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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 EDTTail measure and spectral tail process of regularly varying time serieshttps://projecteuclid.org/euclid.aoap/1538985638<strong>Clément Dombry</strong>, <strong>Enkelejd Hashorva</strong>, <strong>Philippe Soulier</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 28, Number 6, 3884--3921.</p><p><strong>Abstract:</strong><br/>
The goal of this paper is an exhaustive investigation of the link between the tail measure of a regularly varying time series and its spectral tail process, independently introduced in [Owada and Samorodnitsky (2012)] and [ Stochastic Process. Appl. 119 (2009) 1055–1080]. Our main result is to prove in an abstract framework that there is a one-to-one correspondence between these two objects, and given one of them to show that it is always possible to build a time series of which it will be the tail measure or the spectral tail process. For nonnegative time series, we recover results explicitly or implicitly known in the theory of max-stable processes.
</p>projecteuclid.org/euclid.aoap/1538985638_20181008040046Mon, 08 Oct 2018 04:00 EDTMutation frequencies in a birth–death branching processhttps://projecteuclid.org/euclid.aoap/1538985639<strong>David Cheek</strong>, <strong>Tibor Antal</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 28, Number 6, 3922--3947.</p><p><strong>Abstract:</strong><br/>
First, we revisit the stochastic Luria–Delbrück model: a classic two-type branching process which describes cell proliferation and mutation. We prove limit theorems and exact results for the mutation times, clone sizes and number of mutants. Second, we extend the framework to consider mutations at multiple sites along the genome. The number of mutants in the two-type model characterises the mean site frequency spectrum in the multiple-site model. Our predictions are consistent with previously published cancer genomic data.
</p>projecteuclid.org/euclid.aoap/1538985639_20181008040046Mon, 08 Oct 2018 04:00 EDTRate control under heavy traffic with strategic servershttps://projecteuclid.org/euclid.aoap/1544000424<strong>Erhan Bayraktar</strong>, <strong>Amarjit Budhiraja</strong>, <strong>Asaf Cohen</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 1, 1--35.</p><p><strong>Abstract:</strong><br/>
We consider a large queueing system that consists of many strategic servers that are weakly interacting. Each server processes jobs from its unique critically loaded buffer and controls the rate of arrivals and departures associated with its queue to minimize its expected cost. The rates and the cost functions in addition to depending on the control action, can depend, in a symmetric fashion, on the size of the individual queue and the empirical measure of the states of all queues in the system. In order to determine an approximate Nash equilibrium for this finite player game, we construct a Lasry–Lions-type mean-field game (MFG) for certain reflected diffusions that governs the limiting behavior. Under conditions, we establish the convergence of the Nash-equilibrium value for the finite size queuing system to the value of the MFG.
</p>projecteuclid.org/euclid.aoap/1544000424_20181205040111Wed, 05 Dec 2018 04:01 ESTNonconvex homogenization for one-dimensional controlled random walks in random potentialhttps://projecteuclid.org/euclid.aoap/1544000425<strong>Atilla Yilmaz</strong>, <strong>Ofer Zeitouni</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 1, 36--88.</p><p><strong>Abstract:</strong><br/>
We consider a finite horizon stochastic optimal control problem for nearest-neighbor random walk $\{X_{i}\}$ on the set of integers. The cost function is the expectation of the exponential of the path sum of a random stationary and ergodic bounded potential plus $\theta X_{n}$. The random walk policies are measurable with respect to the random potential, and are adapted, with their drifts uniformly bounded in magnitude by a parameter $\delta\in[0,1]$. Under natural conditions on the potential, we prove that the normalized logarithm of the optimal cost function converges. The proof is constructive in the sense that we identify asymptotically optimal policies given the value of the parameter $\delta$, as well as the law of the potential. It relies on correctors from large deviation theory as opposed to arguments based on subadditivity which do not seem to work except when $\delta=0$.
The Bellman equation associated to this control problem is a second-order Hamilton–Jacobi (HJ) partial difference equation with a separable random Hamiltonian which is nonconvex in $\theta$ unless $\delta=0$. We prove that this equation homogenizes under linear initial data to a first-order HJ equation with a deterministic effective Hamiltonian. When $\delta=0$, the effective Hamiltonian is the tilted free energy of random walk in random potential and it is convex in $\theta$. In contrast, when $\delta=1$, the effective Hamiltonian is piecewise linear and nonconvex in $\theta$. Finally, when $\delta\in(0,1)$, the effective Hamiltonian is expressed completely in terms of the tilted free energy for the $\delta=0$ case and its convexity/nonconvexity in $\theta$ is characterized by a simple inequality involving $\delta$ and the magnitude of the potential, thereby marking two qualitatively distinct control regimes.
</p>projecteuclid.org/euclid.aoap/1544000425_20181205040111Wed, 05 Dec 2018 04:01 ESTParticle systems with singular interaction through hitting times: Application in systemic risk modelinghttps://projecteuclid.org/euclid.aoap/1544000426<strong>Sergey Nadtochiy</strong>, <strong>Mykhaylo Shkolnikov</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 1, 89--129.</p><p><strong>Abstract:</strong><br/>
We propose an interacting particle system to model the evolution of a system of banks with mutual exposures. In this model, a bank defaults when its normalized asset value hits a lower threshold, and its default causes instantaneous losses to other banks, possibly triggering a cascade of defaults. The strength of this interaction is determined by the level of the so-called noncore exposure . We show that, when the size of the system becomes large, the cumulative loss process of a bank resulting from the defaults of other banks exhibits discontinuities. These discontinuities are naturally interpreted as systemic events , and we characterize them explicitly in terms of the level of noncore exposure and the fraction of banks that are “about to default.” The main mathematical challenges of our work stem from the very singular nature of the interaction between the particles, which is inherited by the limiting system. A similar particle system is analyzed in [ Ann. Appl. Probab. 25 (2015) 2096–2133] and [ Stochastic Process. Appl. 125 (2015) 2451–2492], and we build on and extend their results. In particular, we characterize the large-population limit of the system and analyze the jump times, the regularity between jumps, and the local uniqueness of the limiting process.
</p>projecteuclid.org/euclid.aoap/1544000426_20181205040111Wed, 05 Dec 2018 04:01 ESTCubature on Wiener space for McKean–Vlasov SDEs with smooth scalar interactionhttps://projecteuclid.org/euclid.aoap/1544000427<strong>Dan Crisan</strong>, <strong>Eamon McMurray</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 1, 130--177.</p><p><strong>Abstract:</strong><br/>
We present two cubature on Wiener space algorithms for the numerical solution of McKean–Vlasov SDEs with smooth scalar interaction. First, we consider a method introduced in [ Stochastic Process. Appl. 125 (2015) 2206–2255] under a uniformly elliptic assumption and extend the analysis to a uniform strong Hörmander assumption. Then we introduce a new method based on Lagrange polynomial interpolation. The analysis hinges on sharp gradient to time-inhomogeneous parabolic PDEs bounds. These bounds may be of independent interest. They extend the classical results of Kusuoka and Stroock [ J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 32 (1985) 1–76] and Kusuoka [ J. Math. Sci. Univ. Tokyo 10 (2003) 261–277] further developed in [ J. Funct. Anal. 263 (2012) 3024–3101; J. Funct. Anal. 268 (2015) 1928–1971; Cubature Methods and Applications (2013), Springer, Cham] and, more recently, in [ Probab. Theory Related Fields 171 (2016) 97–148]. Both algorithms are tested through two numerical examples.
</p>projecteuclid.org/euclid.aoap/1544000427_20181205040111Wed, 05 Dec 2018 04:01 ESTLower error bounds for strong approximation of scalar SDEs with non-Lipschitzian coefficientshttps://projecteuclid.org/euclid.aoap/1544000428<strong>Mario Hefter</strong>, <strong>André Herzwurm</strong>, <strong>Thomas Müller-Gronbach</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 1, 178--216.</p><p><strong>Abstract:</strong><br/>
We study pathwise approximation of scalar stochastic differential equations at a single time point or globally in time by means of methods that are based on finitely many observations of the driving Brownian motion. We prove lower error bounds in terms of the average number of evaluations of the driving Brownian motion that hold for every such method under rather mild assumptions on the coefficients of the equation. The underlying simple idea of our analysis is as follows: the lower error bounds known for equations with coefficients that have sufficient regularity globally in space should still apply in the case of coefficients that have this regularity in space only locally, in a small neighborhood of the initial value. Our results apply to a huge variety of equations with coefficients that are not globally Lipschitz continuous in space including Cox–Ingersoll–Ross processes, equations with superlinearly growing coefficients, and equations with discontinuous coefficients. In many of these cases, the resulting lower error bounds even turn out to be sharp.
</p>projecteuclid.org/euclid.aoap/1544000428_20181205040111Wed, 05 Dec 2018 04:01 ESTKnudsen gas in flat tirehttps://projecteuclid.org/euclid.aoap/1544000429<strong>Krzysztof Burdzy</strong>, <strong>Carl-Erik Gauthier</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 1, 217--263.</p><p><strong>Abstract:</strong><br/>
We consider random reflections (according to the Lambertian distribution) of a light ray in a thin variable width (but almost circular) tube. As the width of the tube goes to zero, properly rescaled angular component of the light ray position converges in distribution to a diffusion whose parameters (diffusivity and drift) are given explicitly in terms of the tube width.
</p>projecteuclid.org/euclid.aoap/1544000429_20181205040111Wed, 05 Dec 2018 04:01 ESTThe Bouchaud–Anderson model with double-exponential potentialhttps://projecteuclid.org/euclid.aoap/1544000430<strong>S. Muirhead</strong>, <strong>R. Pymar</strong>, <strong>R. S. dos Santos</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 1, 264--325.</p><p><strong>Abstract:</strong><br/>
The Bouchaud–Anderson model (BAM) is a generalisation of the parabolic Anderson model (PAM) in which the driving simple random walk is replaced by a random walk in an inhomogeneous trapping landscape; the BAM reduces to the PAM in the case of constant traps. In this paper, we study the BAM with double-exponential potential. We prove the complete localisation of the model whenever the distribution of the traps is unbounded. This may be contrasted with the case of constant traps (i.e., the PAM), for which it is known that complete localisation fails. This shows that the presence of an inhomogeneous trapping landscape may cause a system of branching particles to exhibit qualitatively distinct concentration behaviour.
</p>projecteuclid.org/euclid.aoap/1544000430_20181205040111Wed, 05 Dec 2018 04:01 ESTRandom switching between vector fields having a common zerohttps://projecteuclid.org/euclid.aoap/1544000431<strong>Michel Benaïm</strong>, <strong>Edouard Strickler</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 1, 326--375.</p><p><strong>Abstract:</strong><br/>
Let $E$ be a finite set, $\{F^{i}\}_{i\in E}$ a family of vector fields on $\mathbb{R}^{d}$ leaving positively invariant a compact set $M$ and having a common zero $p\in M$. We consider a piecewise deterministic Markov process $(X,I)$ on $M\times E$ defined by $\dot{X}_{t}=F^{I_{t}}(X_{t})$ where $I$ is a jump process controlled by $X$: ${\mathsf{P}}(I_{t+s}=j|(X_{u},I_{u})_{u\leq t})=a_{ij}(X_{t})s+o(s)$ for $i\neq j$ on $\{I_{t}=i\}$.
We show that the behaviour of $(X,I)$ is mainly determined by the behaviour of the linearized process $(Y,J)$ where $\dot{Y}_{t}=A^{J_{t}}Y_{t}$, $A^{i}$ is the Jacobian matrix of $F^{i}$ at $p$ and $J$ is the jump process with rates $(a_{ij}(p))$. We introduce two quantities $\Lambda^{-}$ and $\Lambda^{+}$, respectively, defined as the minimal (resp., maximal ) growth rate of $\|Y_{t}\|$, where the minimum (resp., maximum) is taken over all the ergodic measures of the angular process $(\Theta,J)$ with $\Theta_{t}=\frac{Y_{t}}{\|Y_{t}\|}$. It is shown that $\Lambda^{+}$ coincides with the top Lyapunov exponent (in the sense of ergodic theory) of $(Y,J)$ and that under general assumptions $\Lambda^{-}=\Lambda^{+}$. We then prove that, under certain irreducibility conditions, $X_{t}\rightarrow p$ exponentially fast when $\Lambda^{+}<0$ and $(X,I)$ converges in distribution at an exponential rate toward a (unique) invariant measure supported by $M\setminus \{p\}\times E$ when $\Lambda^{-}>0$. Some applications to certain epidemic models in a fluctuating environment are discussed and illustrate our results.
</p>projecteuclid.org/euclid.aoap/1544000431_20181205040111Wed, 05 Dec 2018 04:01 ESTMulti-scale Lipschitz percolation of increasing events for Poisson random walkshttps://projecteuclid.org/euclid.aoap/1544000432<strong>Peter Gracar</strong>, <strong>Alexandre Stauffer</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 1, 376--433.</p><p><strong>Abstract:</strong><br/>
Consider the graph induced by $\mathbb{Z}^{d}$, equipped with uniformly elliptic random conductances. At time $0$, place a Poisson point process of particles on $\mathbb{Z}^{d}$ and let them perform independent simple random walks. Tessellate the graph into cubes indexed by $i\in\mathbb{Z}^{d}$ and tessellate time into intervals indexed by $\tau$. Given a local event $E(i,\tau)$ that depends only on the particles inside the space time region given by the cube $i$ and the time interval $\tau$, we prove the existence of a Lipschitz connected surface of cells $(i,\tau)$ that separates the origin from infinity on which $E(i,\tau)$ holds. This gives a directly applicable and robust framework for proving results in this setting that need a multi-scale argument. For example, this allows us to prove that an infection spreads with positive speed among the particles.
</p>projecteuclid.org/euclid.aoap/1544000432_20181205040111Wed, 05 Dec 2018 04:01 ESTTheoretical properties of quasi-stationary Monte Carlo methodshttps://projecteuclid.org/euclid.aoap/1544000433<strong>Andi Q. Wang</strong>, <strong>Martin Kolb</strong>, <strong>Gareth O. Roberts</strong>, <strong>David Steinsaltz</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 1, 434--457.</p><p><strong>Abstract:</strong><br/>
This paper gives foundational results for the application of quasi-stationarity to Monte Carlo inference problems. We prove natural sufficient conditions for the quasi-limiting distribution of a killed diffusion to coincide with a target density of interest. We also quantify the rate of convergence to quasi-stationarity by relating the killed diffusion to an appropriate Langevin diffusion. As an example, we consider in detail a killed Ornstein–Uhlenbeck process with Gaussian quasi-stationary distribution.
</p>projecteuclid.org/euclid.aoap/1544000433_20181205040111Wed, 05 Dec 2018 04:01 ESTApproximation of stable law in Wasserstein-1 distance by Stein’s methodhttps://projecteuclid.org/euclid.aoap/1544000434<strong>Lihu Xu</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 1, 458--504.</p><p><strong>Abstract:</strong><br/>
Let $n\in\mathbb{N}$, let $\zeta_{n,1},\ldots,\zeta_{n,n}$ be a sequence of independent random variables with $\mathbb{E}\zeta_{n,i}=0$ and $\mathbb{E}|\zeta_{n,i}|<\infty$ for each $i$, and let $\mu$ be an $\alpha$-stable distribution having characteristic function $e^{-|\lambda|^{\alpha}}$ with $\alpha\in(1,2)$. Denote $S_{n}=\zeta_{n,1}+\cdots+\zeta_{n,n}$ and its distribution by $\mathcal{L}(S_{n})$, we bound the Wasserstein-1 distance of $\mathcal{L} (S_{n})$ and $\mu$ essentially by an $L^{1}$ discrepancy between two kernels. More precisely, we prove the following inequality: \[d_{W}\big(\mathcal{L}(S_{n}),\mu\big)\le C\Bigg[\sum_{i=1}^{n}\int_{-N}^{N}\bigg\vert \frac{\mathcal{K}_{\alpha}(t,N)}{n}-\frac{K_{i}(t,N)}{\alpha}\bigg\vert \,\mathrm{d}t+\mathcal{R}_{N,n}\Bigg],\] where $d_{W}$ is the Wasserstein-1 distance of probability measures, $\mathcal{K}_{\alpha}(t,N)$ is the kernel of a decomposition of the fractional Laplacian $\Delta^{\frac{\alpha}{2}}$, $K_{i}(t,N)$ is a $K$ function ( Normal Approximation by Stein’s Method (2011) Springer) with a truncation and $\mathcal{R}_{N,n}$ is a small remainder. The integral term \[\sum_{i=1}^{n}\int_{-N}^{N}\bigg\vert \frac{\mathcal{K}_{\alpha}(t,N)}{n}-\frac{K_{i}(t,N)}{\alpha}\bigg\vert \,\mathrm{d}t\] can be interpreted as an $L^{1}$ discrepancy.
As an application, we prove a general theorem of stable law convergence rate when $\zeta_{n,i}$ are i.i.d. and the distribution falls in the normal domain of attraction of $\mu$. To test our results, we compare our convergence rates with those known in the literature for four given examples, among which the distribution in the fourth example is not in the normal domain of attraction of $\mu$.
</p>projecteuclid.org/euclid.aoap/1544000434_20181205040111Wed, 05 Dec 2018 04:01 ESTContinuity of the optimal stopping boundary for two-dimensional diffusionshttps://projecteuclid.org/euclid.aoap/1544000435<strong>Goran Peskir</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 1, 505--530.</p><p><strong>Abstract:</strong><br/>
We first show that a smooth fit between the value function and the gain function at the optimal stopping boundary for a two-dimensional diffusion process implies the absence of boundary’s discontinuities of the first kind (the right-hand and left-hand limits exist but differ). We then show that the smooth fit itself is satisfied over the flat portion of the optimal stopping boundary arising from any of its hypothesised jumps. Combining the two facts we obtain that the optimal stopping boundary is continuous whenever it has no discontinuities of the second kind. The derived fact holds both in the parabolic and elliptic case under the sole hypothesis of Hölder continuous coefficients, thus improving upon all known results in the parabolic case, and establishing the fact for the first time in the elliptic case. The method of proof relies upon regularity results for the second-order parabolic/elliptic PDEs and makes use of the local time-space calculus techniques.
</p>projecteuclid.org/euclid.aoap/1544000435_20181205040111Wed, 05 Dec 2018 04:01 ESTRobust hedging of options on a leveraged exchange traded fundhttps://projecteuclid.org/euclid.aoap/1544000436<strong>Alexander M. G. Cox</strong>, <strong>Sam M. Kinsley</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 1, 531--576.</p><p><strong>Abstract:</strong><br/>
A leveraged exchange traded fund (LETF) is an exchange traded fund that uses financial derivatives to amplify the price changes of a basket of goods. In this paper, we consider the robust hedging of European options on a LETF, finding model-free bounds on the price of these options.
To obtain an upper bound, we establish a new optimal solution to the Skorokhod embedding problem (SEP) using methods introduced in Beiglböck–Cox–Huesmann. This stopping time can be represented as the hitting time of some region by a Brownian motion, but unlike other solutions of, for example, Root, this region is not unique. Much of this paper is dedicated to characterising the choice of the embedding region that gives the required optimality property. Notably, this appears to be the first solution to the SEP where the solution is not uniquely characterised by its geometric structure, and an additional condition is needed on the stopping region to guarantee that it is the optimiser. An important part of determining the optimal region is identifying the correct form of the dual solution, which has a financial interpretation as a model-independent superhedging strategy.
</p>projecteuclid.org/euclid.aoap/1544000436_20181205040111Wed, 05 Dec 2018 04:01 ESTExponential utility maximization under model uncertainty for unbounded endowmentshttps://projecteuclid.org/euclid.aoap/1544000437<strong>Daniel Bartl</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 1, 577--612.</p><p><strong>Abstract:</strong><br/>
We consider the robust exponential utility maximization problem in discrete time: An investor maximizes the worst case expected exponential utility with respect to a family of nondominated probabilistic models of her endowment by dynamically investing in a financial market, and statically in available options.
We show that, for any measurable random endowment (regardless of whether the problem is finite or not) an optimal strategy exists, a dual representation in terms of (calibrated) martingale measures holds true, and that the problem satisfies the dynamic programming principle (in case of no options). Further, it is shown that the value of the utility maximization problem converges to the robust superhedging price as the risk aversion parameter gets large, and examples of nondominated probabilistic models are discussed.
</p>projecteuclid.org/euclid.aoap/1544000437_20181205040111Wed, 05 Dec 2018 04:01 ESTServe the shortest queue and Walsh Brownian motionhttps://projecteuclid.org/euclid.aoap/1544000438<strong>Rami Atar</strong>, <strong>Asaf Cohen</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 1, 613--651.</p><p><strong>Abstract:</strong><br/>
We study a single-server Markovian queueing model with $N$ customer classes in which priority is given to the shortest queue. Under a critical load condition, we establish the diffusion limit of the nominal workload and queue length processes in the form of a Walsh Brownian motion (WBM) living in the union of the $N$ nonnegative coordinate axes in $\mathbb{R}^{N}$ and a linear transformation thereof. This reveals the following asymptotic behavior. Each time that queues begin to build starting from an empty system, one of them becomes dominant in the sense that it contains nearly all the workload in the system, and it remains so until the system becomes (nearly) empty again. The radial part of the WBM, given as a reflected Brownian motion (RBM) on the half-line, captures the total workload asymptotics, whereas its angular distribution expresses how likely it is for each class to become dominant on excursions.
As a heavy traffic result, it is nonstandard in three ways: (i) In the terminology of Harrison (In Stochastic Networks (1995) 1–20 Springer), it is unconventional , in that the limit is not a RBM. (ii) It does not constitute an invariance principle , in that the limit law (specifically, the angular distribution) is not determined solely by the first two moments of the data, and is sensitive even to tie breaking rules. (iii) The proof method does not fully characterize the limit law (specifically, it gives no information on the angular distribution).
</p>projecteuclid.org/euclid.aoap/1544000438_20181205040111Wed, 05 Dec 2018 04:01 ESTWeak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficientshttps://projecteuclid.org/euclid.aoap/1548298927<strong>Daniel Conus</strong>, <strong>Arnulf Jentzen</strong>, <strong>Ryan Kurniawan</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 2, 653--716.</p><p><strong>Abstract:</strong><br/>
Strong convergence rates for (temporal, spatial, and noise) numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the scientific literature. Weak convergence rates for numerical approximations of such SEEs have been investigated for about two decades and are far away from being well understood: roughly speaking, no essentially sharp weak convergence rates are known for parabolic SEEs with nonlinear diffusion coefficient functions; see Remark 2.3 in [ Math. Comp. 80 (2011) 89–117] for details. In this article, we solve the weak convergence problem emerged from Debussche’s article in the case of spectral Galerkin approximations and establish essentially sharp weak convergence rates for spatial spectral Galerkin approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Our solution to the weak convergence problem does not use Malliavin calculus. Rather, key ingredients in our solution to the weak convergence problem emerged from Debussche’s article are the use of appropriately modified versions of the spatial Galerkin approximation processes and applications of a mild Itô-type formula for solutions and numerical approximations of semilinear SEEs. This article solves the weak convergence problem emerged from Debussche’s article merely in the case of spatial spectral Galerkin approximations instead of other more complicated numerical approximations. Our method of proof extends, however, to a number of other kinds of spatial and temporal numerical approximations for semilinear SEEs.
</p>projecteuclid.org/euclid.aoap/1548298927_20190123220226Wed, 23 Jan 2019 22:02 ESTChange-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 EDT