org/euclid.aoap/1550566832<strong>Ke Jin</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 3, 1311--1355.</p><p><strong>Abstract:</strong><br/>
The Mallows measure is a probability measure on $S_{n}$ where the probability of a permutation $\pi$ is proportional to $q^{l(\pi)}$ with $q>0$ being a parameter and $l(\pi)$ the number of inversions in $\pi$. We prove a weak law of large numbers for the length of the longest common subsequences of two independent permutations drawn from the Mallows measure, when $q$ is a function of $n$ and $n(1-q)$ has limit in $\mathbb{R}$ as $n\to\infty$.
</p>projecteuclid.org/euclid.aoap/1550566832_20190219040044Tue, 19 Feb 2019 04:00 ESTDeterminant of sample correlation matrix with applicationhttps://projecteuclid.org/euclid.aoap/1550566833<strong>Tiefeng Jiang</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 3, 1356--1397.</p><p><strong>Abstract:</strong><br/>
Let $\mathbf{x}_{1},\ldots ,\mathbf{x}_{n}$ be independent random vectors of a common $p$-dimensional normal distribution with population correlation matrix $\mathbf{R}_{n}$. The sample correlation matrix $\hat{\mathbf {R}}_{n}=(\hat{r}_{ij})_{p\times p}$ is generated from $\mathbf{x}_{1},\ldots ,\mathbf{x}_{n}$ such that $\hat{r}_{ij}$ is the Pearson correlation coefficient between the $i$th column and the $j$th column of the data matrix $(\mathbf{x}_{1},\ldots ,\mathbf{x}_{n})'$. The matrix $\hat{\mathbf {R}}_{n}$ is a popular object in multivariate analysis and it has many connections to other problems. We derive a central limit theorem (CLT) for the logarithm of the determinant of $\hat{\mathbf {R}}_{n}$ for a big class of $\mathbf{R}_{n}$. The expressions of mean and the variance in the CLT are not obvious, and they are not known before. In particular, the CLT holds if $p/n$ has a nonzero limit and the smallest eigenvalue of $\mathbf{R}_{n}$ is larger than $1/2$. Besides, a formula of the moments of $\vert \hat{\mathbf {R}}_{n}\vert $ and a new method of showing weak convergence are introduced. We apply the CLT to a high-dimensional statistical test.
</p>projecteuclid.org/euclid.aoap/1550566833_20190219040044Tue, 19 Feb 2019 04:00 ESTAnnealed limit theorems for the Ising model on random regular graphshttps://projecteuclid.org/euclid.aoap/1550566834<strong>Van Hao Can</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 3, 1398--1445.</p><p><strong>Abstract:</strong><br/>
In a recent paper, Giardinà et al. [ ALEA Lat. Am. J. Probab. Math. Stat. 13 (2016) 121–161] have proved a law of large number and a central limit theorem with respect to the annealed measure for the magnetization of the Ising model on some random graphs, including the random 2-regular graph. In this paper, we present a new proof of their results which applies to all random regular graphs. In addition, we prove the existence of annealed pressure in the case of configuration model random graphs.
</p>projecteuclid.org/euclid.aoap/1550566834_20190219040044Tue, 19 Feb 2019 04:00 ESTApproximating geodesics via random pointshttps://projecteuclid.org/euclid.aoap/1550566835<strong>Erik Davis</strong>, <strong>Sunder Sethuraman</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 3, 1446--1486.</p><p><strong>Abstract:</strong><br/>
Given a cost functional $F$ on paths $\gamma$ in a domain $D\subset\mathbb{R}^{d}$, in the form $F(\gamma)=\int_{0}^{1}f(\gamma(t),\dot{\gamma}(t))\,dt$, it is of interest to approximate its minimum cost and geodesic paths. Let $X_{1},\ldots,X_{n}$ be points drawn independently from $D$ according to a distribution with a density. Form a random geometric graph on the points where $X_{i}$ and $X_{j}$ are connected when $0<|X_{i}-X_{j}|<\varepsilon$, and the length scale $\varepsilon=\varepsilon_{n}$ vanishes at a suitable rate.
For a general class of functionals $F$, associated to Finsler and other distances on $D$, using a probabilistic form of Gamma convergence, we show that the minimum costs and geodesic paths, with respect to types of approximating discrete cost functionals, built from the random geometric graph, converge almost surely in various senses to those corresponding to the continuum cost $F$, as the number of sample points diverges. In particular, the geodesic path convergence shown appears to be among the first results of its kind.
</p>projecteuclid.org/euclid.aoap/1550566835_20190219040044Tue, 19 Feb 2019 04:00 ESTFreidlin–Wentzell LDP in path space for McKean–Vlasov equations and the functional iterated logarithm lawhttps://projecteuclid.org/euclid.aoap/1550566836<strong>Gonçalo dos Reis</strong>, <strong>William Salkeld</strong>, <strong>Julian Tugaut</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 3, 1487--1540.</p><p><strong>Abstract:</strong><br/>
We show two Freidlin–Wentzell-type Large Deviations Principles (LDP) in path space topologies (uniform and Hölder) for the solution process of McKean–Vlasov Stochastic Differential Equations (MV-SDEs) using techniques which directly address the presence of the law in the coefficients and altogether avoiding decoupling arguments or limits of particle systems. We provide existence and uniqueness results along with several properties for a class of MV-SDEs having random coefficients and drifts of superlinear growth.
As an application of our results, we establish a functional Strassen-type result (law of iterated logarithm) for the solution process of a MV-SDE.
</p>projecteuclid.org/euclid.aoap/1550566836_20190219040044Tue, 19 Feb 2019 04:00 ESTA constrained Langevin approximation for chemical reaction networkshttps://projecteuclid.org/euclid.aoap/1550566837<strong>Saul C. Leite</strong>, <strong>Ruth J. Williams</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 3, 1541--1608.</p><p><strong>Abstract:</strong><br/>
Continuous-time Markov chain models are often used to describe the stochastic dynamics of networks of reacting chemical species, especially in the growing field of systems biology. These Markov chain models are often studied by simulating sample paths in order to generate Monte-Carlo estimates. However, discrete-event stochastic simulation of these models rapidly becomes computationally intensive. Consequently, more tractable diffusion approximations are commonly used in numerical computation, even for modest-sized networks. However, existing approximations either do not respect the constraint that chemical concentrations are never negative (linear noise approximation) or are typically only valid until the concentration of some chemical species first becomes zero (Langevin approximation).
In this paper, we propose an approximation for such Markov chains via reflected diffusion processes that respect the fact that concentrations of chemical species are never negative. We call this a constrained Langevin approximation because it behaves like the Langevin approximation in the interior of the positive orthant, to which it is constrained by instantaneous reflection at the boundary of the orthant. An additional advantage of our approximation is that it can be written down immediately from the chemical reactions. This contrasts with the linear noise approximation, which involves a two-stage procedure—first solve a deterministic reaction rate ordinary differential equation, followed by a stochastic differential equation for fluctuations around those solutions. Our approximation also captures the interaction of nonlinearities in the reaction rate function with the driving noise. In simulations, we have found the computation time for our approximation to be at least comparable to, and often better than, that for the linear noise approximation.
Under mild assumptions, we first prove that our proposed approximation is well defined for all time. Then we prove that it can be obtained as the weak limit of a sequence of jump-diffusion processes that behave like the Langevin approximation in the interior of the positive orthant and like a rescaled version of the Markov chain on the boundary of the orthant. For this limit theorem, we adapt an invariance principle for reflected diffusions, due to Kang and Williams [ Ann. Appl. Probab. 17 (2007) 741–779], and modify a result on pathwise uniqueness for reflected diffusions due to Dupuis and Ishii [ Ann. Probab. 21 (1993) 554–580]. Some numerical examples illustrate the advantages of our approximation over direct simulation of the Markov chain or use of the linear noise approximation.
</p>projecteuclid.org/euclid.aoap/1550566837_20190219040044Tue, 19 Feb 2019 04:00 ESTOn a Wasserstein-type distance between solutions to stochastic differential equationshttps://projecteuclid.org/euclid.aoap/1550566838<strong>Jocelyne Bion–Nadal</strong>, <strong>Denis Talay</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 3, 1609--1639.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce a Wasserstein-type distance on the set of the probability distributions of strong solutions to stochastic differential equations. This new distance is defined by restricting the set of possible coupling measures. We prove that it may also be defined by means of the value function of a stochastic control problem whose Hamilton–Jacobi–Bellman equation has a smooth solution, which allows one to deduce a priori estimates or to obtain numerical evaluations. We exhibit an optimal coupling measure and characterize it as a weak solution to an explicit stochastic differential equation, and we finally describe procedures to approximate this optimal coupling measure.
A notable application concerns the following modeling issue: given an exact diffusion model, how to select a simplified diffusion model within a class of admissible models under the constraint that the probability distribution of the exact model is preserved as much as possible?
</p>projecteuclid.org/euclid.aoap/1550566838_20190219040044Tue, 19 Feb 2019 04:00 ESTNumerical method for FBSDEs of McKean–Vlasov typehttps://projecteuclid.org/euclid.aoap/1550566839<strong>Jean-François Chassagneux</strong>, <strong>Dan Crisan</strong>, <strong>François Delarue</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 3, 1640--1684.</p><p><strong>Abstract:</strong><br/>
This paper is dedicated to the presentation and the analysis of a numerical scheme for forward–backward SDEs of the McKean–Vlasov type, or equivalently for solutions to PDEs on the Wasserstein space. Because of the mean field structure of the equation, earlier methods for classical forward–backward systems fail. The scheme is based on a variation of the method of continuation. The principle is to implement recursively local Picard iterations on small time intervals.
We establish a bound for the rate of convergence under the assumption that the decoupling field of the forward–backward SDE (or equivalently the solution of the PDE) satisfies mild regularity conditions. We also provide numerical illustrations.
</p>projecteuclid.org/euclid.aoap/1550566839_20190219040044Tue, 19 Feb 2019 04:00 ESTSecond-order BSDE under monotonicity condition and liquidation problem under uncertaintyhttps://projecteuclid.org/euclid.aoap/1550566840<strong>Alexandre Popier</strong>, <strong>Chao Zhou</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 3, 1685--1739.</p><p><strong>Abstract:</strong><br/>
In this work, we investigate an optimal liquidation problem under Knightian uncertainty. We obtain the value function and an optimal control characterised by the solution of a second-order BSDE with monotone generator and with a singular terminal condition.
</p>projecteuclid.org/euclid.aoap/1550566840_20190219040044Tue, 19 Feb 2019 04:00 ESTSupermarket model on graphshttps://projecteuclid.org/euclid.aoap/1550566841<strong>Amarjit Budhiraja</strong>, <strong>Debankur Mukherjee</strong>, <strong>Ruoyu Wu</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 3, 1740--1777.</p><p><strong>Abstract:</strong><br/>
We consider a variation of the supermarket model in which the servers can communicate with their neighbors and where the neighborhood relationships are described in terms of a suitable graph. Tasks with unit-exponential service time distributions arrive at each vertex as independent Poisson processes with rate $\lambda$, and each task is irrevocably assigned to the shortest queue among the one it first appears and its $d-1$ randomly selected neighbors. This model has been extensively studied when the underlying graph is a clique in which case it reduces to the well-known power-of-$d$ scheme. In particular, results of Mitzenmacher (1996) and Vvedenskaya et al. (1996) show that as the size of the clique gets large, the occupancy process associated with the queue-lengths at the various servers converges to a deterministic limit described by an infinite system of ordinary differential equations (ODE). In this work, we consider settings where the underlying graph need not be a clique and is allowed to be suitably sparse. We show that if the minimum degree approaches infinity (however slowly) as the number of servers $N$ approaches infinity, and the ratio between the maximum degree and the minimum degree in each connected component approaches $1$ uniformly, the occupancy process converges to the same system of ODE as the classical supermarket model. In particular, the asymptotic behavior of the occupancy process is insensitive to the precise network topology. We also study the case where the graph sequence is random, with the $N$th graph given as an Erdős–Rényi random graph on $N$ vertices with average degree $c(N)$. Annealed convergence of the occupancy process to the same deterministic limit is established under the condition $c(N)\to\infty$, and under a stronger condition $c(N)/\ln N\to\infty$, convergence (in probability) is shown for almost every realization of the random graph.
</p>projecteuclid.org/euclid.aoap/1550566841_20190219040044Tue, 19 Feb 2019 04:00 ESTEffective Berry–Esseen and concentration bounds for Markov chains with a spectral gaphttps://projecteuclid.org/euclid.aoap/1550566842<strong>Benoît Kloeckner</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 3, 1778--1807.</p><p><strong>Abstract:</strong><br/>
Applying quantitative perturbation theory for linear operators, we prove nonasymptotic bounds for Markov chains whose transition kernel has a spectral gap in an arbitrary Banach algebra of functions $\mathscr{X}$. The main results are concentration inequalities and Berry–Esseen bounds, obtained assuming neither reversibility nor “warm start” hypothesis: the law of the first term of the chain can be arbitrary. The spectral gap hypothesis is basically a uniform $\mathscr{X}$-ergodicity hypothesis, and when $\mathscr{X}$ consist in regular functions this is weaker than uniform ergodicity. We show on a few examples how the flexibility in the choice of function space can be used. The constants are completely explicit and reasonable enough to make the results usable in practice, notably in MCMC methods.
</p>projecteuclid.org/euclid.aoap/1550566842_20190219040044Tue, 19 Feb 2019 04:00 ESTThe nested Kingman coalescent: Speed of coming down from infinityhttps://projecteuclid.org/euclid.aoap/1550566843<strong>Airam Blancas</strong>, <strong>Tim Rogers</strong>, <strong>Jason Schweinsberg</strong>, <strong>Arno Siri-Jégousse</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 3, 1808--1836.</p><p><strong>Abstract:</strong><br/>
The nested Kingman coalescent describes the ancestral tree of a population undergoing neutral evolution at the level of individuals and at the level of species, simultaneously. We study the speed at which the number of lineages descends from infinity in this hierarchical coalescent process and prove the existence of an early-time phase during which the number of lineages at time $t$ decays as $2\gamma/ct^{2}$, where $c$ is the ratio of the coalescence rates at the individual and species levels, and the constant $\gamma\approx3.45$ is derived from a recursive distributional equation for the number of lineages contained within a species at a typical time.
</p>projecteuclid.org/euclid.aoap/1550566843_20190219040044Tue, 19 Feb 2019 04:00 ESTCondensation in critical Cauchy Bienaymé–Galton–Watson treeshttps://projecteuclid.org/euclid.aoap/1550566844<strong>Igor Kortchemski</strong>, <strong>Loïc Richier</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 3, 1837--1877.</p><p><strong>Abstract:</strong><br/>
We are interested in the structure of large Bienaymé–Galton–Watson random trees whose offspring distribution is critical and falls within the domain of attraction of a stable law of index $\alpha=1$. In stark contrast to the case $\alpha\in(1,2]$, we show that a condensation phenomenon occurs: in such trees, one vertex with macroscopic degree emerges (see Figure 1). To this end, we establish limit theorems for centered downwards skip-free random walks whose steps are in the domain of attraction of a Cauchy distribution, when conditioned on a late entrance in the negative real line. These results are of independent interest. As an application, we study the geometry of the boundary of random planar maps in a specific regime (called nongeneric of parameter $3/2$). This supports the conjecture that faces in Le Gall and Miermont’s $3/2$-stable maps are self-avoiding.
</p>projecteuclid.org/euclid.aoap/1550566844_20190219040044Tue, 19 Feb 2019 04:00 ESTEntropy-controlled Last-Passage Percolationhttps://projecteuclid.org/euclid.aoap/1550566845<strong>Quentin Berger</strong>, <strong>Niccolò Torri</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 3, 1878--1903.</p><p><strong>Abstract:</strong><br/>
We introduce a natural generalization of Hammersley’s Last-Passage Percolation (LPP) called Entropy-controlled Last-Passage Percolation (E-LPP), where points can be collected by paths with a global (path-entropy) constraint which takes into account the whole structure of the path, instead of a local ($1$-Lipschitz) constraint as in Hammersley’s LPP. Our main result is to prove quantitative tail estimates on the maximal number of points that can be collected by a path with entropy bounded by a prescribed constant. The E-LPP turns out to be a key ingredient in the context of the directed polymer model when the environment is heavy-tailed, which we consider in (Berger and Torri (2018)). We give applications in this context, which are essentials tools in (Berger and Torri (2018)): we show that the limiting variational problem conjectured in ( Ann. Probab. 44 (2016) 4006–4048), Conjecture 1.7, is finite, and we prove that the discrete variational problem converges to the continuous one, generalizing techniques used in ( Comm. Pure Appl. Math. 64 (2011) 183–204; Probab. Theory Related Fields 137 (2007) 227–275).
</p>projecteuclid.org/euclid.aoap/1550566845_20190219040044Tue, 19 Feb 2019 04:00 ESTThe left-curtain martingale coupling in the presence of atomshttps://projecteuclid.org/euclid.aoap/1550566846<strong>David G. Hobson</strong>, <strong>Dominykas Norgilas</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 3, 1904--1928.</p><p><strong>Abstract:</strong><br/>
Beiglböck and Juillet ( Ann. Probab. 44 (2016) 42–106) introduced the left-curtain martingale coupling of probability measures $\mu$ and $\nu$, and proved that, when the initial law $\mu$ is continuous, it is supported by the graphs of two functions. We extend the later result by constructing the generalised left-curtain martingale coupling and show that for an arbitrary starting law $\mu$ it is characterised by two appropriately defined lower and upper functions.
As an application of this result, we derive the model-independent upper bound of an American put option. This extends recent results of Hobson and Norgilas (2017) on the atom-free case.
</p>projecteuclid.org/euclid.aoap/1550566846_20190219040044Tue, 19 Feb 2019 04:00 ESTUpper bounds for the function solution of the homogeneous $2D$ Boltzmann equation with hard potentialhttps://projecteuclid.org/euclid.aoap/1550566847<strong>Vlad Bally</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 3, 1929--1961.</p><p><strong>Abstract:</strong><br/>
We deal with $f_{t}(dv)$, the solution of the homogeneous $2D$ Boltzmann equation without cutoff. The initial condition $f_{0}(dv)$ may be any probability distribution (except a Dirac mass). However, for sufficiently hard potentials, the semigroup has a regularization property (see Probab. Theory Related Fields 151 (2011) 659–704): $f_{t}(dv)=f_{t}(v)\,dv$ for every $t>0$. The aim of this paper is to give upper bounds for $f_{t}(v)$, the most significant one being of type $f_{t}(v)\leq Ct^{-\eta}e^{-\vert v\vert^{\lambda}}$ for some $\eta,\lambda>0$.
</p>projecteuclid.org/euclid.aoap/1550566847_20190219040044Tue, 19 Feb 2019 04:00 ESTWhen multiplicative noise stymies controlhttps://projecteuclid.org/euclid.aoap/1563869034<strong>Jian Ding</strong>, <strong>Yuval Peres</strong>, <strong>Gireeja Ranade</strong>, <strong>Alex Zhai</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 4, 1963--1992.</p><p><strong>Abstract:</strong><br/>
We consider the stabilization of an unstable discrete-time linear system that is observed over a channel corrupted by continuous multiplicative noise. Our main result shows that if the system growth is large enough, then the system cannot be stabilized. This is done by showing that the probability that the state magnitude remains bounded must go to zero with time. Our proof technique recursively bounds the conditional density of the system state to bound the progress the controller can make. This sidesteps the difficulty encountered in using the standard data-rate theorem style approach; that approach does not work because the mutual information per round between the system state and the observation is potentially unbounded.
It was known that a system with multiplicative observation noise can be stabilized using a simple memoryless linear strategy if the system growth is suitably bounded. The second main result in this paper shows that while memory cannot improve the performance of a linear scheme, a simple nonlinear scheme that uses one-step memory can do better than the best linear scheme.
</p>projecteuclid.org/euclid.aoap/1563869034_20190723040421Tue, 23 Jul 2019 04:04 EDTLarge deviations for fast transport stochastic RDEs with applications to the exit problemhttps://projecteuclid.org/euclid.aoap/1563869035<strong>Sandra Cerrai</strong>, <strong>Nicholas Paskal</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 4, 1993--2032.</p><p><strong>Abstract:</strong><br/>
We study reaction diffusion equations with a deterministic reaction term as well as two random reaction terms, one that acts on the interior of the domain, and another that acts only on the boundary of the domain. We are interested in the regime where the relative sizes of the diffusion and reaction terms are different. Specifically, we consider the case where the diffusion rate is much larger than the rate of reaction, and the deterministic rate of reaction is much larger than either of the random rate of reactions.
</p>projecteuclid.org/euclid.aoap/1563869035_20190723040421Tue, 23 Jul 2019 04:04 EDTStochastic approximation with random step sizes and urn models with random replacement matrices having finite meanhttps://projecteuclid.org/euclid.aoap/1563869036<strong>Ujan Gangopadhyay</strong>, <strong>Krishanu Maulik</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 4, 2033--2066.</p><p><strong>Abstract:</strong><br/>
The stochastic approximation algorithm is a useful technique which has been exploited successfully in probability theory and statistics for a long time. The step sizes used in stochastic approximation are generally taken to be deterministic and same is true for the drift. However, the specific application of urn models with random replacement matrices motivates us to consider stochastic approximation in a setup where both the step sizes and the drift are random, but the sequence is uniformly bounded. The problem becomes interesting when the negligibility conditions on the errors hold only in probability. We first prove a result on stochastic approximation in this setup, which is new in the literature. Then, as an application, we study urn models with random replacement matrices.
In the urn model, the replacement matrices need neither be independent, nor identically distributed. We assume that the replacement matrices are only independent of the color drawn in the same round conditioned on the entire past. We relax the usual second moment assumption on the replacement matrices in the literature and require only first moment to be finite. We require the conditional expectation of the replacement matrix given the past to be close to an irreducible matrix, in an appropriate sense. We do not require any of the matrices to be balanced or nonrandom. We prove convergence of the proportion vector, the composition vector and the count vector in $L^{1}$, and hence in probability. It is to be noted that the related differential equation is of Lotka–Volterra type and can be analyzed directly.
</p>projecteuclid.org/euclid.aoap/1563869036_20190723040421Tue, 23 Jul 2019 04:04 EDTCritical point for infinite cycles in a random loop model on treeshttps://projecteuclid.org/euclid.aoap/1563869037<strong>Alan Hammond</strong>, <strong>Milind Hegde</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 4, 2067--2088.</p><p><strong>Abstract:</strong><br/>
We study a spatial model of random permutations on trees with a time parameter $T>0$, a special case of which is the random stirring process. The model on trees was first analysed by Björnberg and Ueltschi [ Ann. Appl. Probab. 28 (2018) 2063–2082], who established the existence of infinite cycles for $T$ slightly above a putatively identified critical value but left open behaviour at arbitrarily high values of $T$. We show the existence of infinite cycles for all $T$ greater than a constant, thus classifying behaviour for all values of $T$ and establishing the existence of a sharp phase transition. Numerical studies [ J. Phys. A 48 Article ID 345002] of the model on $\mathbb{Z}^{d}$ have shown behaviour with strong similarities to what is proven for trees.
</p>projecteuclid.org/euclid.aoap/1563869037_20190723040421Tue, 23 Jul 2019 04:04 EDTParking on transitive unimodular graphshttps://projecteuclid.org/euclid.aoap/1563869038<strong>Michael Damron</strong>, <strong>Janko Gravner</strong>, <strong>Matthew Junge</strong>, <strong>Hanbaek Lyu</strong>, <strong>David Sivakoff</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 4, 2089--2113.</p><p><strong>Abstract:</strong><br/>
Place a car independently with probability $p$ at each site of a graph. Each initially vacant site is a parking spot that can fit one car. Cars simultaneously perform independent random walks. When a car encounters an available parking spot it parks there. Other cars can still drive over the site, but cannot park there. For a large class of transitive and unimodular graphs, we show that the root is almost surely visited infinitely many times when $p\geq1/2$, and only finitely many times otherwise.
</p>projecteuclid.org/euclid.aoap/1563869038_20190723040421Tue, 23 Jul 2019 04:04 EDTThe hydrodynamic limit of a randomized load balancing networkhttps://projecteuclid.org/euclid.aoap/1563869039<strong>Reza Aghajani</strong>, <strong>Kavita Ramanan</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 4, 2114--2174.</p><p><strong>Abstract:</strong><br/>
Randomized load balancing networks arise in a variety of applications, and allow for efficient sharing of resources, while being relatively easy to implement. We consider a network of parallel queues in which incoming jobs with independent and identically distributed service times are assigned to the shortest queue among a subset of $d$ queues chosen uniformly at random, and leave the network on completion of service. Prior work on dynamical properties of this model has focused on the case of exponential service distributions. In this work, we analyze the more realistic case of general service distributions. We first introduce a novel particle representation of the state of the network, and characterize the state dynamics via a countable sequence of interacting stochastic measure-valued evolution equations. Under mild assumptions, we show that the sequence of scaled state processes converges, as the number of servers goes to infinity, to a hydrodynamic limit that is characterized as the unique solution to a countable system of coupled deterministic measure-valued equations. As a simple corollary, we also establish a propagation of chaos result that shows that finite collections of queues are asymptotically independent. The general framework developed here is potentially useful for analyzing a larger class of models arising in diverse fields including biology and materials science.
</p>projecteuclid.org/euclid.aoap/1563869039_20190723040421Tue, 23 Jul 2019 04:04 EDTA probabilistic approach to Dirac concentration in nonlocal models of adaptation with several resourceshttps://projecteuclid.org/euclid.aoap/1563869040<strong>Nicolas Champagnat</strong>, <strong>Benoit Henry</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 4, 2175--2216.</p><p><strong>Abstract:</strong><br/>
This work is devoted to the study of scaling limits in small mutations and large time of the solutions $u^{\varepsilon}$ of two deterministic models of phenotypic adaptation, where the parameter $\varepsilon>0$ scales the size or frequency of mutations. The second model is the so-called Lotka–Volterra parabolic PDE in $\mathbb{R}^{d}$ with an arbitrary number of resources and the first one is a version of the second model with finite phenotype space. The solutions of such systems typically concentrate as Dirac masses in the limit $\varepsilon\to0$. Our main results are, in both cases, the representation of the limits of $\varepsilon\log u^{\varepsilon}$ as solutions of variational problems and regularity results for these limits. The method mainly relies on Feynman–Kac-type representations of $u^{\varepsilon}$ and Varadhan’s lemma. Our probabilistic approach applies to multiresources situations not covered by standard analytical methods and makes the link between variational limit problems and Hamilton–Jacobi equations with irregular Hamiltonians that arise naturally from analytical methods. The finite case presents substantial difficulties since the rate function of the associated large deviation principle (LDP) has noncompact level sets. In that case, we are also able to obtain uniqueness of the solution of the variational problem and of the associated differential problem which can be interpreted as a Hamilton–Jacobi equation in finite state space.
</p>projecteuclid.org/euclid.aoap/1563869040_20190723040421Tue, 23 Jul 2019 04:04 EDTA version of Aldous’ spectral-gap conjecture for the zero range processhttps://projecteuclid.org/euclid.aoap/1563869041<strong>Jonathan Hermon</strong>, <strong>Justin Salez</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 4, 2217--2229.</p><p><strong>Abstract:</strong><br/>
We show that the spectral gap of a general zero range process can be controlled in terms of the spectral gap for a single particle. This is in the spirit of Aldous’ famous spectral-gap conjecture for the interchange process, now resolved by Caputo et al. Our main inequality decouples the role of the geometry (defined by the jump matrix) from that of the kinetics (specified by the exit rates). Among other consequences, the various spectral gap estimates that were so far only available on the complete graph or the $d$-dimensional torus now extend effortlessly to arbitrary geometries. As an illustration, we determine the exact order of magnitude of the spectral gap of the rate-one zero-range process on any regular graph and, more generally, for any doubly stochastic jump matrix.
</p>projecteuclid.org/euclid.aoap/1563869041_20190723040421Tue, 23 Jul 2019 04:04 EDTIterative multilevel particle approximation for McKean–Vlasov SDEshttps://projecteuclid.org/euclid.aoap/1563869042<strong>Lukasz Szpruch</strong>, <strong>Shuren Tan</strong>, <strong>Alvin Tse</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 4, 2230--2265.</p><p><strong>Abstract:</strong><br/>
The mean field limits of systems of interacting diffusions (also called stochastic interacting particle systems (SIPS)) have been intensively studied since McKean ( Proc. Natl. Acad. Sci. USA 56 (1966) 1907–1911) as they pave a way to probabilistic representations for many important nonlinear/nonlocal PDEs. The fact that particles are not independent render classical variance reduction techniques not directly applicable, and consequently make simulations of interacting diffusions prohibitive.
In this article, we provide an alternative iterative particle representation, inspired by the fixed-point argument by Sznitman (In École D’Été de Probabilités de Saint-Flour XIX—1989 (1991) 165–251, Springer). The representation enjoys suitable conditional independence property that is leveraged in our analysis. We establish weak convergence of iterative particle system to the McKean–Vlasov SDEs (McKV–SDEs). One of the immediate advantages of the iterative particle system is that it can be combined with the Multilevel Monte Carlo (MLMC) approach for the simulation of McKV–SDEs. We proved that the MLMC approach reduces the computational complexity of calculating expectations by an order of magnitude. Another perspective on this work is that we analyse the error of nested Multilevel Monte Carlo estimators, which is of independent interest. Furthermore, we work with state dependent functionals, unlike scalar outputs which are common in literature on MLMC. The error analysis is carried out in uniform, and what seems to be new, weighted norms.
</p>projecteuclid.org/euclid.aoap/1563869042_20190723040421Tue, 23 Jul 2019 04:04 EDTErgodicity of the zigzag processhttps://projecteuclid.org/euclid.aoap/1563869043<strong>Joris Bierkens</strong>, <strong>Gareth O. Roberts</strong>, <strong>Pierre-André Zitt</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 4, 2266--2301.</p><p><strong>Abstract:</strong><br/>
The zigzag process is a piecewise deterministic Markov process which can be used in a MCMC framework to sample from a given target distribution. We prove the convergence of this process to its target under very weak assumptions, and establish a central limit theorem for empirical averages under stronger assumptions on the decay of the target measure. We use the classical “Meyn–Tweedie” approach ( Markov Chains and Stochastic Stability (2009) Cambridge Univ. Press; Adv. in Appl. Probab. 25 (1993) 487–517). The main difficulty turns out to be the proof that the process can indeed reach all the points in the space, even if we consider the minimal switching rates.
</p>projecteuclid.org/euclid.aoap/1563869043_20190723040421Tue, 23 Jul 2019 04:04 EDTOn Skorokhod embeddings and Poisson equationshttps://projecteuclid.org/euclid.aoap/1563869044<strong>Leif Döring</strong>, <strong>Lukas Gonon</strong>, <strong>David J. Prömel</strong>, <strong>Oleg Reichmann</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 4, 2302--2337.</p><p><strong>Abstract:</strong><br/>
The classical Skorokhod embedding problem for a Brownian motion $W$ asks to find a stopping time $\tau $ so that $W_{\tau }$ is distributed according to a prescribed probability distribution $\mu $. Many solutions have been proposed during the past 50 years and applications in different fields emerged. This article deals with a generalized Skorokhod embedding problem (SEP): Let $X$ be a Markov process with initial marginal distribution $\mu_{0}$ and let $\mu_{1}$ be a probability measure. The task is to find a stopping time $\tau $ such that $X_{\tau }$ is distributed according to $\mu_{1}$. More precisely, we study the question of deciding if a finite mean solution to the SEP can exist for given $\mu_{0}$, $\mu_{1}$ and the task of giving a solution which is as explicit as possible.
If $\mu_{0}$ and $\mu_{1}$ have positive densities $h_{0}$ and $h_{1}$ and the generator $\mathcal{A}$ of $X$ has a formal adjoint operator $\mathcal{A}^{*}$, then we propose necessary and sufficient conditions for the existence of an embedding in terms of the Poisson equation $\mathcal{A}^{*}H=h_{1}-h_{0}$ and give a fairly explicit construction of the stopping time using the solution of the Poisson equation. For the class of Lévy processes, we carry out the procedure and extend a result of Bertoin and Le Jan to Lévy processes without local times.
</p>projecteuclid.org/euclid.aoap/1563869044_20190723040421Tue, 23 Jul 2019 04:04 EDTA McKean–Vlasov equation with positive feedback and blow-upshttps://projecteuclid.org/euclid.aoap/1563869045<strong>Ben Hambly</strong>, <strong>Sean Ledger</strong>, <strong>Andreas Søjmark</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 4, 2338--2373.</p><p><strong>Abstract:</strong><br/>
We study a McKean–Vlasov equation arising from a mean-field model of a particle system with positive feedback. As particles hit a barrier, they cause the other particles to jump in the direction of the barrier and this feedback mechanism leads to the possibility that the system can exhibit contagious blow-ups. Using a fixed-point argument, we construct a differentiable solution up to a first explosion time. Our main contribution is a proof of uniqueness in the class of càdlàg functions, which confirms the validity of related propagation-of-chaos results in the literature. We extend the allowed initial conditions to include densities with any power law decay at the boundary, and connect the exponent of decay with the growth exponent of the solution in small time in a precise way. This takes us asymptotically close to the control on initial conditions required for a global solution theory. A novel minimality result and trapping technique are introduced to prove uniqueness.
</p>projecteuclid.org/euclid.aoap/1563869045_20190723040421Tue, 23 Jul 2019 04:04 EDTApproximation of stochastic processes by nonexpansive flows and coming down from infinityhttps://projecteuclid.org/euclid.aoap/1563869046<strong>Vincent Bansaye</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 4, 2374--2438.</p><p><strong>Abstract:</strong><br/>
This paper deals with the approximation of semimartingales in finite dimension by dynamical systems. We give trajectorial estimates uniform with respect to the initial condition for a well-chosen distance. This relies on a nonexpansivity property of the flow and allows to consider non-Lipschitz vector fields. The fluctuations of the process are controlled using the martingale technics and stochastic calculus.
Our main motivation is the trajectorial description of stochastic processes starting from large initial values. We state general properties on the coming down from infinity of one-dimensional SDEs, with a focus on stochastically monotone processes. In particular, we recover and complement known results on $\Lambda $-coalescent and birth and death processes. Moreover, using Poincaré’s compactification techniques for flows close to infinity, we develop this approach in two dimensions for competitive stochastic models. We thus classify the coming down from infinity of Lotka–Volterra diffusions and provide uniform estimates for the scaling limits of competitive birth and death processes.
</p>projecteuclid.org/euclid.aoap/1563869046_20190723040421Tue, 23 Jul 2019 04:04 EDTMixing time estimation in reversible Markov chains from a single sample pathhttps://projecteuclid.org/euclid.aoap/1563869047<strong>Daniel Hsu</strong>, <strong>Aryeh Kontorovich</strong>, <strong>David A. Levin</strong>, <strong>Yuval Peres</strong>, <strong>Csaba Szepesvári</strong>, <strong>Geoffrey Wolfer</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 4, 2439--2480.</p><p><strong>Abstract:</strong><br/>
The spectral gap $\gamma_{\star}$ of a finite, ergodic and reversible Markov chain is an important parameter measuring the asymptotic rate of convergence. In applications, the transition matrix $\mathbf{{P}}$ may be unknown, yet one sample of the chain up to a fixed time $n$ may be observed. We consider here the problem of estimating $\gamma_{\star}$ from this data. Let $\boldsymbol{\pi}$ be the stationary distribution of $\mathbf{{P}}$, and $\pi_{\star}=\min_{x}\pi (x)$. We show that if $n$ is at least $\frac{1}{\gamma_{\star}\pi_{\star}}$ times a logarithmic correction, then $\gamma_{\star}$ can be estimated to within a multiplicative factor with high probability. When $\pi $ is uniform on $d$ states, this nearly matches a lower bound of $\frac{d}{\gamma_{\star}}$ steps required for precise estimation of $\gamma_{\star}$. Moreover, we provide the first procedure for computing a fully data-dependent interval, from a single finite-length trajectory of the chain, that traps the mixing time $t_{\mathrm{mix}}$ of the chain at a prescribed confidence level. The interval does not require the knowledge of any parameters of the chain. This stands in contrast to previous approaches, which either only provide point estimates, or require a reset mechanism, or additional prior knowledge. The interval is constructed around the relaxation time $t_{\mathrm{relax}}=1/\gamma_{\star}$, which is strongly related to the mixing time, and the width of the interval converges to zero roughly at a $1/\sqrt{n}$ rate, where $n$ is the length of the sample path.
</p>projecteuclid.org/euclid.aoap/1563869047_20190723040421Tue, 23 Jul 2019 04:04 EDTReduced-form framework under model uncertaintyhttps://projecteuclid.org/euclid.aoap/1563869048<strong>Francesca Biagini</strong>, <strong>Yinglin Zhang</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 4, 2481--2522.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce a sublinear conditional expectation with respect to a family of possibly nondominated probability measures on a progressively enlarged filtration. In this way, we extend the classic reduced-form setting for credit and insurance markets to the case under model uncertainty, when we consider a family of priors possibly mutually singular to each other. Furthermore, we study the superhedging approach in continuous time for payment streams under model uncertainty, and establish several equivalent versions of dynamic robust superhedging duality. These results close the gap between robust framework for financial market, which is recently studied in an intensive way, and the one for credit and insurance markets, which is limited in the present literature only to some very specific cases.
</p>projecteuclid.org/euclid.aoap/1563869048_20190723040421Tue, 23 Jul 2019 04:04 EDTA general continuous-state nonlinear branching processhttps://projecteuclid.org/euclid.aoap/1563869049<strong>Pei-Sen Li</strong>, <strong>Xu Yang</strong>, <strong>Xiaowen Zhou</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 4, 2523--2555.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider the unique nonnegative solution to the following generalized version of the stochastic differential equation for a continuous-state branching process: \begin{eqnarray*}X_{t}&=&x+\int_{0}^{t}\gamma_{0}(X_{s})\,\mathrm{d}s+\int_{0}^{t}\int_{0}^{\gamma_{1}(X_{s-})}W(\mathrm{d}s,\mathrm{d}u)\\&&{}+\int_{0}^{t}\int_{0}^{\infty}\int_{0}^{\gamma_{2}(X_{s-})}z\tilde{N}(\mathrm{d}s,\mathrm{d}z,\mathrm{d}u),\end{eqnarray*} where $W(\mathrm{d}t,\mathrm{d}u)$ and $\tilde{N}(\mathrm{d}s,\mathrm{d}z,\mathrm{d}u)$ denote a Gaussian white noise and an independent compensated spectrally positive Poisson random measure, respectively, and $\gamma_{0},\gamma_{1}$ and $\gamma_{2}$ are functions on $\mathbb{R}_{+}$ with both $\gamma_{1}$ and $\gamma_{2}$ taking nonnegative values. Intuitively, this process can be identified as a continuous-state branching process with population-size-dependent branching rates and with competition. Using martingale techniques we find rather sharp conditions on extinction, explosion and coming down from infinity behaviors of the process. Some Foster–Lyapunov-type criteria are also developed for such a process. More explicit results are obtained when $\gamma_{i}$, $i=0,1,2$ are power functions.
</p>projecteuclid.org/euclid.aoap/1563869049_20190723040421Tue, 23 Jul 2019 04:04 EDTEquilibrium interfaces of biased voter modelshttps://projecteuclid.org/euclid.aoap/1563869050<strong>Rongfeng Sun</strong>, <strong>Jan M. Swart</strong>, <strong>Jinjiong Yu</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 4, 2556--2593.</p><p><strong>Abstract:</strong><br/>
A one-dimensional interacting particle system is said to exhibit interface tightness if starting in an initial condition describing the interface between two constant configurations of different types, the process modulo translations is positive recurrent. In a biological setting, this describes two populations that do not mix, and it is believed to be a common phenomenon in one-dimensional particle systems. Interface tightness has been proved for voter models satisfying a finite second moment condition on the rates. We extend this to biased voter models. Furthermore, we show that the distribution of the equilibrium interface for the biased voter model converges to that of the voter model when the bias parameter tends to zero. A key ingredient is an identity for the expected number of boundaries in the equilibrium voter model interface, which is of independent interest.
</p>projecteuclid.org/euclid.aoap/1563869050_20190723040421Tue, 23 Jul 2019 04:04 EDTPropagation of chaos for topological interactionshttps://projecteuclid.org/euclid.aoap/1563869051<strong>P. Degond</strong>, <strong>M. Pulvirenti</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 4, 2594--2612.</p><p><strong>Abstract:</strong><br/>
We consider a $N$-particle model describing an alignment mechanism due to a topological interaction among the agents. We show that the kinetic equation, expected to hold in the mean-field limit $N\to \infty $, as following from the previous analysis in ( J. Stat. Phys. 163 (2016) 41–60) can be rigorously derived. This means that the statistical independence (propagation of chaos) is indeed recovered in the limit, provided it is assumed at time zero.
</p>projecteuclid.org/euclid.aoap/1563869051_20190723040421Tue, 23 Jul 2019 04:04 EDTNormal convergence of nonlocalised geometric functionals and shot-noise excursionshttps://projecteuclid.org/euclid.aoap/1571385618<strong>Raphaël Lachièze-Rey</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 2613--2653.</p><p><strong>Abstract:</strong><br/>
This article presents a complete second-order theory for a large class of geometric functionals on homogeneous Poisson input. In particular, the results do not require the existence of a radius of stabilisation. Hence they can be applied to geometric functionals of spatial shot-noise fields excursions such as volume, perimeter, or Euler characteristic (the method still applies to stabilising functionals). More generally, it must be checked that a local contribution to the functional is not strongly affected under a perturbation of the input far away. In this case, the exact asymptotic variance is given, as well as the likely optimal speed of convergence in the central limit theorem. This goes through a general mixing-type condition that adapts nicely to both proving asymptotic normality and that variance is of volume order.
</p>projecteuclid.org/euclid.aoap/1571385618_20191018040042Fri, 18 Oct 2019 04:00 EDTMetastability of the contact process on fast evolving scale-free networkshttps://projecteuclid.org/euclid.aoap/1571385619<strong>Emmanuel Jacob</strong>, <strong>Amitai Linker</strong>, <strong>Peter Mörters</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 2654--2699.</p><p><strong>Abstract:</strong><br/>
We study the contact process in the regime of small infection rates on finite scale-free networks with stationary dynamics based on simultaneous updating of all connections of a vertex. We allow the update rates of individual vertices to increase with the strength of a vertex, leading to a fast evolution of the network. We first develop an approach for inhomogeneous networks with general kernel and then focus on two canonical cases, the factor kernel and the preferential attachment kernel. For these specific networks, we identify and analyse four possible strategies how the infection can survive for a long time. We show that there is fast extinction of the infection when neither of the strategies is successful, otherwise there is slow extinction and the most successful strategy determines the asymptotics of the metastable density as the infection rate goes to zero. We identify the domains in which these strategies dominate in terms of phase diagrams for the exponent describing the decay of the metastable density.
</p>projecteuclid.org/euclid.aoap/1571385619_20191018040042Fri, 18 Oct 2019 04:00 EDTTree lengths for general $\Lambda $-coalescents and the asymptotic site frequency spectrum around the Bolthausen–Sznitman coalescenthttps://projecteuclid.org/euclid.aoap/1571385620<strong>Christina S. Diehl</strong>, <strong>Götz Kersting</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 2700--2743.</p><p><strong>Abstract:</strong><br/>
We study tree lengths in $\Lambda $-coalescents without a dust component from a sample of $n$ individuals. For the total length of all branches and the total length of all external branches, we present laws of large numbers in full generality. The other results treat regularly varying coalescents with exponent 1, which cover the Bolthausen–Sznitman coalescent. The theorems contain laws of large numbers for the total length of all internal branches and of internal branches of order $a$ (i.e., branches carrying $a$ individuals out of the sample). These results immediately transform to sampling formulas in the infinite sites model. In particular, we obtain the asymptotic site frequency spectrum of the Bolthausen–Sznitman coalescent. The proofs rely on a new technique to obtain laws of large numbers for certain functionals of decreasing Markov chains.
</p>projecteuclid.org/euclid.aoap/1571385620_20191018040042Fri, 18 Oct 2019 04:00 EDTEmpirical optimal transport on countable metric spaces: Distributional limits and statistical applicationshttps://projecteuclid.org/euclid.aoap/1571385621<strong>Carla Tameling</strong>, <strong>Max Sommerfeld</strong>, <strong>Axel Munk</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 2744--2781.</p><p><strong>Abstract:</strong><br/>
We derive distributional limits for empirical transport distances between probability measures supported on countable sets. Our approach is based on sensitivity analysis of optimal values of infinite dimensional mathematical programs and a delta method for nonlinear derivatives. A careful calibration of the norm on the space of probability measures is needed in order to combine differentiability and weak convergence of the underlying empirical process. Based on this, we provide a sufficient and necessary condition for the underlying distribution on the countable metric space for such a distributional limit to hold. We give an explicit form of the limiting distribution for tree spaces.
Finally, we apply our findings to optimal transport based inference in large scale problems. An application to nanoscale microscopy is given.
</p>projecteuclid.org/euclid.aoap/1571385621_20191018040042Fri, 18 Oct 2019 04:00 EDTExtinction in lower Hessenberg branching processes with countably many typeshttps://projecteuclid.org/euclid.aoap/1571385622<strong>Peter Braunsteins</strong>, <strong>Sophie Hautphenne</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 2782--2818.</p><p><strong>Abstract:</strong><br/>
We consider a class of branching processes with countably many types which we refer to as Lower Hessenberg branching processes . These are multitype Galton–Watson processes with typeset $\mathcal{X}=\{0,1,2,\ldots\}$, in which individuals of type $i$ may give birth to offspring of type $j\leq i+1$ only. For this class of processes, we study the set $S$ of fixed points of the progeny generating function. In particular, we highlight the existence of a continuum of fixed points whose minimum is the global extinction probability vector $\boldsymbol{q}$ and whose maximum is the partial extinction probability vector $\boldsymbol{\tilde{q}}$. In the case where $\boldsymbol{\tilde{q}}=\boldsymbol{1}$, we derive a global extinction criterion which holds under second moment conditions, and when $\boldsymbol{\tilde{q}}<\boldsymbol{1}$ we develop necessary and sufficient conditions for $\boldsymbol{q}=\boldsymbol{\tilde{q}}$. We also correct a result in the literature on a sequence of finite extinction probability vectors that converge to the infinite vector $\boldsymbol{\tilde{q}}$.
</p>projecteuclid.org/euclid.aoap/1571385622_20191018040042Fri, 18 Oct 2019 04:00 EDTControlled reflected SDEs and Neumann problem for backward SPDEshttps://projecteuclid.org/euclid.aoap/1571385623<strong>Erhan Bayraktar</strong>, <strong>Jinniao Qiu</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 2819--2848.</p><p><strong>Abstract:</strong><br/>
We solve the optimal control problem of a one-dimensional reflected stochastic differential equation, whose coefficients can be path dependent. The value function of this problem is characterized by a backward stochastic partial differential equation (BSPDE) with Neumann boundary conditions. We prove the existence and uniqueness of a sufficiently regular solution for this BSPDE, which is then used to construct the optimal feedback control. In fact, we prove a more general result: the existence and uniqueness of strong solution for the Neumann problem for general nonlinear BSPDEs, which might be of interest even out of the current context.
</p>projecteuclid.org/euclid.aoap/1571385623_20191018040042Fri, 18 Oct 2019 04:00 EDTDynamics of observables in rank-based models and performance of functionally generated portfolioshttps://projecteuclid.org/euclid.aoap/1571385624<strong>Sergio A. Almada Monter</strong>, <strong>Mykhaylo Shkolnikov</strong>, <strong>Jiacheng Zhang</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 2849--2883.</p><p><strong>Abstract:</strong><br/>
In the seminal work ( Stochastic Portfolio Theory: Stochastic Modelling and Applied Probability (2002) Springer), several macroscopic market observables have been introduced, in an attempt to find characteristics capturing the diversity of a financial market. Despite the crucial importance of such observables for investment decisions, a concise mathematical description of their dynamics has been missing. We fill this gap in the setting of rank-based models. The results are then used to study the performance of multiplicatively and additively functionally generated portfolios.
</p>projecteuclid.org/euclid.aoap/1571385624_20191018040042Fri, 18 Oct 2019 04:00 EDTMeasuring sample quality with diffusionshttps://projecteuclid.org/euclid.aoap/1571385625<strong>Jackson Gorham</strong>, <strong>Andrew B. Duncan</strong>, <strong>Sebastian J. Vollmer</strong>, <strong>Lester Mackey</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 2884--2928.</p><p><strong>Abstract:</strong><br/>
Stein’s method for measuring convergence to a continuous target distribution relies on an operator characterizing the target and Stein factor bounds on the solutions of an associated differential equation. While such operators and bounds are readily available for a diversity of univariate targets, few multivariate targets have been analyzed. We introduce a new class of characterizing operators based on Itô diffusions and develop explicit multivariate Stein factor bounds for any target with a fast-coupling Itô diffusion. As example applications, we develop computable and convergence-determining diffusion Stein discrepancies for log-concave, heavy-tailed and multimodal targets and use these quality measures to select the hyperparameters of biased Markov chain Monte Carlo (MCMC) samplers, compare random and deterministic quadrature rules and quantify bias-variance tradeoffs in approximate MCMC. Our results establish a near-linear relationship between diffusion Stein discrepancies and Wasserstein distances, improving upon past work even for strongly log-concave targets. The exposed relationship between Stein factors and Markov process coupling may be of independent interest.
</p>projecteuclid.org/euclid.aoap/1571385625_20191018040042Fri, 18 Oct 2019 04:00 EDTInterference queueing networks on gridshttps://projecteuclid.org/euclid.aoap/1571385626<strong>Abishek Sankararaman</strong>, <strong>François Baccelli</strong>, <strong>Sergey Foss</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 2929--2987.</p><p><strong>Abstract:</strong><br/>
Consider a countably infinite collection of interacting queues, with a queue located at each point of the $d$-dimensional integer grid, having independent Poisson arrivals, but dependent service rates. The service discipline is of the processor sharing type, with the service rate in each queue slowed down, when the neighboring queues have a larger workload. The interactions are translation invariant in space and is neither of the Jackson Networks type, nor of the mean-field type. Coupling and percolation techniques are first used to show that this dynamics has well-defined trajectories. Coupling from the past techniques are then proposed to build its minimal stationary regime. The rate conservation principle of Palm calculus is then used to identify the stability condition of this system, where the notion of stability is appropriately defined for an infinite dimensional process. We show that the identified condition is also necessary in certain special cases and conjecture it to be true in all cases. Remarkably, the rate conservation principle also provides a closed-form expression for the mean queue size. When the stability condition holds, this minimal solution is the unique translation invariant stationary regime. In addition, there exists a range of small initial conditions for which the dynamics is attracted to the minimal regime. Nevertheless, there exists another range of larger though finite initial conditions for which the dynamics diverges, even though stability criterion holds.
</p>projecteuclid.org/euclid.aoap/1571385626_20191018040042Fri, 18 Oct 2019 04:00 EDTApproximating mixed Hölder functions using random sampleshttps://projecteuclid.org/euclid.aoap/1571385627<strong>Nicholas F. Marshall</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 2988--3005.</p><p><strong>Abstract:</strong><br/>
Suppose $f:[0,1]^{2}\rightarrow \mathbb{R}$ is a $(c,\alpha )$-mixed Hölder function that we sample at $l$ points $X_{1},\ldots ,X_{l}$ chosen uniformly at random from the unit square. Let the location of these points and the function values $f(X_{1}),\ldots ,f(X_{l})$ be given. If $l\ge c_{1}n\log^{2}n$, then we can compute an approximation $\tilde{f}$ such that \begin{equation*}\|f-\tilde{f}\|_{L^{2}}=\mathcal{O}\big(n^{-\alpha}\log^{3/2}n\big),\end{equation*} with probability at least $1-n^{2-c_{1}}$, where the implicit constant only depends on the constants $c>0$ and $c_{1}>0$.
</p>projecteuclid.org/euclid.aoap/1571385627_20191018040042Fri, 18 Oct 2019 04:00 EDTLocal law and Tracy–Widom limit for sparse sample covariance matriceshttps://projecteuclid.org/euclid.aoap/1571385628<strong>Jong Yun Hwang</strong>, <strong>Ji Oon Lee</strong>, <strong>Kevin Schnelli</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 3006--3036.</p><p><strong>Abstract:</strong><br/>
We consider spectral properties of sparse sample covariance matrices, which includes biadjacency matrices of the bipartite Erdős–Rényi graph model. We prove a local law for the eigenvalue density up to the upper spectral edge. Under a suitable condition on the sparsity, we also prove that the limiting distribution of the rescaled, shifted extremal eigenvalues is given by the GOE Tracy–Widom law with an explicit formula on the deterministic shift of the spectral edge. For the biadjacency matrix of an Erdős–Rényi graph with two vertex sets of comparable sizes $M$ and $N$, this establishes Tracy–Widom fluctuations of the second largest eigenvalue when the connection probability $p$ is much larger than $N^{-2/3}$ with a deterministic shift of order $(Np)^{-1}$.
</p>projecteuclid.org/euclid.aoap/1571385628_20191018040042Fri, 18 Oct 2019 04:00 EDTAnother look into the Wong–Zakai theorem for stochastic heat equationhttps://projecteuclid.org/euclid.aoap/1571385629<strong>Yu Gu</strong>, <strong>Li-Cheng Tsai</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 3037--3061.</p><p><strong>Abstract:</strong><br/>
For the heat equation driven by a smooth, Gaussian random potential: \begin{equation*}\partial_{t}u_{\varepsilon}=\frac{1}{2}\Delta u_{\varepsilon}+u_{\varepsilon}(\xi_{\varepsilon}-c_{\varepsilon}),\quad t>0,x\in\mathbb{R},\end{equation*} where $\xi_{\varepsilon}$ converges to a spacetime white noise, and $c_{\varepsilon}$ is a diverging constant chosen properly, we prove that $u_{\varepsilon}$ converges in $L^{n}$ to the solution of the stochastic heat equation for any $n\geq1$. Our proof is probabilistic, hence provides another perspective of the general result of Hairer and Pardoux ( J. Math. Soc. Japan 67 (2015) 1551–1604), for the special case of the stochastic heat equation. We also discuss the transition from homogenization to stochasticity.
</p>projecteuclid.org/euclid.aoap/1571385629_20191018040042Fri, 18 Oct 2019 04:00 EDTPathwise convergence of the hard spheres Kac processhttps://projecteuclid.org/euclid.aoap/1571385630<strong>Daniel Heydecker</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 3062--3127.</p><p><strong>Abstract:</strong><br/>
We derive two estimates for the deviation of the $N$-particle, hard-spheres Kac process from the corresponding Boltzmann equation, measured in expected Wasserstein distance. Particular care is paid to the long-time properties of our estimates, exploiting the stability properties of the limiting Boltzmann equation at the level of realisations of the interacting particle system. As a consequence, we obtain an estimate for the propagation of chaos, uniformly in time and with polynomial rates, as soon as the initial data has a $k$th moment, $k>2$. Our approach is similar to Kac’s proposal of relating the long-time behaviour of the particle system to that of the limit equation. Along the way, we prove a new estimate for the continuity of the Boltzmann flow measured in Wasserstein distance.
</p>projecteuclid.org/euclid.aoap/1571385630_20191018040042Fri, 18 Oct 2019 04:00 EDTThe zealot voter modelhttps://projecteuclid.org/euclid.aoap/1571385631<strong>Ran Huo</strong>, <strong>Rick Durrett</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 3128--3154.</p><p><strong>Abstract:</strong><br/>
Inspired by the spread of discontent as in the 2016 presidential election, we consider a voter model in which 0’s are ordinary voters and 1’s are zealots. Thinking of a social network, but desiring the simplicity of an infinite object that can have a nontrivial stationary distribution, space is represented by a tree. The dynamics are a variant of the biased voter: if $x$ has degree $d(x)$ then at rate $d(x)p_{k}$ the individual at $x$ consults $k\ge 1$ neighbors. If at least one neighbor is 1, they adopt state 1, otherwise they become 0. In addition at rate $p_{0}$ individuals with opinion 1 change to 0. As in the contact process on trees, we are interested in determining when the zealots survive and when they will survive locally.
</p>projecteuclid.org/euclid.aoap/1571385631_20191018040042Fri, 18 Oct 2019 04:00 EDTAffine Volterra processeshttps://projecteuclid.org/euclid.aoap/1571385632<strong>Eduardo Abi Jaber</strong>, <strong>Martin Larsson</strong>, <strong>Sergio Pulido</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 3155--3200.</p><p><strong>Abstract:</strong><br/>
We introduce affine Volterra processes, defined as solutions of certain stochastic convolution equations with affine coefficients. Classical affine diffusions constitute a special case, but affine Volterra processes are neither semimartingales, nor Markov processes in general. We provide explicit exponential-affine representations of the Fourier–Laplace functional in terms of the solution of an associated system of deterministic integral equations of convolution type, extending well-known formulas for classical affine diffusions. For specific state spaces, we prove existence, uniqueness, and invariance properties of solutions of the corresponding stochastic convolution equations. Our arguments avoid infinite-dimensional stochastic analysis as well as stochastic integration with respect to non-semimartingales, relying instead on tools from the theory of finite-dimensional deterministic convolution equations. Our findings generalize and clarify recent results in the literature on rough volatility models in finance.
</p>projecteuclid.org/euclid.aoap/1571385632_20191018040042Fri, 18 Oct 2019 04:00 EDTApproximating stationary distributions of fast mixing Glauber dynamics, with applications to exponential random graphshttps://projecteuclid.org/euclid.aoap/1571385633<strong>Gesine Reinert</strong>, <strong>Nathan Ross</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 3201--3229.</p><p><strong>Abstract:</strong><br/>
We provide a general bound on the Wasserstein distance between two arbitrary distributions of sequences of Bernoulli random variables. The bound is in terms of a mixing quantity for the Glauber dynamics of one of the sequences, and a simple expectation of the other. The result is applied to estimate, with explicit error, expectations of functions of random vectors for some Ising models and exponential random graphs in “high temperature” regimes.
</p>projecteuclid.org/euclid.aoap/1571385633_20191018040042Fri, 18 Oct 2019 04:00 EDTStein’s method for stationary distributions of Markov chains and application to Ising modelshttps://projecteuclid.org/euclid.aoap/1571385634<strong>Guy Bresler</strong>, <strong>Dheeraj Nagaraj</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 3230--3265.</p><p><strong>Abstract:</strong><br/>
We develop a new technique, based on Stein’s method, for comparing two stationary distributions of irreducible Markov chains whose update rules are close in a certain sense. We apply this technique to compare Ising models on $d$-regular expander graphs to the Curie–Weiss model (complete graph) in terms of pairwise correlations and more generally $k$th order moments. Concretely, we show that $d$-regular Ramanujan graphs approximate the $k$th order moments of the Curie–Weiss model to within average error $k/\sqrt{d}$ (averaged over size $k$ subsets), independent of graph size . The result applies even in the low-temperature regime; we also derive simpler approximation results for functionals of Ising models that hold only at high temperatures.
</p>projecteuclid.org/euclid.aoap/1571385634_20191018040042Fri, 18 Oct 2019 04:00 EDTCorrection note: A strong order 1/2 method for multidimensional SDEs with discontinuous drifthttps://projecteuclid.org/euclid.aoap/1571385635<strong>Gunther Leobacher</strong>, <strong>Michaela Szölgyenyi</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 3266--3269.</p>projecteuclid.org/euclid.aoap/1571385635_20191018040042Fri, 18 Oct 2019 04:00 EDT