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The latest articles from The Annals of Applied Probability on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTThu, 02 Jun 2011 09:14 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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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 EDTLarge deviation principles for first-order scalar conservation laws with stochastic forcinghttps://projecteuclid.org/euclid.aoap/1582621226<strong>Zhao Dong</strong>, <strong>Jiang-Lun Wu</strong>, <strong>Rangrang Zhang</strong>, <strong>Tusheng Zhang</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 30, Number 1, 324--367.</p><p><strong>Abstract:</strong><br/>
In this paper, we established the Freidlin–Wentzell-type large deviation principles for first-order scalar conservation laws perturbed by small multiplicative noise. Due to the lack of the viscous terms in the stochastic equations, the kinetic solution to the Cauchy problem for these first-order conservation laws is studied. Then, based on the well-posedness of the kinetic solutions, we show that the large deviations holds by utilising the weak convergence approach.
</p>projecteuclid.org/euclid.aoap/1582621226_20200225040028Tue, 25 Feb 2020 04:00 ESTMonte Carlo with determinantal point processeshttps://projecteuclid.org/euclid.aoap/1582621227<strong>Rémi Bardenet</strong>, <strong>Adrien Hardy</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 30, Number 1, 368--417.</p><p><strong>Abstract:</strong><br/>
We show that repulsive random variables can yield Monte Carlo methods with faster convergence rates than the typical $N^{-1/2}$, where $N$ is the number of integrand evaluations. More precisely, we propose stochastic numerical quadratures involving determinantal point processes associated with multivariate orthogonal polynomials, and we obtain root mean square errors that decrease as $N^{-(1+1/d)/2}$, where $d$ is the dimension of the ambient space. First, we prove a central limit theorem (CLT) for the linear statistics of a class of determinantal point processes, when the reference measure is a product measure supported on a hypercube, which satisfies the Nevai-class regularity condition; a result which may be of independent interest. Next, we introduce a Monte Carlo method based on these determinantal point processes, and prove a CLT with explicit limiting variance for the quadrature error, when the reference measure satisfies a stronger regularity condition. As a corollary, by taking a specific reference measure and using a construction similar to importance sampling, we obtain a general Monte Carlo method, which applies to any measure with continuously derivable density. Loosely speaking, our method can be interpreted as a stochastic counterpart to Gaussian quadrature, which at the price of some convergence rate, is easily generalizable to any dimension and has a more explicit error term.
</p>projecteuclid.org/euclid.aoap/1582621227_20200225040028Tue, 25 Feb 2020 04:00 ESTRandom-cluster dynamics in $\mathbb{Z}^{2}$: Rapid mixing with general boundary conditionshttps://projecteuclid.org/euclid.aoap/1582621228<strong>Antonio Blanca</strong>, <strong>Reza Gheissari</strong>, <strong>Eric Vigoda</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 30, Number 1, 418--459.</p><p><strong>Abstract:</strong><br/>
The random-cluster model with parameters $(p,q)$ is a random graph model that generalizes bond percolation ($q=1$) and the Ising and Potts models ($q\geq 2$). We study its Glauber dynamics on $n\times n$ boxes $\Lambda_{n}$ of the integer lattice graph $\mathbb{Z}^{2}$, where the model exhibits a sharp phase transition at $p=p_{c}(q)$. Unlike traditional spin systems like the Ising and Potts models, the random-cluster model has non-local interactions. Long-range interactions can be imposed as external connections in the boundary of $\Lambda_{n}$, known as boundary conditions . For select boundary conditions that do not carry long-range information (namely, wired and free), Blanca and Sinclair proved that when $q>1$ and $p\neq p_{c}(q)$, the Glauber dynamics on $\Lambda_{n}$ mixes in optimal $O(n^{2}\log n)$ time. In this paper, we prove that this mixing time is polynomial in $n$ for every boundary condition that is realizable as a configuration on $\mathbb{Z}^{2}\setminus\Lambda_{n}$. We then use this to prove near-optimal $\tilde{O}(n^{2})$ mixing time for “typical” boundary conditions. As a complementary result, we construct classes of nonrealizable (nonplanar) boundary conditions inducing slow (stretched-exponential) mixing at $p\ll p_{c}$.
</p>projecteuclid.org/euclid.aoap/1582621228_20200225040028Tue, 25 Feb 2020 04:00 ESTThe largest real eigenvalue in the real Ginibre ensemble and its relation to the Zakharov–Shabat systemhttps://projecteuclid.org/euclid.aoap/1582621229<strong>Jinho Baik</strong>, <strong>Thomas Bothner</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 30, Number 1, 460--501.</p><p><strong>Abstract:</strong><br/>
The real Ginibre ensemble consists of $n\times n$ real matrices $\mathbf{X}$ whose entries are i.i.d. standard normal random variables. In sharp contrast to the complex and quaternion Ginibre ensemble, real eigenvalues in the real Ginibre ensemble attain positive likelihood. In turn, the spectral radius $R_{n}=\max_{1\leq j\leq n}|z_{j}(\mathbf{X})|$ of the eigenvalues $z_{j}(\mathbf{X})\in \mathbb{C}$ of a real Ginibre matrix $\mathbf{X}$ follows a different limiting law (as $n\rightarrow \infty $) for $z_{j}(\mathbf{X})\in \mathbb{R}$ than for $z_{j}(\mathbf{X})\in \mathbb{C}\setminus \mathbb{R}$. Building on previous work by Rider and Sinclair ( Ann. Appl. Probab. 24 (2014) 1621–1651) and Poplavskyi, Tribe and Zaboronski ( Ann. Appl. Probab. 27 (2017) 1395–1413), we show that the limiting distribution of $\max_{j:z_{j}\in \mathbb{R}}z_{j}(\mathbf{X})$ admits a closed-form expression in terms of a distinguished solution to an inverse scattering problem for the Zakharov–Shabat system. As byproducts of our analysis, we also obtain a new determinantal representation for the limiting distribution of $\max_{j:z_{j}\in \mathbb{R}}z_{j}(\mathbf{X})$ and extend recent tail estimates in ( Ann. Appl. Probab. 27 (2017) 1395–1413) via nonlinear steepest descent techniques.
</p>projecteuclid.org/euclid.aoap/1582621229_20200225040028Tue, 25 Feb 2020 04:00 ESTLower bounds for trace reconstructionhttps://projecteuclid.org/euclid.aoap/1591603214<strong>Nina Holden</strong>, <strong>Russell Lyons</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 503--525.</p><p><strong>Abstract:</strong><br/>
In the trace reconstruction problem, an unknown bit string ${\mathbf{x}}\in\{0,1\}^{n}$ is sent through a deletion channel where each bit is deleted independently with some probability $q\in(0,1)$, yielding a contracted string ${\widetilde{\mathbf{x}}}$. How many i.i.d. samples of ${\widetilde{\mathbf{x}}}$ are needed to reconstruct ${\mathbf{x}}$ with high probability? We prove that there exist ${\mathbf{x}},{\mathbf{y}}\in\{0,1\}^{n}$ such that at least $cn^{5/4}/\sqrt{\log n}$ traces are required to distinguish between ${\mathbf{x}}$ and ${\mathbf{y}}$ for some absolute constant $c$, improving the previous lower bound of $cn$. Furthermore, our result improves the previously known lower bound for reconstruction of random strings from $c\log^{2}n$ to $c\log^{9/4}n/\sqrt{\log\log n}$.
</p>projecteuclid.org/euclid.aoap/1591603214_20200608040026Mon, 08 Jun 2020 04:00 EDTAdaptive Euler–Maruyama method for SDEs with nonglobally Lipschitz drifthttps://projecteuclid.org/euclid.aoap/1591603215<strong>Wei Fang</strong>, <strong>Michael B. Giles</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 526--560.</p><p><strong>Abstract:</strong><br/>
This paper proposes an adaptive timestep construction for an Euler–Maruyama approximation of SDEs with nonglobally Lipschitz drift. It is proved that if the timestep is bounded appropriately, then over a finite time interval the numerical approximation is stable, and the expected number of timesteps is finite. Furthermore, the order of strong convergence is the same as usual, that is, order $\frac{1}{2}$ for SDEs with a nonuniform globally Lipschitz volatility, and order $1$ for Langevin SDEs with unit volatility and a drift with sufficient smoothness. For a class of ergodic SDEs, we also show that the bound for the moments and the strong error of the numerical solution are uniform in $T$, which allow us to introduce the adaptive multilevel Monte Carlo method to compute the expectations with respect to the invariant distribution. The analysis is supported by numerical experiments.
</p>projecteuclid.org/euclid.aoap/1591603215_20200608040026Mon, 08 Jun 2020 04:00 EDTMean geometry for 2D random fields: Level perimeter and level total curvature integralshttps://projecteuclid.org/euclid.aoap/1591603216<strong>Hermine Biermé</strong>, <strong>Agnès Desolneux</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 561--607.</p><p><strong>Abstract:</strong><br/>
We introduce the level perimeter integral and the total curvature integral associated with a real-valued function $f$ defined on the plane $\mathbb{R}^{2}$, as integrals allowing to compute the perimeter of the excursion set of $f$ above level $t$ and the total (signed) curvature of its boundary for almost every level $t$. Thanks to the Gauss–Bonnet theorem, the total curvature is directly related to the Euler characteristic of the excursion set. We show that the level perimeter and the total curvature integrals can be computed in two different frameworks: smooth (at least $C^{2}$) functions and piecewise constant functions (also called here elementary functions). Considering 2D random fields (in particular shot noise random fields), we compute their mean perimeter and total curvature integrals, and this provides new “explicit” computations of the mean perimeter and Euler characteristic densities of excursion sets, beyond the Gaussian framework: for piecewise constant shot noise random fields, we give some examples of completely explicit formulas, and for smooth shot noise random fields the provided examples are only partly explicit, since the formulas are given under the form of integrals of some special functions.
</p>projecteuclid.org/euclid.aoap/1591603216_20200608040026Mon, 08 Jun 2020 04:00 EDTBounds for the asymptotic distribution of the likelihood ratiohttps://projecteuclid.org/euclid.aoap/1591603217<strong>Andreas Anastasiou</strong>, <strong>Gesine Reinert</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 608--643.</p><p><strong>Abstract:</strong><br/>
In this paper, we give an explicit bound on the distance to chi-square for the likelihood ratio statistic when the data are realisations of independent and identically distributed random elements. To our knowledge, this is the first explicit bound which is available in the literature. The bound depends on the number of samples as well as on the dimension of the parameter space. We illustrate the bound with three examples: samples from an exponential distribution, samples from a normal distribution and logistic regression.
</p>projecteuclid.org/euclid.aoap/1591603217_20200608040026Mon, 08 Jun 2020 04:00 EDTOptimal position targeting via decoupling fieldshttps://projecteuclid.org/euclid.aoap/1591603218<strong>Stefan Ankirchner</strong>, <strong>Alexander Fromm</strong>, <strong>Thomas Kruse</strong>, <strong>Alexandre Popier</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 644--672.</p><p><strong>Abstract:</strong><br/>
We consider a variant of the basic problem of the calculus of variations, where the Lagrangian is convex and subject to randomness adapted to a Brownian filtration. We solve the problem by reducing it, via a limiting argument, to an unconstrained control problem that consists in finding an absolutely continuous process minimizing the expected sum of the Lagrangian and the deviation of the terminal state from a given target position. Using the Pontryagin maximum principle, we characterize a solution of the unconstrained control problem in terms of a fully coupled forward–backward stochastic differential equation (FBSDE). We use the method of decoupling fields for proving that the FBSDE has a unique solution. We exploit a monotonicity property of the decoupling field for solving the original constrained problem and characterize its solution in terms of an FBSDE with a free backward part.
</p>projecteuclid.org/euclid.aoap/1591603218_20200608040026Mon, 08 Jun 2020 04:00 EDTSemi-implicit Euler–Maruyama approximation for noncolliding particle systemshttps://projecteuclid.org/euclid.aoap/1591603219<strong>Hoang-Long Ngo</strong>, <strong>Dai Taguchi</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 673--705.</p><p><strong>Abstract:</strong><br/>
We introduce a semi-implicit Euler–Maruyama approximation which preserves the noncolliding property for some class of noncolliding particle systems such as Dyson–Brownian motions, Dyson–Ornstein–Uhlenbeck processes and Brownian particle systems with nearest neighbor repulsion, and study its rates of convergence in both $L^{p}$-norm and pathwise sense.
</p>projecteuclid.org/euclid.aoap/1591603219_20200608040026Mon, 08 Jun 2020 04:00 EDTTrading with small nonlinear price impacthttps://projecteuclid.org/euclid.aoap/1591603220<strong>Thomas Cayé</strong>, <strong>Martin Herdegen</strong>, <strong>Johannes Muhle-Karbe</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 706--746.</p><p><strong>Abstract:</strong><br/>
We study portfolio choice with small nonlinear price impact on general market dynamics. Using probabilistic techniques and convex duality, we show that the asymptotic optimum can be described explicitly up to the solution of a nonlinear ODE, which identifies the optimal trading speed and the performance loss due to the trading friction. Previous asymptotic results for proportional and quadratic trading costs are obtained as limiting cases. As an illustration, we discuss how nonlinear trading costs affect the pricing and hedging of derivative securities and active portfolio management.
</p>projecteuclid.org/euclid.aoap/1591603220_20200608040026Mon, 08 Jun 2020 04:00 EDTOptimal investment and consumption with labor income in incomplete marketshttps://projecteuclid.org/euclid.aoap/1591603221<strong>Oleksii Mostovyi</strong>, <strong>Mihai Sîrbu</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 747--787.</p><p><strong>Abstract:</strong><br/>
We consider the problem of optimal consumption from labor income and investment in a general incomplete semimartingale market. The economic agent cannot borrow against future income, so the total wealth is required to be positive at (all or some) previous times. Under very general conditions, we show that an optimal consumption and investment plan exists and is unique, and provide a dual characterization in terms of an optional strong supermartingale deflator and a decreasing part, which charges only the times when the no-borrowing constraint is binding. The analysis relies on the infinite-dimensional parametrization of the income/liability streams and, therefore, provides the first-order dependence of the optimal investment and consumption plans on future income/liabilities (as well as a pricing rule).
</p>projecteuclid.org/euclid.aoap/1591603221_20200608040026Mon, 08 Jun 2020 04:00 EDTOriented first passage percolation in the mean field limit, 2. The extremal processhttps://projecteuclid.org/euclid.aoap/1591603222<strong>Nicola Kistler</strong>, <strong>Adrien Schertzer</strong>, <strong>Marius A. Schmidt</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 788--811.</p><p><strong>Abstract:</strong><br/>
This is the second, and last paper in which we address the behavior of oriented first passage percolation on the hypercube in the limit of large dimensions. We prove here that the extremal process converges to a Cox process with exponential intensity. This entails, in particular, that the first passage time converges weakly to a random shift of the Gumbel distribution. The random shift, which has an explicit, universal distribution related to modified Bessel functions of the second kind, is the sole manifestation of correlations ensuing from the geometry of Euclidean space in infinite dimensions. The proof combines the multiscale refinement of the second moment method with a conditional version of the Chen–Stein bounds, and a contraction principle.
</p>projecteuclid.org/euclid.aoap/1591603222_20200608040026Mon, 08 Jun 2020 04:00 EDTNonlinear large deviations: Beyond the hypercubehttps://projecteuclid.org/euclid.aoap/1591603223<strong>Jun Yan</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 812--846.</p><p><strong>Abstract:</strong><br/>
By extending ( Adv. Math. 299 (2016) 396–450), we present a framework to calculate large deviations for nonlinear functions of independent random variables supported on compact sets in Banach spaces. Previous research on nonlinear large deviations has only focused on random variables supported on $\{-1,+1\}^{n}$, and accordingly we build theory for random variables with general distributions, increasing flexibility in the applications. As examples, we compute the large deviation rate functions for monochromatic subgraph counts in edge-colored complete graphs, and for triangle counts in dense random graphs with continuous edge weights. Moreover, we verify the mean field approximation for a class of vector spin models.
</p>projecteuclid.org/euclid.aoap/1591603223_20200608040026Mon, 08 Jun 2020 04:00 EDTA Berry–Esseen theorem for Pitman’s $\alpha $-diversityhttps://projecteuclid.org/euclid.aoap/1591603224<strong>Emanuele Dolera</strong>, <strong>Stefano Favaro</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 847--869.</p><p><strong>Abstract:</strong><br/>
This paper contributes to the study of the random number $K_{n}$ of blocks in the random partition of $\{1,\ldots,n\}$ induced by random sampling from the celebrated two parameter Poisson–Dirichlet process. For any $\alpha \in (0,1)$ and $\theta >-\alpha $ Pitman ( Combinatorial Stochastic Processes (2006) Springer, Berlin) showed that $n^{-\alpha }K_{n}\stackrel{\text{a.s.}}{\longrightarrow }S_{\alpha,\theta }$ as $n\rightarrow +\infty $, where the limiting random variable, referred to as Pitman’s $\alpha $-diversity, is distributed according to a polynomially scaled Mittag–Leffler distribution function. Our main result is a Berry–Esseen theorem for Pitman’s $\alpha $-diversity $S_{\alpha,\theta }$, namely we show that \[\mathop{\mathrm{sup}}_{x\geq 0}\biggl\vert \mathsf{P}\biggl[\frac{K_{n}}{n^{\alpha }}\leq x\biggr]-\mathsf{P}[S_{\alpha,\theta }\leq x]\biggr\vert \leq\frac{C(\alpha,\theta )}{n^{\alpha }}\] holds for every $n\in \mathbb{N}$ with an explicit constant term $C(\alpha,\theta )$, for $\alpha \in (0,1)$ and $\theta >0$. The proof relies on three intermediate novel results which are of independent interest: (i) a probabilistic representation of the distribution of $K_{n}$ in terms of a compound distribution; (ii) a quantitative version of the Laplace’s approximation method for integrals; (iii) a refined quantitative bound for Poisson approximation. An application of our Berry–Esseen theorem is presented in the context of Bayesian nonparametric inference for species sampling problems, quantifying the error of a posterior approximation that has been extensively applied to infer the number of unseen species in a population.
</p>projecteuclid.org/euclid.aoap/1591603224_20200608040026Mon, 08 Jun 2020 04:00 EDTA lower bound on the queueing delay in resource constrained load balancinghttps://projecteuclid.org/euclid.aoap/1591603225<strong>David Gamarnik</strong>, <strong>John N. Tsitsiklis</strong>, <strong>Martin Zubeldia</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 870--901.</p><p><strong>Abstract:</strong><br/>
We consider the following distributed service model: jobs with unit mean, general distribution, and independent processing times arrive as a renewal process of rate $\lambda n$, with $0<\lambda <1$, and are immediately dispatched to one of several queues associated with $n$ identical servers with unit processing rate. We assume that the dispatching decisions are made by a central dispatcher endowed with a finite memory, and with the ability to exchange messages with the servers.
We study the fundamental resource requirements (memory bits and message exchange rate), in order to drive the expected queueing delay in steady-state of a typical job to zero, as $n$ increases. We develop a novel approach to show that, within a certain broad class of “symmetric” policies, every dispatching policy with a message rate of the order of $n$, and with a memory of the order of $\log n$ bits, results in an expected queueing delay which is bounded away from zero, uniformly as $n\to \infty $. This complements existing results which show that, in the absence of such limitations on the memory or the message rate, there exist policies with vanishing queueing delay (at least with Poisson arrivals and exponential service times).
</p>projecteuclid.org/euclid.aoap/1591603225_20200608040026Mon, 08 Jun 2020 04:00 EDTThe social network model on infinite graphshttps://projecteuclid.org/euclid.aoap/1591603226<strong>Jonathan Hermon</strong>, <strong>Ben Morris</strong>, <strong>Chuan Qin</strong>, <strong>Allan Sly</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 902--935.</p><p><strong>Abstract:</strong><br/>
Given an infinite connected regular graph $G=(V,E)$, place at each vertex $\operatorname{Poisson}(\lambda)$ walkers performing independent lazy simple random walks on $G$ simultaneously. When two walkers visit the same vertex at the same time they are declared to be acquainted. We show that when $G$ is vertex-transitive and amenable, for all $\lambda>0$ a.s. any pair of walkers will eventually have a path of acquaintances between them. In contrast, we show that when $G$ is nonamenable (not necessarily transitive) there is always a phase transition at some $\lambda_{\mathrm{c}}(G)>0$. We give general bounds on $\lambda_{\mathrm{c}}(G)$ and study the case that $G$ is the $d$-regular tree in more detail. Finally, we show that in the nonamenable setup, for every $\lambda$ there exists a finite time $t_{\lambda}(G)$ such that a.s. there exists an infinite set of walkers having a path of acquaintances between them by time $t_{\lambda}(G)$.
</p>projecteuclid.org/euclid.aoap/1591603226_20200608040026Mon, 08 Jun 2020 04:00 EDTViscosity solutions to parabolic master equations and McKean–Vlasov SDEs with closed-loop controlshttps://projecteuclid.org/euclid.aoap/1591603227<strong>Cong Wu</strong>, <strong>Jianfeng Zhang</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 936--986.</p><p><strong>Abstract:</strong><br/>
The master equation is a type of PDE whose state variable involves the distribution of certain underlying state process. It is a powerful tool for studying the limit behavior of large interacting systems, including mean field games and systemic risk. It also appears naturally in stochastic control problems with partial information and in time inconsistent problems. In this paper we propose a novel notion of viscosity solution for parabolic master equations, arising mainly from control problems, and establish its wellposedness. Our main innovation is to restrict the involved measures to a certain set of semimartingale measures which satisfy the desired compactness. As an important example, we study the HJB master equation associated with the control problems for McKean–Vlasov SDEs. Due to practical considerations, we consider closed-loop controls. It turns out that the regularity of the value function becomes much more involved in this framework than the counterpart in the standard control problems. Finally, we build the whole theory in the path dependent setting, which is often seen in applications. The main result in this part is an extension of Dupire’s (2009) functional Itô formula. This Itô formula requires a special structure of the derivatives with respect to the measures, which was originally due to Lions in the state dependent case. We provided an elementary proof for this well known result in the short note (2017), and the same arguments work in the path dependent setting here.
</p>projecteuclid.org/euclid.aoap/1591603227_20200608040026Mon, 08 Jun 2020 04:00 EDTKinetically constrained models with random constraintshttps://projecteuclid.org/euclid.aoap/1591603228<strong>Assaf Shapira</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 987--1006.</p><p><strong>Abstract:</strong><br/>
We study two kinetically constrained models in a quenched random environment. The first model is a mixed threshold Fredrickson–Andersen model on $\mathbb{Z}^{2}$, where the update threshold is either $1$ or $2$. The second is a mixture of the Fredrickson–Andersen $1$-spin facilitated constraint and the North-East constraint in $\mathbb{Z}^{2}$. We compare three time scales related to these models—the bootstrap percolation time for emptying the origin, the relaxation time of the kinetically constrained model, and the time for emptying the origin of the kinetically constrained model—and understand the effect of the random environment on each of them.
</p>projecteuclid.org/euclid.aoap/1591603228_20200608040026Mon, 08 Jun 2020 04:00 EDTGlobal $C^{1}$ regularity of the value function in optimal stopping problemshttps://projecteuclid.org/euclid.aoap/1596009615<strong>T. De Angelis</strong>, <strong>G. Peskir</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 3, 1007--1031.</p><p><strong>Abstract:</strong><br/>
We show that if either the process is strong Feller and the boundary point is probabilistically regular for the stopping set, or the process is strong Markov and the boundary point is probabilistically regular for the interior of the stopping set, then the boundary point is Green regular for the stopping set. Combining this implication with the existence of a continuously differentiable flow of the process we show that the value function is continuously differentiable at the optimal stopping boundary whenever the gain function is so. The derived fact holds both in the parabolic and elliptic case of the boundary value problem under the sole hypothesis of probabilistic regularity of the optimal stopping boundary, thus improving upon known analytic results in the PDE literature, and establishing the fact for the first time in the case of integro-differential equations. The method of proof is purely probabilistic and conceptually simple. Examples of application include the first known probabilistic proof of the fact that the time derivative of the value function in the American put problem is continuous across the optimal stopping boundary.
</p>projecteuclid.org/euclid.aoap/1596009615_20200729040025Wed, 29 Jul 2020 04:00 EDTOptimal real-time detection of a drifting Brownian coordinatehttps://projecteuclid.org/euclid.aoap/1596009616<strong>P. A. Ernst</strong>, <strong>G. Peskir</strong>, <strong>Q. Zhou</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 3, 1032--1065.</p><p><strong>Abstract:</strong><br/>
Consider the motion of a Brownian particle in three dimensions, whose two spatial coordinates are standard Brownian motions with zero drift, and the remaining (unknown) spatial coordinate is a standard Brownian motion with a (known) nonzero drift. Given that the position of the Brownian particle is being observed in real time, the problem is to detect as soon as possible and with minimal probabilities of the wrong terminal decisions, which spatial coordinate has the nonzero drift. We solve this problem in the Bayesian formulation, under any prior probabilities of the nonzero drift being in any of the three spatial coordinates, when the passage of time is penalised linearly. Finding the exact solution to the problem in three dimensions, including a rigorous treatment of its nonmonotone optimal stopping boundaries, is the main contribution of the present paper. To our knowledge this is the first time that such a problem has been solved in the literature.
</p>projecteuclid.org/euclid.aoap/1596009616_20200729040025Wed, 29 Jul 2020 04:00 EDTAn information-percolation bound for spin synchronization on general graphshttps://projecteuclid.org/euclid.aoap/1596009617<strong>Emmanuel Abbe</strong>, <strong>Enric Boix-Adserà</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 3, 1066--1090.</p><p><strong>Abstract:</strong><br/>
This paper considers the problem of reconstructing $n$ independent uniform spins $X_{1},\dots,X_{n}$ living on the vertices of an $n$-vertex graph $G$, by observing their interactions on the edges of the graph. This captures instances of models such as (i) broadcasting on trees, (ii) block models, (iii) synchronization on grids, (iv) spiked Wigner models. The paper gives an upper bound on the mutual information between two vertices in terms of a bond percolation estimate. Namely, the information between two vertices’ spins is bounded by the probability that these vertices are connected when edges are opened with a probability that “emulates” the edge-information. Both the information and the open-probability are based on the Chi-squared mutual information. The main results allow us to re-derive known results for information-theoretic nonreconstruction in models (i)–(iv), with more direct or improved bounds in some cases, and to obtain new results, such as for a spiked Wigner model on grids. The main result also implies a new subadditivity property for the Chi-squared mutual information for symmetric channels and general graphs, extending the subadditivity property obtained by Evans–Kenyon–Peres–Schulman ( Ann. Appl. Probab. 10 (2000) 410–433) for trees. Some cases of nonsymmetrical channels are also discussed.
</p>projecteuclid.org/euclid.aoap/1596009617_20200729040025Wed, 29 Jul 2020 04:00 EDTSpatial growth processes with long range dispersion: Microscopics, mesoscopics and discrepancy in spread ratehttps://projecteuclid.org/euclid.aoap/1596009618<strong>Viktor Bezborodov</strong>, <strong>Luca Di Persio</strong>, <strong>Tyll Krueger</strong>, <strong>Pasha Tkachov</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 3, 1091--1129.</p><p><strong>Abstract:</strong><br/>
We consider the speed of propagation of a continuous-time continuous-space branching random walk with the additional restriction that the birth rate at any spatial point cannot exceed $1$. The dispersion kernel is taken to have density that decays polynomially as $|x|^{-2\alpha }$, $x\to \infty $. We show that if $\alpha >2$, then the system spreads at a linear speed, while for $\alpha \in (\frac{1}{2},2]$ the spread is faster than linear. We also consider the mesoscopic equation corresponding to the microscopic stochastic system. We show that in contrast to the microscopic process, the solution to the mesoscopic equation spreads exponentially fast for every $\alpha >\frac{1}{2}$.
</p>projecteuclid.org/euclid.aoap/1596009618_20200729040025Wed, 29 Jul 2020 04:00 EDTOn the exit time from open sets of some semi-Markov processeshttps://projecteuclid.org/euclid.aoap/1596009619<strong>Giacomo Ascione</strong>, <strong>Enrica Pirozzi</strong>, <strong>Bruno Toaldo</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 3, 1130--1163.</p><p><strong>Abstract:</strong><br/>
In this paper we characterize the distribution of the first exit time from an arbitrary open set for a class of semi-Markov processes obtained as time-changed Markov processes. We estimate the asymptotic behaviour of the survival function (for large $t$) and of the distribution function (for small $t$) and we provide some conditions for absolute continuity. We have been inspired by a problem of neurophyshiology and our results are particularly usefull in this field, precisely for the so-called Leaky Integrate-and-Fire (LIF) models: the use of semi-Markov processes in these models appear to be realistic under several aspects, for example, it makes the intertimes between spikes a r.v. with infinite expectation, which is a desiderable property. Hence, after the theoretical part, we provide a LIF model based on semi-Markov processes.
</p>projecteuclid.org/euclid.aoap/1596009619_20200729040025Wed, 29 Jul 2020 04:00 EDTThermalisation for small random perturbations of dynamical systemshttps://projecteuclid.org/euclid.aoap/1596009620<strong>Gerardo Barrera</strong>, <strong>Milton Jara</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 3, 1164--1208.</p><p><strong>Abstract:</strong><br/>
We consider an ordinary differential equation with a unique hyperbolic attractor at the origin, to which we add a small random perturbation. It is known that under general conditions, the solution of this stochastic differential equation converges exponentially fast to an equilibrium distribution. We show that the convergence occurs abruptly: in a time window of small size compared to the natural time scale of the process, the distance to equilibrium drops from its maximal possible value to near zero, and only after this time window the convergence is exponentially fast. This is what is known as the cut-off phenomenon in the context of Markov chains of increasing complexity. In addition, we are able to give general conditions to decide whether the distance to equilibrium converges in this time window to a universal function, a fact known as profile cut-off.
</p>projecteuclid.org/euclid.aoap/1596009620_20200729040025Wed, 29 Jul 2020 04:00 EDTCoupling and convergence for Hamiltonian Monte Carlohttps://projecteuclid.org/euclid.aoap/1596009621<strong>Nawaf Bou-Rabee</strong>, <strong>Andreas Eberle</strong>, <strong>Raphael Zimmer</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 3, 1209--1250.</p><p><strong>Abstract:</strong><br/>
Based on a new coupling approach, we prove that the transition step of the Hamiltonian Monte Carlo algorithm is contractive w.r.t. a carefully designed Kantorovich ($L^{1}$ Wasserstein) distance. The lower bound for the contraction rate is explicit. Global convexity of the potential is not required, and thus multimodal target distributions are included. Explicit quantitative bounds for the number of steps required to approximate the stationary distribution up to a given error $\epsilon $ are a direct consequence of contractivity. These bounds show that HMC can overcome diffusive behavior if the duration of the Hamiltonian dynamics is adjusted appropriately.
</p>projecteuclid.org/euclid.aoap/1596009621_20200729040025Wed, 29 Jul 2020 04:00 EDTThe inverse first passage time problem for killed Brownian motionhttps://projecteuclid.org/euclid.aoap/1596009622<strong>Boris Ettinger</strong>, <strong>Alexandru Hening</strong>, <strong>Tak Kwong Wong</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 3, 1251--1275.</p><p><strong>Abstract:</strong><br/>
The classical inverse first passage time problem asks whether, for a Brownian motion $(B_{t})_{t\geq0}$ and a positive random variable $\xi$, there exists a barrier $b:\mathbb{R}_{+}\to\mathbb{R}$ such that $\mathbb{P}\{B_{s}>b(s),0\leq s\leq t\}=\mathbb{P}\{\xi>t\}$, for all $t\geq0$. We study a variant of the inverse first passage time problem for killed Brownian motion. We show that if $\lambda>0$ is a killing rate parameter and $𝟙_{(-\infty,0]}$ is the indicator of the set $(-\infty,0]$ then, under certain compatibility assumptions, there exists a unique continuous function $b:\mathbb{R}_{+}\to\mathbb{R}$ such that $\mathbb{E}[-\lambda\int_{0}^{t}𝟙_{(-\infty,0]}(B_{s}-b(s))\,ds]=\mathbb{P}\{\zeta>t\}$ holds for all $t\geq0$. This is a significant improvement of a result of the first two authors ( Ann. Appl. Probab. 24 (2014) 1–33).
The main difficulty arises because $𝟙_{(-\infty,0]}$ is discontinuous. We associate a semilinear parabolic partial differential equation (PDE) coupled with an integral constraint to this version of the inverse first passage time problem. We prove the existence and uniqueness of weak solutions to this constrained PDE system. In addition, we use the recent Feynman–Kac representation results of Glau ( Finance Stoch. 20 (2016) 1021–1059) to prove that the weak solutions give the correct probabilistic interpretation.
</p>projecteuclid.org/euclid.aoap/1596009622_20200729040025Wed, 29 Jul 2020 04:00 EDTTransport-information inequalities for Markov chainshttps://projecteuclid.org/euclid.aoap/1596009623<strong>Neng-Yi Wang</strong>, <strong>Liming Wu</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 3, 1276--1320.</p><p><strong>Abstract:</strong><br/>
This paper is the discrete time counterpart of the previous work in the continuous time case by Guillin, Léonard, the second named author and Yao [ Probab. Theory Related Fields 144 (2009), 669–695]. We investigate the following transport-information $T_{\mathcal{V}}I$ inequality: $\alpha (T_{\mathcal{V}}(\nu ,\mu ))\le I(\nu |P,\mu )$ for all probability measures $\nu $ on some metric space $(\mathcal{X},d)$, where $\mu $ is an invariant and ergodic probability measure of some given transition kernel $P(x,dy)$, $T_{\mathcal{V}}(\nu ,\mu )$ is some transportation cost from $\nu $ to $\mu $, $I(\nu |P,\mu )$ is the Donsker–Varadhan information of $\nu $ with respect to $(P,\mu )$ and $\alpha :[0,\infty )\to [0,\infty ]$ is some left continuous increasing function. Using large deviation techniques, we show that $T_{\mathcal{V}}I$ is equivalent to some concentration inequality for the occupation measure of the $\mu $-reversible Markov chain $(X_{n})_{n\ge 0}$ with transition probability $P(x,dy)$. Its relationships with the transport-entropy inequalities are discussed. Several easy-to-check sufficient conditions are provided for $T_{\mathcal{V}}I$. We show the usefulness and sharpness of our general results by a number of applications and examples. The main difficulty resides in the fact that the information $I(\nu |P,\mu )$ has no closed expression, contrary to the continuous time or independent and identically distributed case.
</p>projecteuclid.org/euclid.aoap/1596009623_20200729040025Wed, 29 Jul 2020 04:00 EDTNonexponential Sanov and Schilder theorems on Wiener space: BSDEs, Schrödinger problems and controlhttps://projecteuclid.org/euclid.aoap/1596009624<strong>Julio Backhoff-Veraguas</strong>, <strong>Daniel Lacker</strong>, <strong>Ludovic Tangpi</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 3, 1321--1367.</p><p><strong>Abstract:</strong><br/>
We derive new limit theorems for Brownian motion, which can be seen as nonexponential analogues of the large deviation theorems of Sanov and Schilder in their Laplace principle forms. As a first application, we obtain novel scaling limits of backward stochastic differential equations and their related partial differential equations. As a second application, we extend prior results on the small-noise limit of the Schrödinger problem as an optimal transport cost, unifying the control-theoretic and probabilistic approaches initiated respectively by T. Mikami and C. Léonard. Lastly, our results suggest a new scheme for the computation of mean field optimal control problems, distinct from the conventional particle approximation. A key ingredient in our analysis is an extension of the classical variational formula (often attributed to Borell or Boué–Dupuis) for the Laplace transform of Wiener measure.
</p>projecteuclid.org/euclid.aoap/1596009624_20200729040025Wed, 29 Jul 2020 04:00 EDTThe coalescent structure of continuous-time Galton–Watson treeshttps://projecteuclid.org/euclid.aoap/1596009625<strong>Simon C. Harris</strong>, <strong>Samuel G. G. Johnston</strong>, <strong>Matthew I. Roberts</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 3, 1368--1414.</p><p><strong>Abstract:</strong><br/>
Take a continuous-time Galton–Watson tree. If the system survives until a large time $T$, then choose $k$ particles uniformly from those alive. What does the ancestral tree drawn out by these $k$ particles look like? Some special cases are known but we give a more complete answer. We concentrate on near-critical cases where the mean number of offspring is $1+\mu/T$ for some $\mu\in\mathbb{R}$, and show that a scaling limit exists as $T\to\infty$. Viewed backwards in time, the resulting coalescent process is topologically equivalent to Kingman’s coalescent, but the times of coalescence have an interesting and highly nontrivial structure. The randomly fluctuating population size, as opposed to constant size populations where the Kingman coalescent more usually arises, have a pronounced effect on both the results and the method of proof required. We give explicit formulas for the distribution of the coalescent times, as well as a construction of the genealogical tree involving a mixture of independent and identically distributed random variables. In general subcritical and supercritical cases it is not possible to give such explicit formulas, but we highlight the special case of birth–death processes.
</p>projecteuclid.org/euclid.aoap/1596009625_20200729040025Wed, 29 Jul 2020 04:00 EDTZero-sum path-dependent stochastic differential games in weak formulationhttps://projecteuclid.org/euclid.aoap/1596009626<strong>Dylan Possamaï</strong>, <strong>Nizar Touzi</strong>, <strong>Jianfeng Zhang</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 3, 1415--1457.</p><p><strong>Abstract:</strong><br/>
We consider zero-sum stochastic differential games with possibly path-dependent volatility controls. Unlike the previous literature, we allow for weak solutions of the state equation so that the players’ controls are automatically of feedback type. In particular, we do not require the controls to be “simple,” which has fundamental importance for the possible existence of saddle-points. Under some restrictions, needed for the a priori regularity of the upper and lower value functions of the game, we show that the game value exists when both the appropriate path-dependent Isaacs condition, and the uniqueness of viscosity solutions of the corresponding path-dependent Isaacs-HJB equation hold. We also provide a general verification argument and a characterisation of saddle-points by means of an appropriate notion of second-order backward SDE.
</p>projecteuclid.org/euclid.aoap/1596009626_20200729040025Wed, 29 Jul 2020 04:00 EDTOptimal Cheeger cuts and bisections of random geometric graphshttps://projecteuclid.org/euclid.aoap/1596009627<strong>Tobias Müller</strong>, <strong>Mathew D. Penrose</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 3, 1458--1483.</p><p><strong>Abstract:</strong><br/>
Let $d\geq 2$. The Cheeger constant of a graph is the minimum surface-to-volume ratio of all subsets of the vertex set with relative volume at most 1/2. There are several ways to define surface and volume here: the simplest method is to count boundary edges (for the surface) and vertices (for the volume). We show that for a geometric (possibly weighted) graph on $n$ random points in a $d$-dimensional domain with Lipschitz boundary and with distance parameter decaying more slowly (as a function of $n$) than the connectivity threshold, the Cheeger constant (under several possible definitions of surface and volume), also known as conductance, suitably rescaled, converges for large $n$ to an analogous Cheeger-type constant of the domain. Previously, García Trillos et al. had shown this for $d\geq 3$ but had required an extra condition on the distance parameter when $d=2$.
</p>projecteuclid.org/euclid.aoap/1596009627_20200729040025Wed, 29 Jul 2020 04:00 EDTRandom permutations without macroscopic cycleshttps://projecteuclid.org/euclid.aoap/1596009628<strong>Volker Betz</strong>, <strong>Helge Schäfer</strong>, <strong>Dirk Zeindler</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 3, 1484--1505.</p><p><strong>Abstract:</strong><br/>
We consider uniform random permutations of length $n$ conditioned to have no cycle longer than $n^{\beta }$ with $0<\beta <1$, in the limit of large $n$. Since in unconstrained uniform random permutations most of the indices are in cycles of macroscopic length, this is a singular conditioning in the limit. Nevertheless, we obtain a fairly complete picture about the cycle number distribution at various lengths. Depending on the scale at which cycle numbers are studied, our results include Poisson convergence, a central limit theorem, a shape theorem and two different functional central limit theorems.
</p>projecteuclid.org/euclid.aoap/1596009628_20200729040025Wed, 29 Jul 2020 04:00 EDTSensitivity analysis for rare events based on Rényi divergencehttps://projecteuclid.org/euclid.aoap/1596528014<strong>Paul Dupuis</strong>, <strong>Markos A. Katsoulakis</strong>, <strong>Yannis Pantazis</strong>, <strong>Luc Rey-Bellet</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 4, 1507--1533.</p><p><strong>Abstract:</strong><br/>
Rare events play a key role in many applications and numerous algorithms have been proposed for estimating the probability of a rare event. However, relatively little is known on how to quantify the sensitivity of the rare event’s probability with respect to model parameters. In this paper, instead of the direct statistical estimation of rare event sensitivities, we develop novel and general uncertainty quantification and sensitivity bounds which are not tied to specific rare event simulation methods and which apply to families of rare events. Our method is based on a recently derived variational representation for the family of Rényi divergences in terms of risk sensitive functionals associated with the rare events under consideration. Inspired by the derived bounds, we propose new sensitivity indices for rare events and relate them to the moment generating function of the score function. The bounds scale in such a way that we additionally develop sensitivity indices for large deviation rate functions.
</p>projecteuclid.org/euclid.aoap/1596528014_20200804040033Tue, 04 Aug 2020 04:00 EDTNonasymptotic bounds for sampling algorithms without log-concavityhttps://projecteuclid.org/euclid.aoap/1596528015<strong>Mateusz B. Majka</strong>, <strong>Aleksandar Mijatović</strong>, <strong>Łukasz Szpruch</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 4, 1534--1581.</p><p><strong>Abstract:</strong><br/>
Discrete time analogues of ergodic stochastic differential equations (SDEs) are one of the most popular and flexible tools for sampling high-dimensional probability measures. Non-asymptotic analysis in the $L^{2}$ Wasserstein distance of sampling algorithms based on Euler discretisations of SDEs has been recently developed by several authors for log-concave probability distributions. In this work we replace the log-concavity assumption with a log-concavity at infinity condition. We provide novel $L^{2}$ convergence rates for Euler schemes, expressed explicitly in terms of problem parameters. From there we derive nonasymptotic bounds on the distance between the laws induced by Euler schemes and the invariant laws of SDEs, both for schemes with standard and with randomised (inaccurate) drifts. We also obtain bounds for the hierarchy of discretisation, which enables us to deploy a multi-level Monte Carlo estimator. Our proof relies on a novel construction of a coupling for the Markov chains that can be used to control both the $L^{1}$ and $L^{2}$ Wasserstein distances simultaneously. Finally, we provide a weak convergence analysis that covers both the standard and the randomised (inaccurate) drift case. In particular, we reveal that the variance of the randomised drift does not influence the rate of weak convergence of the Euler scheme to the SDE.
</p>projecteuclid.org/euclid.aoap/1596528015_20200804040033Tue, 04 Aug 2020 04:00 EDTHydrodynamic limit and propagation of chaos for Brownian particles reflecting from a Newtonian barrierhttps://projecteuclid.org/euclid.aoap/1596528016<strong>Clayton L. Barnes</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 4, 1582--1613.</p><p><strong>Abstract:</strong><br/>
In 2001, Frank Knight constructed a stochastic process modeling the one-dimensional interaction of two particles, one being Newtonian in the sense that it obeys Newton’s laws of motion, and the other particle being Brownian. We construct a multi-particle analog, using Skorohod map estimates in proving a propagation of chaos, and characterizing the hydrodynamic limit as the solution to a PDE with free boundary condition. This PDE resembles the Stefan problem but has a Neumann type boundary condition. Stochastic methods are used to show existence and uniqueness for this free boundary problem.
</p>projecteuclid.org/euclid.aoap/1596528016_20200804040033Tue, 04 Aug 2020 04:00 EDTRandom walk on random walks: Low densitieshttps://projecteuclid.org/euclid.aoap/1596528017<strong>Oriane Blondel</strong>, <strong>Marcelo R. Hilário</strong>, <strong>Renato S. dos Santos</strong>, <strong>Vladas Sidoravicius</strong>, <strong>Augusto Teixeira</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 4, 1614--1641.</p><p><strong>Abstract:</strong><br/>
We consider a random walker in a dynamic random environment given by a system of independent discrete-time simple symmetric random walks. We obtain ballisticity results under two types of perturbations: low particle density, and strong local drift on particles. Surprisingly, the random walker may behave very differently depending on whether the underlying environment particles perform lazy or nonlazy random walks, which is related to a notion of permeability of the system. We also provide a strong law of large numbers, a functional central limit theorem and large deviation bounds under an ellipticity condition.
</p>projecteuclid.org/euclid.aoap/1596528017_20200804040033Tue, 04 Aug 2020 04:00 EDTHigh-dimensional limits of eigenvalue distributions for general Wishart processhttps://projecteuclid.org/euclid.aoap/1596528018<strong>Jian Song</strong>, <strong>Jianfeng Yao</strong>, <strong>Wangjun Yuan</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 4, 1642--1668.</p><p><strong>Abstract:</strong><br/>
In this article, we obtain an equation for the high-dimensional limit measure of eigenvalues of generalized Wishart processes, and the results are extended to random particle systems that generalize SDEs of eigenvalues. We also introduce a new set of conditions on the coefficient matrices for the existence and uniqueness of a strong solution for the SDEs of eigenvalues. The equation of the limit measure is further discussed assuming self-similarity on the eigenvalues.
</p>projecteuclid.org/euclid.aoap/1596528018_20200804040033Tue, 04 Aug 2020 04:00 EDTConditional optimal stopping: A time-inconsistent optimizationhttps://projecteuclid.org/euclid.aoap/1596528019<strong>Marcel Nutz</strong>, <strong>Yuchong Zhang</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 4, 1669--1692.</p><p><strong>Abstract:</strong><br/>
Inspired by recent work of P.-L. Lions on conditional optimal control, we introduce a problem of optimal stopping under bounded rationality: the objective is the expected payoff at the time of stopping, conditioned on another event. For instance, an agent may care only about states where she is still alive at the time of stopping, or a company may condition on not being bankrupt. We observe that conditional optimization is time-inconsistent due to the dynamic change of the conditioning probability and develop an equilibrium approach in the spirit of R. H. Strotz’ work for sophisticated agents in discrete time. Equilibria are found to be essentially unique in the case of a finite time horizon whereas an infinite horizon gives rise to nonuniqueness and other interesting phenomena. We also introduce a theory which generalizes the classical Snell envelope approach for optimal stopping by considering a pair of processes with Snell-type properties.
</p>projecteuclid.org/euclid.aoap/1596528019_20200804040033Tue, 04 Aug 2020 04:00 EDTOn the convergence of closed-loop Nash equilibria to the mean field game limithttps://projecteuclid.org/euclid.aoap/1596528020<strong>Daniel Lacker</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 4, 1693--1761.</p><p><strong>Abstract:</strong><br/>
This paper continues the study of the mean field game (MFG) convergence problem: In what sense do the Nash equilibria of $n$-player stochastic differential games converge to the mean field game as $n\rightarrow\infty $? Previous work on this problem took two forms. First, when the $n$-player equilibria are open-loop, compactness arguments permit a characterization of all limit points of $n$-player equilibria as weak MFG equilibria, which contain additional randomness compared to the standard (strong) equilibrium concept. On the other hand, when the $n$-player equilibria are closed-loop, the convergence to the MFG equilibrium is known only when the MFG equilibrium is unique and the associated “master equation” is solvable and sufficiently smooth. This paper adapts the compactness arguments to the closed-loop case, proving a convergence theorem that holds even when the MFG equilibrium is nonunique. Every limit point of $n$-player equilibria is shown to be the same kind of weak MFG equilibrium as in the open-loop case. Some partial results and examples are discussed for the converse question, regarding which of the weak MFG equilibria can arise as the limit of $n$-player (approximate) equilibria.
</p>projecteuclid.org/euclid.aoap/1596528020_20200804040033Tue, 04 Aug 2020 04:00 EDTOn Bayesian consistency for flows observed through a passive scalarhttps://projecteuclid.org/euclid.aoap/1596528021<strong>Jeff Borggaard</strong>, <strong>Nathan Glatt-Holtz</strong>, <strong>Justin Krometis</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 4, 1762--1783.</p><p><strong>Abstract:</strong><br/>
We consider the statistical inverse problem of estimating a background fluid flow field $\mathbf{v}$ from the partial, noisy observations of the concentration $\theta$ of a substance passively advected by the fluid, so that $\theta$ is governed by the partial differential equation \begin{equation*}\frac{\partial}{\partial t}{\theta}(t,\mathbf{x})=-\mathbf{v}(\mathbf{x})\cdot\nabla\theta(t,\mathbf{x})+\kappa\Delta\theta(t,\mathbf{x}),\quad\theta(0,\mathbf{x})=\theta_{0}(\mathbf{x})\end{equation*} for $t\in[0,T],T>0$ and $\mathbf{x}\in\mathbb{T}^{2}=[0,1]^{2}$. The initial condition $\theta_{0}$ and diffusion coefficient $\kappa$ are assumed to be known and the data consist of point observations of the scalar field $\theta$ corrupted by additive, i.i.d. Gaussian noise. We adopt a Bayesian approach to this estimation problem and establish that the inference is consistent, that is, that the posterior measure identifies the true background flow as the number of scalar observations grows large. Since the inverse map is ill-defined for some classes of problems even for perfect, infinite measurements of $\theta$, multiple experiments (initial conditions) are required to resolve the true fluid flow. Under this assumption, suitable conditions on the observation points, and given support and tail conditions on the prior measure, we show that the posterior measure converges to a Dirac measure centered on the true flow as the number of observations goes to infinity.
</p>projecteuclid.org/euclid.aoap/1596528021_20200804040033Tue, 04 Aug 2020 04:00 EDTApproximation schemes for viscosity solutions of fully nonlinear stochastic partial differential equationshttps://projecteuclid.org/euclid.aoap/1596528022<strong>Benjamin Seeger</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 4, 1784--1823.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to develop a general method for constructing approximation schemes for viscosity solutions of fully nonlinear pathwise stochastic partial differential equations, and for proving their convergence. Our results apply to approximations such as explicit finite difference schemes and Trotter–Kato type mixing formulas. The irregular time dependence disrupts the usual methods from the classical viscosity theory for creating schemes that are both monotone and convergent, an obstacle that cannot be overcome by incorporating higher order correction terms, as is done for numerical approximations of stochastic or rough ordinary differential equations. The novelty here is to regularize those driving paths with nontrivial quadratic variation in order to guarantee both monotonicity and convergence.
We present qualitative and quantitative results, the former covering a wide variety of schemes for second-order equations. An error estimate is established in the Hamilton–Jacobi case, its merit being that it depends on the path only through the modulus of continuity, and not on the derivatives or total variation. As a result, it is possible to choose a regularization of the path so as to obtain efficient rates of convergence. This is demonstrated in the specific setting of equations with multiplicative white noise in time, in which case the convergence holds with probability one. We also present an example using scaled random walks that exhibits convergence in distribution.
</p>projecteuclid.org/euclid.aoap/1596528022_20200804040033Tue, 04 Aug 2020 04:00 EDTA threshold for cutoff in two-community random graphshttps://projecteuclid.org/euclid.aoap/1596528023<strong>Anna Ben-Hamou</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 4, 1824--1846.</p><p><strong>Abstract:</strong><br/>
In this paper, we are interested in the impact of communities on the mixing behavior of the nonbacktracking random walk. We consider sequences of sparse random graphs of size $N$ generated according to a variant of the classical configuration model which incorporates a two-community structure. The strength of the bottleneck is measured by a parameter $\alpha $ which roughly corresponds to the fraction of edges that go from one community to the other. We show that if $\alpha\gg \frac{1}{\log N}$, then the nonbacktracking random walk exhibits cutoff at the same time as in the one-community case, but with a larger cutoff window, and that the distance profile inside this window converges to the Gaussian tail function. On the other hand, if $\alpha \ll \frac{1}{\log N}$ or $\alpha \asymp \frac{1}{\log N}$, then the mixing time is of order $1/\alpha $ and there is no cutoff.
</p>projecteuclid.org/euclid.aoap/1596528023_20200804040033Tue, 04 Aug 2020 04:00 EDTMixing time and cutoff for the weakly asymmetric simple exclusion processhttps://projecteuclid.org/euclid.aoap/1596528024<strong>Cyril Labbé</strong>, <strong>Hubert Lacoin</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 4, 1847--1883.</p><p><strong>Abstract:</strong><br/>
We consider the simple exclusion process with $k$ particles on a segment of length $N$ performing random walks with transition $p>1/2$ to the right and $q=1-p$ to the left. We focus on the case where the asymmetry in the jump rates $b=p-q>0$ vanishes in the limit when $N$ and $k$ tend to infinity, and obtain sharp asymptotics for the mixing times of this sequence of Markov chains in the two cases where the asymmetry is either much larger or much smaller than $(\log k)/N$. We show that in the former case ($b\gg (\log k)/N$), the mixing time corresponds to the time needed to reach macroscopic equilibrium, like for the strongly asymmetric (i.e., constant $b$) case studied in ( Ann. Probab. 47 (2019) 1541–1586), while the latter case ($b\ll (\log k)/N$) macroscopic equilibrium is not sufficient for mixing and one must wait till local fluctuations equilibrate, similarly to what happens in the symmetric case worked out in ( Ann. Probab. 44 (2016) 1426–1487). In both cases, convergence to equilibrium is abrupt: we have a cutoff phenomenon for the total-variation distance. We present a conjecture for the remaining regime when the asymmetry is of order $(\log k)/N$.
</p>projecteuclid.org/euclid.aoap/1596528024_20200804040033Tue, 04 Aug 2020 04:00 EDTParticles systems and numerical schemes for mean reflected stochastic differential equationshttps://projecteuclid.org/euclid.aoap/1596528025<strong>Philippe Briand</strong>, <strong>Paul-Éric Chaudru de Raynal</strong>, <strong>Arnaud Guillin</strong>, <strong>Céline Labart</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 4, 1884--1909.</p><p><strong>Abstract:</strong><br/>
This paper is devoted to the study of reflected Stochastic Differential Equations when the constraint is not on the paths of the solution but acts on its law. These reflected equations have been introduced recently in a backward form by Briand, Elie and Hu ( Ann. Appl. Probab. 28 (2018) 482–510) in the context of risk measures. We here focus on the forward version of such reflected equations. Our main objective is to provide an approximation of the solutions with the help of interacting particles systems. This approximation allows to design a numerical scheme for this kind of equations.
</p>projecteuclid.org/euclid.aoap/1596528025_20200804040033Tue, 04 Aug 2020 04:00 EDTStatistical thresholds for tensor PCAhttps://projecteuclid.org/euclid.aoap/1596528026<strong>Aukosh Jagannath</strong>, <strong>Patrick Lopatto</strong>, <strong>Léo Miolane</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 4, 1910--1933.</p><p><strong>Abstract:</strong><br/>
We study the statistical limits of testing and estimation for a rank one deformation of a Gaussian random tensor. We compute the sharp thresholds for hypothesis testing and estimation by maximum likelihood and show that they are the same. Furthermore, we find that the maximum likelihood estimator achieves the maximal correlation with the planted vector among measurable estimators above the estimation threshold. In this setting, the maximum likelihood estimator exhibits a discontinuous BBP-type transition: below the critical threshold the estimator is orthogonal to the planted vector, but above the critical threshold, it achieves positive correlation which is uniformly bounded away from zero.
</p>projecteuclid.org/euclid.aoap/1596528026_20200804040033Tue, 04 Aug 2020 04:00 EDTExact formulas for two interacting particles and applications in particle systems with dualityhttps://projecteuclid.org/euclid.aoap/1596528027<strong>Gioia Carinci</strong>, <strong>Cristian Giardinà</strong>, <strong>Frank Redig</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 4, 1934--1970.</p><p><strong>Abstract:</strong><br/>
We consider two particles performing continuous-time nearest neighbor random walk on $\mathbb{Z}$ and interacting with each other when they are at neighboring positions. The interaction is either repulsive (partial exclusion process) or attractive (inclusion process). We provide an exact formula for the Laplace–Fourier transform of the transition probabilities of the two-particle dynamics. From this we derive a general scaling limit result, which shows that the possible scaling limits are coalescing Brownian motions, reflected Brownian motions and sticky Brownian motions.
In particle systems with duality, the solution of the dynamics of two dual particles provides relevant information. We apply the exact formula to the the symmetric inclusion process, that is self-dual, in the condensation regime . We thus obtain two results. First, by computing the time-dependent covariance of the particle occupation number at two lattice sites we characterise the time-dependent coarsening in infinite volume when the process is started from a homogeneous product measure. Second, we identify the limiting variance of the density field in the diffusive scaling limit, relating it to the local time of sticky Brownian motion.
</p>projecteuclid.org/euclid.aoap/1596528027_20200804040033Tue, 04 Aug 2020 04:00 EDTEdgeworth expansion for Euler approximation of continuous diffusion processeshttps://projecteuclid.org/euclid.aoap/1596528028<strong>Mark Podolskij</strong>, <strong>Bezirgen Veliyev</strong>, <strong>Nakahiro Yoshida</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 4, 1971--2003.</p><p><strong>Abstract:</strong><br/>
In this paper we present the Edgeworth expansion for the Euler approximation scheme of a continuous diffusion process driven by a Brownian motion. Our methodology is based upon a recent work ( Stochastic Process. Appl. 123 (2013) 887–933), which establishes Edgeworth expansions associated with asymptotic mixed normality using elements of Malliavin calculus. Potential applications of our theoretical results include higher order expansions for weak and strong approximation errors associated to the Euler scheme, and for studentized version of the error process.
</p>projecteuclid.org/euclid.aoap/1596528028_20200804040033Tue, 04 Aug 2020 04:00 EDTParameter and dimension dependence of convergence rates to stationarity for reflecting Brownian motionshttps://projecteuclid.org/euclid.aoap/1600157067<strong>Sayan Banerjee</strong>, <strong>Amarjit Budhiraja</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 5, 2005--2029.</p><p><strong>Abstract:</strong><br/>
We obtain rates of convergence to stationarity in $L^{1}$-Wasserstein distance for a $d$-dimensional reflected Brownian motion (RBM) in the nonnegative orthant that are explicit in the dimension and the system parameters. The results are then applied to a class of RBMs considered in (Blanchet and Xinyun (2016)) and to rank-based diffusions including the Atlas model. In both cases, we obtain explicit rates and bounds on relaxation times. In the first case we improve the relaxation time estimates of $O(d^{4}(\log d)^{2})$ obtained in (Blanchet and Xinyun (2016)) to $O((\log d)^{2})$. In the latter case, we give the first results on explicit parameter and dimension dependent rates under the Wasserstein distance. The proofs do not require an explicit form for the stationary measure or reversibility of the process with respect to this measure, and cover settings where these properties are not available. In the special case of the standard Atlas model (In Stochastic Portfolio Theory (2002) 1–24 Springer), we obtain a bound on the relaxation time of $O(d^{6}(\log d)^{2})$.
</p>projecteuclid.org/euclid.aoap/1600157067_20200915040434Tue, 15 Sep 2020 04:04 EDTA limit theorem for the survival probability of a simple random walk among power-law renewal obstacleshttps://projecteuclid.org/euclid.aoap/1600157068<strong>Julien Poisat</strong>, <strong>François Simenhaus</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 5, 2030--2068.</p><p><strong>Abstract:</strong><br/>
We consider a one-dimensional simple random walk surviving among a field of static soft obstacles: each time it meets an obstacle the walk is killed with probability $1-e^{-\beta}$, where $\beta$ is a positive and fixed parameter. The positions of the obstacles are sampled independently from the walk and according to a renewal process. The increments between consecutive obstacles, or gaps, are assumed to have a power-law decaying tail with exponent $\gamma>0$. We prove convergence in law for the properly rescaled logarithm of the quenched survival probability as time goes to infinity. The normalization exponent is $\gamma/(\gamma+2)$, while the limiting law writes as a variational formula with both universal and nonuniversal features. The latter involves (i) a Poisson point process that emerges as the universal scaling limit of the properly rescaled gaps and (ii) a function of the parameter $\beta$ that we call asymptotic cost of crossing per obstacle and that may, in principle, depend on the details of the gap distribution. Our proof suggests a confinement strategy of the walk in a single large gap. This model may also be seen as a $(1+1)$-directed polymer among many repulsive interfaces, in which case $\beta$ corresponds to the strength of repulsion, the survival probability to the partition function and its logarithm to the finite-volume free energy.
</p>projecteuclid.org/euclid.aoap/1600157068_20200915040434Tue, 15 Sep 2020 04:04 EDTGeometric ergodicity of the Bouncy Particle Samplerhttps://projecteuclid.org/euclid.aoap/1600157069<strong>Alain Durmus</strong>, <strong>Arnaud Guillin</strong>, <strong>Pierre Monmarché</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 5, 2069--2098.</p><p><strong>Abstract:</strong><br/>
The Bouncy Particle Sampler (BPS) is a Monte Carlo Markov chain algorithm to sample from a target density known up to a multiplicative constant. This method is based on a kinetic piecewise deterministic Markov process for which the target measure is invariant. This paper deals with theoretical properties of BPS. First, we establish geometric ergodicity of the associated semi-group under weaker conditions than in ( Ann. Statist. 47 (2019) 1268–1287) both on the target distribution and the velocity probability distribution. This result is based on a new coupling of the process which gives a quantitative minorization condition and yields more insights on the convergence. In addition, we study on a toy model the dependency of the convergence rates on the dimension of the state space. Finally, we apply our results to the analysis of simulated annealing algorithms based on BPS.
</p>projecteuclid.org/euclid.aoap/1600157069_20200915040434Tue, 15 Sep 2020 04:04 EDTImaginary multiplicative chaos: Moments, regularity and connections to the Ising modelhttps://projecteuclid.org/euclid.aoap/1600157070<strong>Janne Junnila</strong>, <strong>Eero Saksman</strong>, <strong>Christian Webb</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 5, 2099--2164.</p><p><strong>Abstract:</strong><br/>
In this article we study imaginary Gaussian multiplicative chaos—namely a family of random generalized functions which can formally be written as $e^{iX(x)}$, where $X$ is a log-correlated real-valued Gaussian field on $\mathbf{R}^{d}$, that is, it has a logarithmic singularity on the diagonal of its covariance. We study basic analytic properties of these random generalized functions, such as what spaces of distributions these objects live in, along with their basic stochastic properties, such as moment and tail estimates.
After this, we discuss connections between imaginary multiplicative chaos and the critical planar Ising model, namely that the scaling limit of the spin field of the critical planar XOR-Ising model can be expressed in terms of the cosine of the Gaussian free field, that is, the real part of an imaginary multiplicative chaos distribution. Moreover, if one adds a magnetic perturbation to the XOR-Ising model, then the scaling limit of the spin field can be expressed in terms of the cosine of the sine-Gordon field, which can also be viewed as the real part of an imaginary multiplicative chaos distribution.
The first sections of the article have been written in the style of a review, and we hope that the text will also serve as an introduction to imaginary chaos for an uninitiated reader.
</p>projecteuclid.org/euclid.aoap/1600157070_20200915040434Tue, 15 Sep 2020 04:04 EDTStochastic equation and exponential ergodicity in Wasserstein distances for affine processeshttps://projecteuclid.org/euclid.aoap/1600157071<strong>Martin Friesen</strong>, <strong>Peng Jin</strong>, <strong>Barbara Rüdiger</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 5, 2165--2195.</p><p><strong>Abstract:</strong><br/>
This work is devoted to the study of conservative affine processes on the canonical state space $D=\mathbb{R}_{+}^{m}\times \mathbb{R}^{n}$, where $m+n>0$. We show that each affine process can be obtained as the pathwise unique strong solution to a stochastic equation driven by Brownian motions and Poisson random measures. Then we study the long-time behavior of affine processes, that is, we show that under first moment condition on the state-dependent and $\log $-moment conditions on the state-independent jump measures, respectively, each subcritical affine process is exponentially ergodic in a suitably chosen Wasserstein distance. Moments of affine processes are studied as well.
</p>projecteuclid.org/euclid.aoap/1600157071_20200915040434Tue, 15 Sep 2020 04:04 EDTSquare permutations are typically rectangularhttps://projecteuclid.org/euclid.aoap/1600157072<strong>Jacopo Borga</strong>, <strong>Erik Slivken</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 5, 2196--2233.</p><p><strong>Abstract:</strong><br/>
We describe the limit (for two topologies) of large uniform random square permutations, that is, permutations where every point is a record. The starting point for all our results is a sampling procedure for asymptotically uniform square permutations. Building on that, we first describe the global behavior by showing that these permutations have a permuton limit which can be described by a random rectangle. We also explore fluctuations about this random rectangle, which we can describe through coupled Brownian motions. Second, we consider the limiting behavior of the neighborhood of a point in the permutation through local limits. As a byproduct, we also determine the random limits of the proportion of occurrences (and consecutive occurrences) of any given pattern in a uniform random square permutation.
</p>projecteuclid.org/euclid.aoap/1600157072_20200915040434Tue, 15 Sep 2020 04:04 EDTHamilton–Jacobi equations for finite-rank matrix inferencehttps://projecteuclid.org/euclid.aoap/1600157073<strong>J.-C. Mourrat</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 5, 2234--2260.</p><p><strong>Abstract:</strong><br/>
We compute the large-scale limit of the free energy associated with the problem of inference of a finite-rank matrix. The method follows the principle put forward in Mourrat (2018) which consists in identifying a suitable Hamilton–Jacobi equation satisfied by the limit free energy. We simplify the approach of Mourrat (2018) using a notion of weak solution of the Hamilton–Jacobi equation which is more convenient to work with and is applicable whenever the nonlinearity in the equation is convex.
</p>projecteuclid.org/euclid.aoap/1600157073_20200915040434Tue, 15 Sep 2020 04:04 EDTThe two-type Richardson model in the half-planehttps://projecteuclid.org/euclid.aoap/1600157074<strong>Daniel Ahlberg</strong>, <strong>Maria Deijfen</strong>, <strong>Christopher Hoffman</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 5, 2261--2273.</p><p><strong>Abstract:</strong><br/>
The two-type Richardson model describes the growth of two competing infection types on the two or higher dimensional integer lattice. For types that spread with the same intensity, it is known that there is a positive probability for infinite coexistence, while for types with different intensities, it is conjectured that infinite coexistence is not possible. In this paper we study the two-type Richardson model in the upper half-plane $\mathbb{Z}\times\mathbb{Z}_{+}$, and prove that coexistence of two types starting on the horizontal axis has positive probability if and only if the types have the same intensity.
</p>projecteuclid.org/euclid.aoap/1600157074_20200915040434Tue, 15 Sep 2020 04:04 EDTPathwise stochastic control with applications to robust filteringhttps://projecteuclid.org/euclid.aoap/1600157075<strong>Andrew L. Allan</strong>, <strong>Samuel N. Cohen</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 5, 2274--2310.</p><p><strong>Abstract:</strong><br/>
We study the problem of pathwise stochastic optimal control, where the optimization is performed for each fixed realisation of the driving noise, by phrasing the problem in terms of the optimal control of rough differential equations. We investigate the degeneracy phenomenon induced by directly controlling the coefficient of the noise term, and propose a simple procedure to resolve this degeneracy whilst retaining dynamic programming. As an application, we use pathwise stochastic control in the context of stochastic filtering to construct filters which are robust to parameter uncertainty, demonstrating an original application of rough path theory to statistics.
</p>projecteuclid.org/euclid.aoap/1600157075_20200915040434Tue, 15 Sep 2020 04:04 EDTPropagation of chaos and the many-demes limit for weakly interacting diffusions in the sparse regimehttps://projecteuclid.org/euclid.aoap/1600157076<strong>Martin Hutzenthaler</strong>, <strong>Daniel Pieper</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 5, 2311--2354.</p><p><strong>Abstract:</strong><br/>
Propagation of chaos is a well-studied phenomenon and shows that weakly interacting diffusions may become independent as the system size converges to infinity. Most of the literature focuses on the case of exchangeable systems where all involved diffusions have the same distribution and are “of the same size”. In this paper, we analyze the case where only a few diffusions start outside of an accessible trap. Our main result shows that in this “sparse regime” the system of weakly interacting diffusions converges in distribution to a forest of excursions from the trap. In particular, initial independence propagates in the limit and results in a forest of independent trees.
</p>projecteuclid.org/euclid.aoap/1600157076_20200915040434Tue, 15 Sep 2020 04:04 EDTPathwise McKean–Vlasov theory with additive noisehttps://projecteuclid.org/euclid.aoap/1600157077<strong>Michele Coghi</strong>, <strong>Jean-Dominique Deuschel</strong>, <strong>Peter K. Friz</strong>, <strong>Mario Maurelli</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 5, 2355--2392.</p><p><strong>Abstract:</strong><br/>
We take a pathwise approach to classical McKean–Vlasov stochastic differential equations with additive noise, as for example, exposed in Sznitmann (In École D’Été de Probabilités de Saint-Flour XIX—1989 (1991) 165–251, Springer). Our study was prompted by some concrete problems in battery modelling ( Contin. Mech. Thermodyn. 30 (2018) 593–628), and also by recent progrss on rough-pathwise McKean–Vlasov theory, notably Cass–Lyons ( Proc. Lond. Math. Soc. (3) 110 (2015) 83–107), and then Bailleul, Catellier and Delarue (Bailleul, Catellier and Delarue (2018)). Such a “pathwise McKean–Vlasov theory” can be traced back to Tanaka (In Stochastic Analysis (Katata/Kyoto, 1982) (1984) 469–488, North-Holland). This paper can be seen as an attempt to advertize the ideas, power and simplicity of the pathwise appproach, not so easily extracted from (Bailleul, Catellier and Delarue (2018); Proc. Lond. Math. Soc. (3) 110 (2015) 83–107; In Stochastic Analysis (Katata/Kyoto, 1982) (1984) 469–488, North-Holland), together with a number of novel applications. These include mean field convergence without a priori independence and exchangeability assumption; common noise, càdlàg noise, and reflecting boundaries. Last not least, we generalize Dawson–Gärtner large deviations and the central limit theorem to a non-Brownian noise setting.
</p>projecteuclid.org/euclid.aoap/1600157077_20200915040434Tue, 15 Sep 2020 04:04 EDTStochastic approximation on noncompact measure spaces and application to measure-valued Pólya processeshttps://projecteuclid.org/euclid.aoap/1600157078<strong>Cécile Mailler</strong>, <strong>Denis Villemonais</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 5, 2393--2438.</p><p><strong>Abstract:</strong><br/>
Our main result is to prove almost-sure convergence of a stochastic-approximation algorithm defined on the space of measures on a noncompact space. Our motivation is to apply this result to measure-valued Pólya processes (MVPPs, also known as infinitely-many Pólya urns). Our main idea is to use Foster–Lyapunov type criteria in a novel way to generalize stochastic-approximation methods to measure-valued Markov processes with a noncompact underlying space, overcoming in a fairly general context one of the major difficulties of existing studies on this subject.
From the MVPPs point of view, our result implies almost-sure convergence of a large class of MVPPs; this convergence was only obtained until now for specific examples, with only convergence in probability established for general classes. Furthermore, our approach allows us to extend the definition of MVPPs by adding “weights” to the different colors of the infinitely-many-color urn. We also exhibit a link between non-“balanced” MVPPs and quasi-stationary distributions of Markovian processes, which allows us to treat, for the first time in the literature, the nonbalanced case.
Finally, we show how our result can be applied to designing stochastic-approximation algorithms for the approximation of quasi-stationary distributions of discrete- and continuous-time Markov processes on noncompact spaces.
</p>projecteuclid.org/euclid.aoap/1600157078_20200915040434Tue, 15 Sep 2020 04:04 EDTRarity of extremal edges in random surfaces and other theoretical applications of cluster algorithmshttps://projecteuclid.org/euclid.aoap/1600157079<strong>Omri Cohen-Alloro</strong>, <strong>Ron Peled</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 5, 2439--2464.</p><p><strong>Abstract:</strong><br/>
Motivated by questions on the delocalization of random surfaces, we prove that random surfaces satisfying a Lipschitz constraint rarely develop extremal gradients. Previous proofs of this fact relied on reflection positivity and were thus limited to random surfaces defined on highly symmetric graphs, whereas our argument applies to general graphs. Our proof makes use of a cluster algorithm and reflection transformation for random surfaces of the type introduced by Swendsen–Wang, Wolff and Evertz et al. We discuss the general framework for such cluster algorithms, reviewing several particular cases with emphasis on their use in obtaining theoretical results. Two additional applications are presented: A reflection principle for random surfaces and a proof that pair correlations in the spin $O(n)$ model have monotone densities, strengthening Griffiths’ first inequality for such correlations.
</p>projecteuclid.org/euclid.aoap/1600157079_20200915040434Tue, 15 Sep 2020 04:04 EDTFunctional large deviations for Cox processes and $\mathit{Cox}/G/\infty$ queues, with a biological applicationhttps://projecteuclid.org/euclid.aoap/1600157080<strong>Justin Dean</strong>, <strong>Ayalvadi Ganesh</strong>, <strong>Edward Crane</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 5, 2465--2490.</p><p><strong>Abstract:</strong><br/>
We consider an infinite-server queue into which customers arrive according to a Cox process and have independent service times with a general distribution. We prove a functional large deviations principle for the equilibrium queue length process. The model is motivated by a linear feed-forward gene regulatory network, in which the rate of protein synthesis is modulated by the number of RNA molecules present in a cell. The system can be modelled as a nonstandard tandem of infinite-server queues, in which the number of customers present in a queue modulates the arrival rate into the next queue in the tandem. We establish large deviation principles for this queueing system in the asymptotic regime in which the arrival process is sped up, while the service process is not scaled.
</p>projecteuclid.org/euclid.aoap/1600157080_20200915040434Tue, 15 Sep 2020 04:04 EDTAsymptotic theory of sparse Bradley–Terry modelhttps://projecteuclid.org/euclid.aoap/1600157081<strong>Ruijian Han</strong>, <strong>Rougang Ye</strong>, <strong>Chunxi Tan</strong>, <strong>Kani Chen</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 5, 2491--2515.</p><p><strong>Abstract:</strong><br/>
The Bradley–Terry model is a fundamental model in the analysis of network data involving paired comparison. Assuming every pair of subjects in the network have an equal number of comparisons, Simons and Yao ( Ann. Statist. 27 (1999) 1041–1060) established an asymptotic theory for statistical estimation in the Bradley–Terry model. In practice, when the size of the network becomes large, the paired comparisons are generally sparse. The sparsity can be characterized by the probability $p_{n}$ that a pair of subjects have at least one comparison, which tends to zero as the size of the network $n$ goes to infinity. In this paper, the asymptotic properties of the maximum likelihood estimate of the Bradley–Terry model are shown under minimal conditions of the sparsity. Specifically, the uniform consistency is proved when $p_{n}$ is as small as the order of $(\log n)^{3}/n$, which is near the theoretical lower bound $\log n/n$ by the theory of the Erdos–Rényi graph. Asymptotic normality and inference are also provided. Evidence in support of the theory is presented in simulation results, along with an application to the analysis of the ATP data.
</p>projecteuclid.org/euclid.aoap/1600157081_20200915040434Tue, 15 Sep 2020 04:04 EDT