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The latest articles from The Annals of Applied Probability on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTThu, 02 Jun 2011 09:14 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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On optimal arbitrage
http://projecteuclid.org/euclid.aoap/1279638783
<strong>Daniel Fernholz</strong>, <strong>Ioannis Karatzas</strong><p><strong>Source: </strong>Ann. Appl. Probab., Volume 20, Number 4, 1179--1204.</p><p><strong>Abstract:</strong><br/>
In a Markovian model for a financial market, we characterize the best arbitrage with respect to the market portfolio that can be achieved using nonanticipative investment strategies, in terms of the smallest positive solution to a parabolic partial differential inequality; this is determined entirely on the basis of the covariance structure of the model. The solution is intimately related to properties of strict local martingales and is used to generate the investment strategy which realizes the best possible arbitrage. Some extensions to non-Markovian situations are also presented.
</p>projecteuclid.org/euclid.aoap/1279638783_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTA general continuous-state nonlinear branching processhttps://projecteuclid.org/euclid.aoap/1563869049<strong>Pei-Sen Li</strong>, <strong>Xu Yang</strong>, <strong>Xiaowen Zhou</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 4, 2523--2555.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider the unique nonnegative solution to the following generalized version of the stochastic differential equation for a continuous-state branching process: \begin{eqnarray*}X_{t}&=&x+\int_{0}^{t}\gamma_{0}(X_{s})\,\mathrm{d}s+\int_{0}^{t}\int_{0}^{\gamma_{1}(X_{s-})}W(\mathrm{d}s,\mathrm{d}u)\\&&{}+\int_{0}^{t}\int_{0}^{\infty}\int_{0}^{\gamma_{2}(X_{s-})}z\tilde{N}(\mathrm{d}s,\mathrm{d}z,\mathrm{d}u),\end{eqnarray*} where $W(\mathrm{d}t,\mathrm{d}u)$ and $\tilde{N}(\mathrm{d}s,\mathrm{d}z,\mathrm{d}u)$ denote a Gaussian white noise and an independent compensated spectrally positive Poisson random measure, respectively, and $\gamma_{0},\gamma_{1}$ and $\gamma_{2}$ are functions on $\mathbb{R}_{+}$ with both $\gamma_{1}$ and $\gamma_{2}$ taking nonnegative values. Intuitively, this process can be identified as a continuous-state branching process with population-size-dependent branching rates and with competition. Using martingale techniques we find rather sharp conditions on extinction, explosion and coming down from infinity behaviors of the process. Some Foster–Lyapunov-type criteria are also developed for such a process. More explicit results are obtained when $\gamma_{i}$, $i=0,1,2$ are power functions.
</p>projecteuclid.org/euclid.aoap/1563869049_20190723040421Tue, 23 Jul 2019 04:04 EDTEquilibrium interfaces of biased voter modelshttps://projecteuclid.org/euclid.aoap/1563869050<strong>Rongfeng Sun</strong>, <strong>Jan M. Swart</strong>, <strong>Jinjiong Yu</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 4, 2556--2593.</p><p><strong>Abstract:</strong><br/>
A one-dimensional interacting particle system is said to exhibit interface tightness if starting in an initial condition describing the interface between two constant configurations of different types, the process modulo translations is positive recurrent. In a biological setting, this describes two populations that do not mix, and it is believed to be a common phenomenon in one-dimensional particle systems. Interface tightness has been proved for voter models satisfying a finite second moment condition on the rates. We extend this to biased voter models. Furthermore, we show that the distribution of the equilibrium interface for the biased voter model converges to that of the voter model when the bias parameter tends to zero. A key ingredient is an identity for the expected number of boundaries in the equilibrium voter model interface, which is of independent interest.
</p>projecteuclid.org/euclid.aoap/1563869050_20190723040421Tue, 23 Jul 2019 04:04 EDTPropagation of chaos for topological interactionshttps://projecteuclid.org/euclid.aoap/1563869051<strong>P. Degond</strong>, <strong>M. Pulvirenti</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 4, 2594--2612.</p><p><strong>Abstract:</strong><br/>
We consider a $N$-particle model describing an alignment mechanism due to a topological interaction among the agents. We show that the kinetic equation, expected to hold in the mean-field limit $N\to \infty $, as following from the previous analysis in ( J. Stat. Phys. 163 (2016) 41–60) can be rigorously derived. This means that the statistical independence (propagation of chaos) is indeed recovered in the limit, provided it is assumed at time zero.
</p>projecteuclid.org/euclid.aoap/1563869051_20190723040421Tue, 23 Jul 2019 04:04 EDTNormal convergence of nonlocalised geometric functionals and shot-noise excursionshttps://projecteuclid.org/euclid.aoap/1571385618<strong>Raphaël Lachièze-Rey</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 2613--2653.</p><p><strong>Abstract:</strong><br/>
This article presents a complete second-order theory for a large class of geometric functionals on homogeneous Poisson input. In particular, the results do not require the existence of a radius of stabilisation. Hence they can be applied to geometric functionals of spatial shot-noise fields excursions such as volume, perimeter, or Euler characteristic (the method still applies to stabilising functionals). More generally, it must be checked that a local contribution to the functional is not strongly affected under a perturbation of the input far away. In this case, the exact asymptotic variance is given, as well as the likely optimal speed of convergence in the central limit theorem. This goes through a general mixing-type condition that adapts nicely to both proving asymptotic normality and that variance is of volume order.
</p>projecteuclid.org/euclid.aoap/1571385618_20191018040042Fri, 18 Oct 2019 04:00 EDTMetastability of the contact process on fast evolving scale-free networkshttps://projecteuclid.org/euclid.aoap/1571385619<strong>Emmanuel Jacob</strong>, <strong>Amitai Linker</strong>, <strong>Peter Mörters</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 2654--2699.</p><p><strong>Abstract:</strong><br/>
We study the contact process in the regime of small infection rates on finite scale-free networks with stationary dynamics based on simultaneous updating of all connections of a vertex. We allow the update rates of individual vertices to increase with the strength of a vertex, leading to a fast evolution of the network. We first develop an approach for inhomogeneous networks with general kernel and then focus on two canonical cases, the factor kernel and the preferential attachment kernel. For these specific networks, we identify and analyse four possible strategies how the infection can survive for a long time. We show that there is fast extinction of the infection when neither of the strategies is successful, otherwise there is slow extinction and the most successful strategy determines the asymptotics of the metastable density as the infection rate goes to zero. We identify the domains in which these strategies dominate in terms of phase diagrams for the exponent describing the decay of the metastable density.
</p>projecteuclid.org/euclid.aoap/1571385619_20191018040042Fri, 18 Oct 2019 04:00 EDTTree lengths for general $\Lambda $-coalescents and the asymptotic site frequency spectrum around the Bolthausen–Sznitman coalescenthttps://projecteuclid.org/euclid.aoap/1571385620<strong>Christina S. Diehl</strong>, <strong>Götz Kersting</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 2700--2743.</p><p><strong>Abstract:</strong><br/>
We study tree lengths in $\Lambda $-coalescents without a dust component from a sample of $n$ individuals. For the total length of all branches and the total length of all external branches, we present laws of large numbers in full generality. The other results treat regularly varying coalescents with exponent 1, which cover the Bolthausen–Sznitman coalescent. The theorems contain laws of large numbers for the total length of all internal branches and of internal branches of order $a$ (i.e., branches carrying $a$ individuals out of the sample). These results immediately transform to sampling formulas in the infinite sites model. In particular, we obtain the asymptotic site frequency spectrum of the Bolthausen–Sznitman coalescent. The proofs rely on a new technique to obtain laws of large numbers for certain functionals of decreasing Markov chains.
</p>projecteuclid.org/euclid.aoap/1571385620_20191018040042Fri, 18 Oct 2019 04:00 EDTEmpirical optimal transport on countable metric spaces: Distributional limits and statistical applicationshttps://projecteuclid.org/euclid.aoap/1571385621<strong>Carla Tameling</strong>, <strong>Max Sommerfeld</strong>, <strong>Axel Munk</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 2744--2781.</p><p><strong>Abstract:</strong><br/>
We derive distributional limits for empirical transport distances between probability measures supported on countable sets. Our approach is based on sensitivity analysis of optimal values of infinite dimensional mathematical programs and a delta method for nonlinear derivatives. A careful calibration of the norm on the space of probability measures is needed in order to combine differentiability and weak convergence of the underlying empirical process. Based on this, we provide a sufficient and necessary condition for the underlying distribution on the countable metric space for such a distributional limit to hold. We give an explicit form of the limiting distribution for tree spaces.
Finally, we apply our findings to optimal transport based inference in large scale problems. An application to nanoscale microscopy is given.
</p>projecteuclid.org/euclid.aoap/1571385621_20191018040042Fri, 18 Oct 2019 04:00 EDTExtinction in lower Hessenberg branching processes with countably many typeshttps://projecteuclid.org/euclid.aoap/1571385622<strong>Peter Braunsteins</strong>, <strong>Sophie Hautphenne</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 2782--2818.</p><p><strong>Abstract:</strong><br/>
We consider a class of branching processes with countably many types which we refer to as Lower Hessenberg branching processes . These are multitype Galton–Watson processes with typeset $\mathcal{X}=\{0,1,2,\ldots\}$, in which individuals of type $i$ may give birth to offspring of type $j\leq i+1$ only. For this class of processes, we study the set $S$ of fixed points of the progeny generating function. In particular, we highlight the existence of a continuum of fixed points whose minimum is the global extinction probability vector $\boldsymbol{q}$ and whose maximum is the partial extinction probability vector $\boldsymbol{\tilde{q}}$. In the case where $\boldsymbol{\tilde{q}}=\boldsymbol{1}$, we derive a global extinction criterion which holds under second moment conditions, and when $\boldsymbol{\tilde{q}}<\boldsymbol{1}$ we develop necessary and sufficient conditions for $\boldsymbol{q}=\boldsymbol{\tilde{q}}$. We also correct a result in the literature on a sequence of finite extinction probability vectors that converge to the infinite vector $\boldsymbol{\tilde{q}}$.
</p>projecteuclid.org/euclid.aoap/1571385622_20191018040042Fri, 18 Oct 2019 04:00 EDTControlled reflected SDEs and Neumann problem for backward SPDEshttps://projecteuclid.org/euclid.aoap/1571385623<strong>Erhan Bayraktar</strong>, <strong>Jinniao Qiu</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 2819--2848.</p><p><strong>Abstract:</strong><br/>
We solve the optimal control problem of a one-dimensional reflected stochastic differential equation, whose coefficients can be path dependent. The value function of this problem is characterized by a backward stochastic partial differential equation (BSPDE) with Neumann boundary conditions. We prove the existence and uniqueness of a sufficiently regular solution for this BSPDE, which is then used to construct the optimal feedback control. In fact, we prove a more general result: the existence and uniqueness of strong solution for the Neumann problem for general nonlinear BSPDEs, which might be of interest even out of the current context.
</p>projecteuclid.org/euclid.aoap/1571385623_20191018040042Fri, 18 Oct 2019 04:00 EDTDynamics of observables in rank-based models and performance of functionally generated portfolioshttps://projecteuclid.org/euclid.aoap/1571385624<strong>Sergio A. Almada Monter</strong>, <strong>Mykhaylo Shkolnikov</strong>, <strong>Jiacheng Zhang</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 2849--2883.</p><p><strong>Abstract:</strong><br/>
In the seminal work ( Stochastic Portfolio Theory: Stochastic Modelling and Applied Probability (2002) Springer), several macroscopic market observables have been introduced, in an attempt to find characteristics capturing the diversity of a financial market. Despite the crucial importance of such observables for investment decisions, a concise mathematical description of their dynamics has been missing. We fill this gap in the setting of rank-based models. The results are then used to study the performance of multiplicatively and additively functionally generated portfolios.
</p>projecteuclid.org/euclid.aoap/1571385624_20191018040042Fri, 18 Oct 2019 04:00 EDTMeasuring sample quality with diffusionshttps://projecteuclid.org/euclid.aoap/1571385625<strong>Jackson Gorham</strong>, <strong>Andrew B. Duncan</strong>, <strong>Sebastian J. Vollmer</strong>, <strong>Lester Mackey</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 2884--2928.</p><p><strong>Abstract:</strong><br/>
Stein’s method for measuring convergence to a continuous target distribution relies on an operator characterizing the target and Stein factor bounds on the solutions of an associated differential equation. While such operators and bounds are readily available for a diversity of univariate targets, few multivariate targets have been analyzed. We introduce a new class of characterizing operators based on Itô diffusions and develop explicit multivariate Stein factor bounds for any target with a fast-coupling Itô diffusion. As example applications, we develop computable and convergence-determining diffusion Stein discrepancies for log-concave, heavy-tailed and multimodal targets and use these quality measures to select the hyperparameters of biased Markov chain Monte Carlo (MCMC) samplers, compare random and deterministic quadrature rules and quantify bias-variance tradeoffs in approximate MCMC. Our results establish a near-linear relationship between diffusion Stein discrepancies and Wasserstein distances, improving upon past work even for strongly log-concave targets. The exposed relationship between Stein factors and Markov process coupling may be of independent interest.
</p>projecteuclid.org/euclid.aoap/1571385625_20191018040042Fri, 18 Oct 2019 04:00 EDTInterference queueing networks on gridshttps://projecteuclid.org/euclid.aoap/1571385626<strong>Abishek Sankararaman</strong>, <strong>François Baccelli</strong>, <strong>Sergey Foss</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 2929--2987.</p><p><strong>Abstract:</strong><br/>
Consider a countably infinite collection of interacting queues, with a queue located at each point of the $d$-dimensional integer grid, having independent Poisson arrivals, but dependent service rates. The service discipline is of the processor sharing type, with the service rate in each queue slowed down, when the neighboring queues have a larger workload. The interactions are translation invariant in space and is neither of the Jackson Networks type, nor of the mean-field type. Coupling and percolation techniques are first used to show that this dynamics has well-defined trajectories. Coupling from the past techniques are then proposed to build its minimal stationary regime. The rate conservation principle of Palm calculus is then used to identify the stability condition of this system, where the notion of stability is appropriately defined for an infinite dimensional process. We show that the identified condition is also necessary in certain special cases and conjecture it to be true in all cases. Remarkably, the rate conservation principle also provides a closed-form expression for the mean queue size. When the stability condition holds, this minimal solution is the unique translation invariant stationary regime. In addition, there exists a range of small initial conditions for which the dynamics is attracted to the minimal regime. Nevertheless, there exists another range of larger though finite initial conditions for which the dynamics diverges, even though stability criterion holds.
</p>projecteuclid.org/euclid.aoap/1571385626_20191018040042Fri, 18 Oct 2019 04:00 EDTApproximating mixed Hölder functions using random sampleshttps://projecteuclid.org/euclid.aoap/1571385627<strong>Nicholas F. Marshall</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 2988--3005.</p><p><strong>Abstract:</strong><br/>
Suppose $f:[0,1]^{2}\rightarrow \mathbb{R}$ is a $(c,\alpha )$-mixed Hölder function that we sample at $l$ points $X_{1},\ldots ,X_{l}$ chosen uniformly at random from the unit square. Let the location of these points and the function values $f(X_{1}),\ldots ,f(X_{l})$ be given. If $l\ge c_{1}n\log^{2}n$, then we can compute an approximation $\tilde{f}$ such that \begin{equation*}\|f-\tilde{f}\|_{L^{2}}=\mathcal{O}\big(n^{-\alpha}\log^{3/2}n\big),\end{equation*} with probability at least $1-n^{2-c_{1}}$, where the implicit constant only depends on the constants $c>0$ and $c_{1}>0$.
</p>projecteuclid.org/euclid.aoap/1571385627_20191018040042Fri, 18 Oct 2019 04:00 EDTLocal law and Tracy–Widom limit for sparse sample covariance matriceshttps://projecteuclid.org/euclid.aoap/1571385628<strong>Jong Yun Hwang</strong>, <strong>Ji Oon Lee</strong>, <strong>Kevin Schnelli</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 3006--3036.</p><p><strong>Abstract:</strong><br/>
We consider spectral properties of sparse sample covariance matrices, which includes biadjacency matrices of the bipartite Erdős–Rényi graph model. We prove a local law for the eigenvalue density up to the upper spectral edge. Under a suitable condition on the sparsity, we also prove that the limiting distribution of the rescaled, shifted extremal eigenvalues is given by the GOE Tracy–Widom law with an explicit formula on the deterministic shift of the spectral edge. For the biadjacency matrix of an Erdős–Rényi graph with two vertex sets of comparable sizes $M$ and $N$, this establishes Tracy–Widom fluctuations of the second largest eigenvalue when the connection probability $p$ is much larger than $N^{-2/3}$ with a deterministic shift of order $(Np)^{-1}$.
</p>projecteuclid.org/euclid.aoap/1571385628_20191018040042Fri, 18 Oct 2019 04:00 EDTAnother look into the Wong–Zakai theorem for stochastic heat equationhttps://projecteuclid.org/euclid.aoap/1571385629<strong>Yu Gu</strong>, <strong>Li-Cheng Tsai</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 3037--3061.</p><p><strong>Abstract:</strong><br/>
For the heat equation driven by a smooth, Gaussian random potential: \begin{equation*}\partial_{t}u_{\varepsilon}=\frac{1}{2}\Delta u_{\varepsilon}+u_{\varepsilon}(\xi_{\varepsilon}-c_{\varepsilon}),\quad t>0,x\in\mathbb{R},\end{equation*} where $\xi_{\varepsilon}$ converges to a spacetime white noise, and $c_{\varepsilon}$ is a diverging constant chosen properly, we prove that $u_{\varepsilon}$ converges in $L^{n}$ to the solution of the stochastic heat equation for any $n\geq1$. Our proof is probabilistic, hence provides another perspective of the general result of Hairer and Pardoux ( J. Math. Soc. Japan 67 (2015) 1551–1604), for the special case of the stochastic heat equation. We also discuss the transition from homogenization to stochasticity.
</p>projecteuclid.org/euclid.aoap/1571385629_20191018040042Fri, 18 Oct 2019 04:00 EDTPathwise convergence of the hard spheres Kac processhttps://projecteuclid.org/euclid.aoap/1571385630<strong>Daniel Heydecker</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 3062--3127.</p><p><strong>Abstract:</strong><br/>
We derive two estimates for the deviation of the $N$-particle, hard-spheres Kac process from the corresponding Boltzmann equation, measured in expected Wasserstein distance. Particular care is paid to the long-time properties of our estimates, exploiting the stability properties of the limiting Boltzmann equation at the level of realisations of the interacting particle system. As a consequence, we obtain an estimate for the propagation of chaos, uniformly in time and with polynomial rates, as soon as the initial data has a $k$th moment, $k>2$. Our approach is similar to Kac’s proposal of relating the long-time behaviour of the particle system to that of the limit equation. Along the way, we prove a new estimate for the continuity of the Boltzmann flow measured in Wasserstein distance.
</p>projecteuclid.org/euclid.aoap/1571385630_20191018040042Fri, 18 Oct 2019 04:00 EDTThe zealot voter modelhttps://projecteuclid.org/euclid.aoap/1571385631<strong>Ran Huo</strong>, <strong>Rick Durrett</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 3128--3154.</p><p><strong>Abstract:</strong><br/>
Inspired by the spread of discontent as in the 2016 presidential election, we consider a voter model in which 0’s are ordinary voters and 1’s are zealots. Thinking of a social network, but desiring the simplicity of an infinite object that can have a nontrivial stationary distribution, space is represented by a tree. The dynamics are a variant of the biased voter: if $x$ has degree $d(x)$ then at rate $d(x)p_{k}$ the individual at $x$ consults $k\ge 1$ neighbors. If at least one neighbor is 1, they adopt state 1, otherwise they become 0. In addition at rate $p_{0}$ individuals with opinion 1 change to 0. As in the contact process on trees, we are interested in determining when the zealots survive and when they will survive locally.
</p>projecteuclid.org/euclid.aoap/1571385631_20191018040042Fri, 18 Oct 2019 04:00 EDTAffine Volterra processeshttps://projecteuclid.org/euclid.aoap/1571385632<strong>Eduardo Abi Jaber</strong>, <strong>Martin Larsson</strong>, <strong>Sergio Pulido</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 3155--3200.</p><p><strong>Abstract:</strong><br/>
We introduce affine Volterra processes, defined as solutions of certain stochastic convolution equations with affine coefficients. Classical affine diffusions constitute a special case, but affine Volterra processes are neither semimartingales, nor Markov processes in general. We provide explicit exponential-affine representations of the Fourier–Laplace functional in terms of the solution of an associated system of deterministic integral equations of convolution type, extending well-known formulas for classical affine diffusions. For specific state spaces, we prove existence, uniqueness, and invariance properties of solutions of the corresponding stochastic convolution equations. Our arguments avoid infinite-dimensional stochastic analysis as well as stochastic integration with respect to non-semimartingales, relying instead on tools from the theory of finite-dimensional deterministic convolution equations. Our findings generalize and clarify recent results in the literature on rough volatility models in finance.
</p>projecteuclid.org/euclid.aoap/1571385632_20191018040042Fri, 18 Oct 2019 04:00 EDTApproximating stationary distributions of fast mixing Glauber dynamics, with applications to exponential random graphshttps://projecteuclid.org/euclid.aoap/1571385633<strong>Gesine Reinert</strong>, <strong>Nathan Ross</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 3201--3229.</p><p><strong>Abstract:</strong><br/>
We provide a general bound on the Wasserstein distance between two arbitrary distributions of sequences of Bernoulli random variables. The bound is in terms of a mixing quantity for the Glauber dynamics of one of the sequences, and a simple expectation of the other. The result is applied to estimate, with explicit error, expectations of functions of random vectors for some Ising models and exponential random graphs in “high temperature” regimes.
</p>projecteuclid.org/euclid.aoap/1571385633_20191018040042Fri, 18 Oct 2019 04:00 EDTStein’s method for stationary distributions of Markov chains and application to Ising modelshttps://projecteuclid.org/euclid.aoap/1571385634<strong>Guy Bresler</strong>, <strong>Dheeraj Nagaraj</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 3230--3265.</p><p><strong>Abstract:</strong><br/>
We develop a new technique, based on Stein’s method, for comparing two stationary distributions of irreducible Markov chains whose update rules are close in a certain sense. We apply this technique to compare Ising models on $d$-regular expander graphs to the Curie–Weiss model (complete graph) in terms of pairwise correlations and more generally $k$th order moments. Concretely, we show that $d$-regular Ramanujan graphs approximate the $k$th order moments of the Curie–Weiss model to within average error $k/\sqrt{d}$ (averaged over size $k$ subsets), independent of graph size . The result applies even in the low-temperature regime; we also derive simpler approximation results for functionals of Ising models that hold only at high temperatures.
</p>projecteuclid.org/euclid.aoap/1571385634_20191018040042Fri, 18 Oct 2019 04:00 EDTCorrection note: A strong order 1/2 method for multidimensional SDEs with discontinuous drifthttps://projecteuclid.org/euclid.aoap/1571385635<strong>Gunther Leobacher</strong>, <strong>Michaela Szölgyenyi</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 5, 3266--3269.</p>projecteuclid.org/euclid.aoap/1571385635_20191018040042Fri, 18 Oct 2019 04:00 EDTStochastic representations for solutions to parabolic Dirichlet problems for nonlocal Bellman equationshttps://projecteuclid.org/euclid.aoap/1578366315<strong>Ruoting Gong</strong>, <strong>Chenchen Mou</strong>, <strong>Andrzej Święch</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 6, 3271--3310.</p><p><strong>Abstract:</strong><br/>
We prove a stochastic representation formula for the viscosity solution of Dirichlet terminal-boundary value problem for a degenerate Hamilton–Jacobi–Bellman integro-partial differential equation in a bounded domain. We show that the unique viscosity solution is the value function of the associated stochastic optimal control problem. We also obtain the dynamic programming principle for the associated stochastic optimal control problem in a bounded domain.
</p>projecteuclid.org/euclid.aoap/1578366315_20200106220527Mon, 06 Jan 2020 22:05 ESTComputational methods for martingale optimal transport problemshttps://projecteuclid.org/euclid.aoap/1578366316<strong>Gaoyue Guo</strong>, <strong>Jan Obłój</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 6, 3311--3347.</p><p><strong>Abstract:</strong><br/>
We develop computational methods for solving the martingale optimal transport (MOT) problem—a version of the classical optimal transport with an additional martingale constraint on the transport’s dynamics. We prove that a general, multi-step multi-dimensional, MOT problem can be approximated through a sequence of linear programming (LP) problems which result from a discretization of the marginal distributions combined with an appropriate relaxation of the martingale condition. Further, we establish two generic approaches for discretising probability distributions, suitable respectively for the cases when we can compute integrals against these distributions or when we can sample from them. These render our main result applicable and lead to an implementable numerical scheme for solving MOT problems. Finally, specialising to the one-step model on real line, we provide an estimate of the convergence rate which, to the best of our knowledge, is the first of its kind in the literature.
</p>projecteuclid.org/euclid.aoap/1578366316_20200106220527Mon, 06 Jan 2020 22:05 ESTRobust pricing and hedging around the globehttps://projecteuclid.org/euclid.aoap/1578366317<strong>Sebastian Herrmann</strong>, <strong>Florian Stebegg</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 6, 3348--3386.</p><p><strong>Abstract:</strong><br/>
We consider the martingale optimal transport duality for càdlàg processes with given initial and terminal laws. Strong duality and existence of dual optimizers (robust semistatic superhedging strategies) are proved for a class of payoffs that includes American, Asian, Bermudan and European options with intermediate maturity. We exhibit an optimal superhedging strategy for which the static part solves an auxiliary problem and the dynamic part is given explicitly in terms of the static part.
</p>projecteuclid.org/euclid.aoap/1578366317_20200106220527Mon, 06 Jan 2020 22:05 ESTAffine processes beyond stochastic continuityhttps://projecteuclid.org/euclid.aoap/1578366318<strong>Martin Keller-Ressel</strong>, <strong>Thorsten Schmidt</strong>, <strong>Robert Wardenga</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 6, 3387--3437.</p><p><strong>Abstract:</strong><br/>
In this paper, we study time-inhomogeneous affine processes beyond the common assumption of stochastic continuity. In this setting, times of jumps can be both inaccessible and predictable. To this end, we develop a general theory of finite dimensional affine semimartingales under very weak assumptions. We show that the corresponding semimartingale characteristics have affine form and that the conditional characteristic function can be represented with solutions to measure differential equations of Riccati type. We prove existence of affine Markov processes and affine semimartingales under mild conditions and elaborate on examples and applications including affine processes in discrete time.
</p>projecteuclid.org/euclid.aoap/1578366318_20200106220527Mon, 06 Jan 2020 22:05 ESTPoincaré and logarithmic Sobolev constants for metastable Markov chains via capacitary inequalitieshttps://projecteuclid.org/euclid.aoap/1578366319<strong>André Schlichting</strong>, <strong>Martin Slowik</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 6, 3438--3488.</p><p><strong>Abstract:</strong><br/>
We investigate the metastable behavior of reversible Markov chains on possibly countable infinite state spaces. Based on a new definition of metastable Markov processes, we compute precisely the mean transition time between metastable sets. Under additional size and regularity properties of metastable sets, we establish asymptotic sharp estimates on the Poincaré and logarithmic Sobolev constant. The main ingredient in the proof is a capacitary inequality along the lines of V. Maz’ya that relates regularity properties of harmonic functions and capacities. We exemplify the usefulness of this new definition in the context of the random field Curie–Weiss model, where metastability and the additional regularity assumptions are verifiable.
</p>projecteuclid.org/euclid.aoap/1578366319_20200106220527Mon, 06 Jan 2020 22:05 ESTA martingale approach for fractional Brownian motions and related path dependent PDEshttps://projecteuclid.org/euclid.aoap/1578366320<strong>Frederi Viens</strong>, <strong>Jianfeng Zhang</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 6, 3489--3540.</p><p><strong>Abstract:</strong><br/>
In this paper, we study dynamic backward problems, with the computation of conditional expectations as a special objective, in a framework where the (forward) state process satisfies a Volterra type SDE, with fractional Brownian motion as a typical example. Such processes are neither Markov processes nor semimartingales, and most notably, they feature a certain time inconsistency which makes any direct application of Markovian ideas, such as flow properties, impossible without passing to a path-dependent framework. Our main result is a functional Itô formula, extending the seminal work of Dupire ( Quant. Finance 19 (2019) 721–729) to our more general framework. In particular, unlike in ( Quant. Finance 19 (2019) 721–729) where one needs only to consider the stopped paths, here we need to concatenate the observed path up to the current time with a certain smooth observable curve derived from the distribution of the future paths. This new feature is due to the time inconsistency involved in this paper. We then derive the path dependent PDEs for the backward problems. Finally, an application to option pricing and hedging in a financial market with rough volatility is presented.
</p>projecteuclid.org/euclid.aoap/1578366320_20200106220527Mon, 06 Jan 2020 22:05 ESTCrossing a fitness valley as a metastable transition in a stochastic population modelhttps://projecteuclid.org/euclid.aoap/1578366321<strong>Anton Bovier</strong>, <strong>Loren Coquille</strong>, <strong>Charline Smadi</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 6, 3541--3589.</p><p><strong>Abstract:</strong><br/>
We consider a stochastic model of population dynamics where each individual is characterised by a trait in $\{0,1,\ldots,L\}$ and has a natural reproduction rate, a logistic death rate due to age or competition and a probability of mutation towards neighbouring traits at each reproduction event. We choose parameters such that the induced fitness landscape exhibits a valley: mutant individuals with negative fitness have to be created in order for the population to reach a trait with positive fitness. We focus on the limit of large population and rare mutations at several speeds. In particular, when the mutation rate is low enough, metastability occurs: the exit time of the valley is an exponentially distributed random variable.
</p>projecteuclid.org/euclid.aoap/1578366321_20200106220527Mon, 06 Jan 2020 22:05 ESTNonparametric spot volatility from optionshttps://projecteuclid.org/euclid.aoap/1578366322<strong>Viktor Todorov</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 6, 3590--3636.</p><p><strong>Abstract:</strong><br/>
We propose a nonparametric estimator of spot volatility from noisy short-dated option data. The estimator is based on forming portfolios of options with different strikes that replicate the (risk-neutral) conditional characteristic function of the underlying price in a model-free way. The separation of volatility from jumps is done by making use of the dominant role of the volatility in the conditional characteristic function over short time intervals and for large values of the characteristic exponent. The latter is chosen in an adaptive way in order to account for the time-varying volatility. We show that the volatility estimator is near rate-optimal in minimax sense. We further derive a feasible joint central limit theorem for the proposed option-based volatility estimator and existing high-frequency return-based volatility estimators. The limit distribution is mixed Gaussian reflecting the time-varying precision in the volatility recovery.
</p>projecteuclid.org/euclid.aoap/1578366322_20200106220527Mon, 06 Jan 2020 22:05 ESTRight marker speeds of solutions to the KPP equation with noisehttps://projecteuclid.org/euclid.aoap/1578366323<strong>Sandra Kliem</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 6, 3637--3694.</p><p><strong>Abstract:</strong><br/>
We consider the one-dimensional KPP-equation driven by space–time white noise. We show that for all parameters above the critical value for survival, there exist stochastic wavelike solutions which travel with a deterministic positive linear speed. We further give a sufficient condition on the initial condition of a solution to attain this speed. Our approach is in the spirit of corresponding results for the nearest-neighbor contact process respectively oriented percolation. Here, the main difficulty arises from the moderate size of the parameter and the long range interaction. Stopping times and averaging techniques are used to overcome this difficulty.
</p>projecteuclid.org/euclid.aoap/1578366323_20200106220527Mon, 06 Jan 2020 22:05 ESTLarge tournament gameshttps://projecteuclid.org/euclid.aoap/1578366324<strong>Erhan Bayraktar</strong>, <strong>Jakša Cvitanić</strong>, <strong>Yuchong Zhang</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 6, 3695--3744.</p><p><strong>Abstract:</strong><br/>
We consider a stochastic tournament game in which each player is rewarded based on her rank in terms of the completion time of her own task and is subject to cost of effort. When players are homogeneous and the rewards are purely rank dependent, the equilibrium has a surprisingly explicit characterization, which allows us to conduct comparative statics and obtain explicit solution to several optimal reward design problems. In the general case when the players are heterogenous and payoffs are not purely rank dependent, we prove the existence, uniqueness and stability of the Nash equilibrium of the associated mean field game, and the existence of an approximate Nash equilibrium of the finite-player game.
</p>projecteuclid.org/euclid.aoap/1578366324_20200106220527Mon, 06 Jan 2020 22:05 ESTLocalization of the Gaussian multiplicative chaos in the Wiener space and the stochastic heat equation in strong disorderhttps://projecteuclid.org/euclid.aoap/1578366325<strong>Yannic Bröker</strong>, <strong>Chiranjib Mukherjee</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 6, 3745--3785.</p><p><strong>Abstract:</strong><br/>
We consider a Gaussian multiplicative chaos (GMC) measure on the classical Wiener space driven by a smoothened (Gaussian) space-time white noise. For $d\geq 3$, it was shown in ( Electron. Commun. Probab. 21 (2016) 61) that for small noise intensity, the total mass of the GMC converges to a strictly positive random variable, while larger disorder strength (i.e., low temperature) forces the total mass to lose uniform integrability, eventually producing a vanishing limit. Inspired by strong localization phenomena for log-correlated Gaussian fields and Gaussian multiplicative chaos in the finite dimensional Euclidean spaces ( Ann. Appl. Probab. 26 (2016) 643–690; Adv. Math. 330 (2018) 589–687), and related results for discrete directed polymers ( Probab. Theory Related Fields 138 (2007) 391–410; Bates and Chatterjee (2016)), we study the endpoint distribution of a Brownian path under the renormalized GMC measure in this setting. We show that in the low temperature regime, the energy landscape of the system freezes and enters the so-called glassy phase as the entire mass of the Cesàro average of the endpoint GMC distribution stays localized in few spatial islands, forcing the endpoint GMC to be asymptotically purely atomic ( Probab. Theory Related Fields 138 (2007) 391–410). The method of our proof is based on the translation-invariant compactification introduced in ( Ann. Probab. 44 (2016) 3934–3964) and a fixed point approach related to the cavity method from spin glasses recently used in (Bates and Chatterjee (2016)) in the context of the directed polymer model in the lattice.
</p>projecteuclid.org/euclid.aoap/1578366325_20200106220527Mon, 06 Jan 2020 22:05 ESTDecompositions of log-correlated fields with applicationshttps://projecteuclid.org/euclid.aoap/1578366326<strong>Janne Junnila</strong>, <strong>Eero Saksman</strong>, <strong>Christian Webb</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 6, 3786--3820.</p><p><strong>Abstract:</strong><br/>
In this article, we establish novel decompositions of Gaussian fields taking values in suitable spaces of generalized functions, and then use these decompositions to prove results about Gaussian multiplicative chaos.
We prove two decomposition theorems. The first one is a global one and says that if the difference between the covariance kernels of two Gaussian fields, taking values in some Sobolev space, has suitable Sobolev regularity, then these fields differ by a Hölder continuous Gaussian process. Our second decomposition theorem is more specialized and is in the setting of Gaussian fields whose covariance kernel has a logarithmic singularity on the diagonal—or log-correlated Gaussian fields. The theorem states that any log-correlated Gaussian field $X$ can be decomposed locally into a sum of a Hölder continuous function and an independent almost $\star $-scale invariant field (a special class of stationary log-correlated fields with ’cone-like’ white noise representations). This decomposition holds whenever the term $g$ in the covariance kernel $C_{X}(x,y)=\log (1/|x-y|)+g(x,y)$ has locally $H^{d+\varepsilon }$ Sobolev smoothness.
We use these decompositions to extend several results that have been known basically only for $\star $-scale invariant fields to general log-correlated fields. These include the existence of critical multiplicative chaos, analytic continuation of the subcritical chaos in the so-called inverse temperature parameter $\beta $, as well as generalised Onsager-type covariance inequalities which play a role in the study of imaginary multiplicative chaos.
</p>projecteuclid.org/euclid.aoap/1578366326_20200106220527Mon, 06 Jan 2020 22:05 ESTZero temperature limit for the Brownian directed polymer among Poissonian disastershttps://projecteuclid.org/euclid.aoap/1578366327<strong>Ryoki Fukushima</strong>, <strong>Stefan Junk</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 6, 3821--3860.</p><p><strong>Abstract:</strong><br/>
We study a continuum model of directed polymer in random environment. The law of the polymer is defined as the Brownian motion conditioned to survive among space-time Poissonian disasters. This model is well studied in the positive temperature regime. However, at zero-temperature, even the existence of the free energy has not been proved. In this article, we show that the free energy exists and is continuous at zero-temperature.
</p>projecteuclid.org/euclid.aoap/1578366327_20200106220527Mon, 06 Jan 2020 22:05 ESTCutoff for the cyclic adjacent transposition shufflehttps://projecteuclid.org/euclid.aoap/1578366328<strong>Danny Nam</strong>, <strong>Evita Nestoridi</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 6, 3861--3892.</p><p><strong>Abstract:</strong><br/>
We study the cyclic adjacent transposition (CAT) shuffle of $n$ cards, which is a systematic scan version of the random adjacent transposition (AT) card shuffle. In this paper, we prove that the CAT shuffle exhibits cutoff at $\frac{n^{3}}{2\pi^{2}}\log n$, which concludes that it is twice as fast as the AT shuffle. This is the first verification of cutoff phenomenon for a time-inhomogeneous card shuffle.
</p>projecteuclid.org/euclid.aoap/1578366328_20200106220527Mon, 06 Jan 2020 22:05 ESTContinuous-time duality for superreplication with transient price impacthttps://projecteuclid.org/euclid.aoap/1578366329<strong>Peter Bank</strong>, <strong>Yan Dolinsky</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 29, Number 6, 3893--3917.</p><p><strong>Abstract:</strong><br/>
We establish a superreplication duality in a continuous-time financial model as in (Bank and Voß (2018)) where an investor’s trades adversely affect bid- and ask-prices for a risky asset and where market resilience drives the resulting spread back towards zero at an exponential rate. Similar to the literature on models with a constant spread (cf., e.g., Math. Finance 6 (1996) 133–165; Ann. Appl. Probab. 20 (2010) 1341–1358; Ann. Appl. Probab. 27 (2017) 1414–1451), our dual description of superreplication prices involves the construction of suitable absolutely continuous measures with martingales close to the unaffected reference price. A novel feature in our duality is a liquidity weighted $L^{2}$-norm that enters as a measurement of this closeness and that accounts for strategy dependent spreads. As applications, we establish optimality of buy-and-hold strategies for the superreplication of call options and we prove a verification theorem for utility maximizing investment strategies.
</p>projecteuclid.org/euclid.aoap/1578366329_20200106220527Mon, 06 Jan 2020 22:05 ESTGeneralized couplings and ergodic rates for SPDEs and other Markov modelshttps://projecteuclid.org/euclid.aoap/1582621218<strong>Oleg Butkovsky</strong>, <strong>Alexei Kulik</strong>, <strong>Michael Scheutzow</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 30, Number 1, 1--39.</p><p><strong>Abstract:</strong><br/>
We establish verifiable general sufficient conditions for exponential or subexponential ergodicity of Markov processes that may lack the strong Feller property. We apply the obtained results to show exponential ergodicity of a variety of nonlinear stochastic partial differential equations with additive forcing, including 2D stochastic Navier–Stokes equations. Our main tool is a new version of the generalized coupling method.
</p>projecteuclid.org/euclid.aoap/1582621218_20200225040028Tue, 25 Feb 2020 04:00 ESTLocal weak convergence for PageRankhttps://projecteuclid.org/euclid.aoap/1582621219<strong>Alessandro Garavaglia</strong>, <strong>Remco van der Hofstad</strong>, <strong>Nelly Litvak</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 30, Number 1, 40--79.</p><p><strong>Abstract:</strong><br/>
PageRank is a well-known algorithm for measuring centrality in networks. It was originally proposed by Google for ranking pages in the World Wide Web. One of the intriguing empirical properties of PageRank is the so-called ‘power-law hypothesis’: in a scale-free network, the PageRank scores follow a power law with the same exponent as the (in-)degrees. To date, this hypothesis has been confirmed empirically and in several specific random graphs models. In contrast, this paper does not focus on one random graph model but investigates the existence of an asymptotic PageRank distribution, when the graph size goes to infinity, using local weak convergence. This may help to identify general network structures in which the power-law hypothesis holds. We start from the definition of local weak convergence for sequences of (random) undirected graphs, and extend this notion to directed graphs. To this end, we define an exploration process in the directed setting that keeps track of in- and out-degrees of vertices. Then we use this to prove the existence of an asymptotic PageRank distribution. As a result, the limiting distribution of PageRank can be computed directly as a function of the limiting object. We apply our results to the directed configuration model and continuous-time branching processes trees, as well as to preferential attachment models.
</p>projecteuclid.org/euclid.aoap/1582621219_20200225040028Tue, 25 Feb 2020 04:00 ESTJoin-the-Shortest Queue diffusion limit in Halfin–Whitt regime: Sensitivity on the heavy-traffic parameterhttps://projecteuclid.org/euclid.aoap/1582621220<strong>Sayan Banerjee</strong>, <strong>Debankur Mukherjee</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 30, Number 1, 80--144.</p><p><strong>Abstract:</strong><br/>
Consider a system of $N$ parallel single-server queues with unit-exponential 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) identified a novel limiting diffusion process that arises as the weak-limit of the appropriately scaled occupancy measure of the system under the JSQ policy in the Halfin–Whitt regime, where $(N-\lambda (N))/\sqrt{N}\to \beta >0$ as $N\to \infty$. The analysis of this diffusion goes beyond the state of the art techniques, and even proving its ergodicity is nontrivial, and was left as an open question. Recently, exploiting a generator expansion framework via the Stein’s method, Braverman (2018) established its exponential ergodicity, and adapting a regenerative approach, Banerjee and Mukherjee ( Ann. Appl. Probab. 29 (2018) 1262–1309) analyzed the tail properties of the stationary distribution and path fluctuations of the diffusion.
However, the analysis of the bulk behavior of the stationary distribution, namely, the moments, remained intractable until this work. In this paper, we perform a thorough analysis of the bulk behavior of the stationary distribution of the diffusion process, and discover that it exhibits different qualitative behavior, depending on the value of the heavy-traffic parameter $\beta$. Moreover, we obtain precise asymptotic laws of the centered and scaled steady-state distribution, as $\beta $ tends to 0 and $\infty$. Of particular interest, we also establish a certain intermittency phenomena in the $\beta \to \infty$ regime and a surprising distributional convergence result in the $\beta \to 0$ regime.
</p>projecteuclid.org/euclid.aoap/1582621220_20200225040028Tue, 25 Feb 2020 04:00 ESTBootstrap percolation on the product of the two-dimensional lattice with a Hamming squarehttps://projecteuclid.org/euclid.aoap/1582621221<strong>Janko Gravner</strong>, <strong>David Sivakoff</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 30, Number 1, 145--174.</p><p><strong>Abstract:</strong><br/>
Bootstrap percolation on a graph is a deterministic process that iteratively enlarges a set of occupied sites by adjoining points with at least $\theta $ occupied neighbors. The initially occupied set is random, given by a uniform product measure with a low density $p$. Our main focus is on this process on the product graph $\mathbb{Z}^{2}\times K_{n}^{2}$, where $K_{n}$ is a complete graph. We investigate how $p$ scales with $n$ so that a typical site is eventually occupied. Under critical scaling, the dynamics with even $\theta $ exhibits a sharp phase transition, while odd $\theta $ yields a gradual percolation transition. We also establish a gradual transition for bootstrap percolation on $\mathbb{Z}^{2}\times K_{n}$. The community structure of the product graphs connects our process to a heterogeneous bootstrap percolation on $\mathbb{Z}^{2}$. This natural relation with a generalization of polluted bootstrap percolation is the leading theme in our analysis.
</p>projecteuclid.org/euclid.aoap/1582621221_20200225040028Tue, 25 Feb 2020 04:00 ESTPropagation of chaos for stochastic spatially structured neuronal networks with delay driven by jump diffusionshttps://projecteuclid.org/euclid.aoap/1582621222<strong>Sima Mehri</strong>, <strong>Michael Scheutzow</strong>, <strong>Wilhelm Stannat</strong>, <strong>Bian Z. Zangeneh</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 30, Number 1, 175--207.</p><p><strong>Abstract:</strong><br/>
Spatially structured neural networks driven by jump diffusion noise with monotone coefficients, fully path dependent delay and with a disorder parameter are considered. Well-posedness for the associated McKean–Vlasov equation and a corresponding propagation of chaos result in the infinite population limit are proven. Our existence result for the McKean–Vlasov equation is based on the Euler approximation that is applied to this type of equation for the first time.
</p>projecteuclid.org/euclid.aoap/1582621222_20200225040028Tue, 25 Feb 2020 04:00 ESTOn an epidemic model on finite graphshttps://projecteuclid.org/euclid.aoap/1582621223<strong>Itai Benjamini</strong>, <strong>Luiz Renato Fontes</strong>, <strong>Jonathan Hermon</strong>, <strong>Fábio Prates Machado</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 30, Number 1, 208--258.</p><p><strong>Abstract:</strong><br/>
We study a system of random walks, known as the frog model, starting from a profile of independent Poisson($\lambda $) particles per site, with one additional active particle planted at some vertex $\mathbf{o}$ of a finite connected simple graph $G=(V,E)$. Initially, only the particles occupying $\mathbf{o}$ are active. Active particles perform $t\in \mathbb{N}\cup \{\infty \}$ steps of the walk they picked before vanishing and activate all inactive particles they hit. This system is often taken as a model for the spread of an epidemic over a population. Let $\mathcal{R}_{t}$ be the set of vertices which are visited by the process, when active particles vanish after $t$ steps. We study the susceptibility of the process on the underlying graph, defined as the random quantity $\mathcal{S}(G):=\inf \{t:\mathcal{R}_{t}=V\}$ (essentially, the shortest particles’ lifespan required for the entire population to get infected). We consider the cases that the underlying graph is either a regular expander or a $d$-dimensional torus of side length $n$ (for all $d\ge 1$) $\mathbb{T}_{d}(n)$ and determine the asymptotic behavior of $\mathcal{S}$ up to a constant factor. In fact, throughout we allow the particle density ${\lambda }$ to depend on $n$ and for $d\ge 2$ we determine the asymptotic behavior of $\mathcal{S}(\mathbb{T}_{d}(n))$ up to smaller order terms for a wide range of ${\lambda }={\lambda }_{n}$.
</p>projecteuclid.org/euclid.aoap/1582621223_20200225040028Tue, 25 Feb 2020 04:00 ESTConvergence to the mean field game limit: A case studyhttps://projecteuclid.org/euclid.aoap/1582621224<strong>Marcel Nutz</strong>, <strong>Jaime San Martin</strong>, <strong>Xiaowei Tan</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 30, Number 1, 259--286.</p><p><strong>Abstract:</strong><br/>
We study the convergence of Nash equilibria in a game of optimal stopping. If the associated mean field game has a unique equilibrium, any sequence of $n$-player equilibria converges to it as $n\to \infty$. However, both the finite and infinite player versions of the game often admit multiple equilibria. We show that mean field equilibria satisfying a transversality condition are limit points of $n$-player equilibria, but we also exhibit a remarkable class of mean field equilibria that are not limits, thus questioning their interpretation as “large $n$” equilibria.
</p>projecteuclid.org/euclid.aoap/1582621224_20200225040028Tue, 25 Feb 2020 04:00 ESTCentral limit theorems for patterns in multiset permutations and set partitionshttps://projecteuclid.org/euclid.aoap/1582621225<strong>Valentin Féray</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 30, Number 1, 287--323.</p><p><strong>Abstract:</strong><br/>
We use the recently developed method of weighted dependency graphs to prove central limit theorems for the number of occurrences of any fixed pattern in multiset permutations and in set partitions. This generalizes results for patterns of size 2 in both settings, obtained by Canfield, Janson and Zeilberger and Chern, Diaconis, Kane and Rhoades, respectively.
</p>projecteuclid.org/euclid.aoap/1582621225_20200225040028Tue, 25 Feb 2020 04:00 ESTLarge deviation principles for first-order scalar conservation laws with stochastic forcinghttps://projecteuclid.org/euclid.aoap/1582621226<strong>Zhao Dong</strong>, <strong>Jiang-Lun Wu</strong>, <strong>Rangrang Zhang</strong>, <strong>Tusheng Zhang</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 30, Number 1, 324--367.</p><p><strong>Abstract:</strong><br/>
In this paper, we established the Freidlin–Wentzell-type large deviation principles for first-order scalar conservation laws perturbed by small multiplicative noise. Due to the lack of the viscous terms in the stochastic equations, the kinetic solution to the Cauchy problem for these first-order conservation laws is studied. Then, based on the well-posedness of the kinetic solutions, we show that the large deviations holds by utilising the weak convergence approach.
</p>projecteuclid.org/euclid.aoap/1582621226_20200225040028Tue, 25 Feb 2020 04:00 ESTMonte Carlo with determinantal point processeshttps://projecteuclid.org/euclid.aoap/1582621227<strong>Rémi Bardenet</strong>, <strong>Adrien Hardy</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 30, Number 1, 368--417.</p><p><strong>Abstract:</strong><br/>
We show that repulsive random variables can yield Monte Carlo methods with faster convergence rates than the typical $N^{-1/2}$, where $N$ is the number of integrand evaluations. More precisely, we propose stochastic numerical quadratures involving determinantal point processes associated with multivariate orthogonal polynomials, and we obtain root mean square errors that decrease as $N^{-(1+1/d)/2}$, where $d$ is the dimension of the ambient space. First, we prove a central limit theorem (CLT) for the linear statistics of a class of determinantal point processes, when the reference measure is a product measure supported on a hypercube, which satisfies the Nevai-class regularity condition; a result which may be of independent interest. Next, we introduce a Monte Carlo method based on these determinantal point processes, and prove a CLT with explicit limiting variance for the quadrature error, when the reference measure satisfies a stronger regularity condition. As a corollary, by taking a specific reference measure and using a construction similar to importance sampling, we obtain a general Monte Carlo method, which applies to any measure with continuously derivable density. Loosely speaking, our method can be interpreted as a stochastic counterpart to Gaussian quadrature, which at the price of some convergence rate, is easily generalizable to any dimension and has a more explicit error term.
</p>projecteuclid.org/euclid.aoap/1582621227_20200225040028Tue, 25 Feb 2020 04:00 ESTRandom-cluster dynamics in $\mathbb{Z}^{2}$: Rapid mixing with general boundary conditionshttps://projecteuclid.org/euclid.aoap/1582621228<strong>Antonio Blanca</strong>, <strong>Reza Gheissari</strong>, <strong>Eric Vigoda</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 30, Number 1, 418--459.</p><p><strong>Abstract:</strong><br/>
The random-cluster model with parameters $(p,q)$ is a random graph model that generalizes bond percolation ($q=1$) and the Ising and Potts models ($q\geq 2$). We study its Glauber dynamics on $n\times n$ boxes $\Lambda_{n}$ of the integer lattice graph $\mathbb{Z}^{2}$, where the model exhibits a sharp phase transition at $p=p_{c}(q)$. Unlike traditional spin systems like the Ising and Potts models, the random-cluster model has non-local interactions. Long-range interactions can be imposed as external connections in the boundary of $\Lambda_{n}$, known as boundary conditions . For select boundary conditions that do not carry long-range information (namely, wired and free), Blanca and Sinclair proved that when $q>1$ and $p\neq p_{c}(q)$, the Glauber dynamics on $\Lambda_{n}$ mixes in optimal $O(n^{2}\log n)$ time. In this paper, we prove that this mixing time is polynomial in $n$ for every boundary condition that is realizable as a configuration on $\mathbb{Z}^{2}\setminus\Lambda_{n}$. We then use this to prove near-optimal $\tilde{O}(n^{2})$ mixing time for “typical” boundary conditions. As a complementary result, we construct classes of nonrealizable (nonplanar) boundary conditions inducing slow (stretched-exponential) mixing at $p\ll p_{c}$.
</p>projecteuclid.org/euclid.aoap/1582621228_20200225040028Tue, 25 Feb 2020 04:00 ESTThe largest real eigenvalue in the real Ginibre ensemble and its relation to the Zakharov–Shabat systemhttps://projecteuclid.org/euclid.aoap/1582621229<strong>Jinho Baik</strong>, <strong>Thomas Bothner</strong>. <p><strong>Source: </strong>The Annals of Applied Probability, Volume 30, Number 1, 460--501.</p><p><strong>Abstract:</strong><br/>
The real Ginibre ensemble consists of $n\times n$ real matrices $\mathbf{X}$ whose entries are i.i.d. standard normal random variables. In sharp contrast to the complex and quaternion Ginibre ensemble, real eigenvalues in the real Ginibre ensemble attain positive likelihood. In turn, the spectral radius $R_{n}=\max_{1\leq j\leq n}|z_{j}(\mathbf{X})|$ of the eigenvalues $z_{j}(\mathbf{X})\in \mathbb{C}$ of a real Ginibre matrix $\mathbf{X}$ follows a different limiting law (as $n\rightarrow \infty $) for $z_{j}(\mathbf{X})\in \mathbb{R}$ than for $z_{j}(\mathbf{X})\in \mathbb{C}\setminus \mathbb{R}$. Building on previous work by Rider and Sinclair ( Ann. Appl. Probab. 24 (2014) 1621–1651) and Poplavskyi, Tribe and Zaboronski ( Ann. Appl. Probab. 27 (2017) 1395–1413), we show that the limiting distribution of $\max_{j:z_{j}\in \mathbb{R}}z_{j}(\mathbf{X})$ admits a closed-form expression in terms of a distinguished solution to an inverse scattering problem for the Zakharov–Shabat system. As byproducts of our analysis, we also obtain a new determinantal representation for the limiting distribution of $\max_{j:z_{j}\in \mathbb{R}}z_{j}(\mathbf{X})$ and extend recent tail estimates in ( Ann. Appl. Probab. 27 (2017) 1395–1413) via nonlinear steepest descent techniques.
</p>projecteuclid.org/euclid.aoap/1582621229_20200225040028Tue, 25 Feb 2020 04:00 ESTLower bounds for trace reconstructionhttps://projecteuclid.org/euclid.aoap/1591603214<strong>Nina Holden</strong>, <strong>Russell Lyons</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 503--525.</p><p><strong>Abstract:</strong><br/>
In the trace reconstruction problem, an unknown bit string ${\mathbf{x}}\in\{0,1\}^{n}$ is sent through a deletion channel where each bit is deleted independently with some probability $q\in(0,1)$, yielding a contracted string ${\widetilde{\mathbf{x}}}$. How many i.i.d. samples of ${\widetilde{\mathbf{x}}}$ are needed to reconstruct ${\mathbf{x}}$ with high probability? We prove that there exist ${\mathbf{x}},{\mathbf{y}}\in\{0,1\}^{n}$ such that at least $cn^{5/4}/\sqrt{\log n}$ traces are required to distinguish between ${\mathbf{x}}$ and ${\mathbf{y}}$ for some absolute constant $c$, improving the previous lower bound of $cn$. Furthermore, our result improves the previously known lower bound for reconstruction of random strings from $c\log^{2}n$ to $c\log^{9/4}n/\sqrt{\log\log n}$.
</p>projecteuclid.org/euclid.aoap/1591603214_20200608040026Mon, 08 Jun 2020 04:00 EDTAdaptive Euler–Maruyama method for SDEs with nonglobally Lipschitz drifthttps://projecteuclid.org/euclid.aoap/1591603215<strong>Wei Fang</strong>, <strong>Michael B. Giles</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 526--560.</p><p><strong>Abstract:</strong><br/>
This paper proposes an adaptive timestep construction for an Euler–Maruyama approximation of SDEs with nonglobally Lipschitz drift. It is proved that if the timestep is bounded appropriately, then over a finite time interval the numerical approximation is stable, and the expected number of timesteps is finite. Furthermore, the order of strong convergence is the same as usual, that is, order $\frac{1}{2}$ for SDEs with a nonuniform globally Lipschitz volatility, and order $1$ for Langevin SDEs with unit volatility and a drift with sufficient smoothness. For a class of ergodic SDEs, we also show that the bound for the moments and the strong error of the numerical solution are uniform in $T$, which allow us to introduce the adaptive multilevel Monte Carlo method to compute the expectations with respect to the invariant distribution. The analysis is supported by numerical experiments.
</p>projecteuclid.org/euclid.aoap/1591603215_20200608040026Mon, 08 Jun 2020 04:00 EDTMean geometry for 2D random fields: Level perimeter and level total curvature integralshttps://projecteuclid.org/euclid.aoap/1591603216<strong>Hermine Biermé</strong>, <strong>Agnès Desolneux</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 561--607.</p><p><strong>Abstract:</strong><br/>
We introduce the level perimeter integral and the total curvature integral associated with a real-valued function $f$ defined on the plane $\mathbb{R}^{2}$, as integrals allowing to compute the perimeter of the excursion set of $f$ above level $t$ and the total (signed) curvature of its boundary for almost every level $t$. Thanks to the Gauss–Bonnet theorem, the total curvature is directly related to the Euler characteristic of the excursion set. We show that the level perimeter and the total curvature integrals can be computed in two different frameworks: smooth (at least $C^{2}$) functions and piecewise constant functions (also called here elementary functions). Considering 2D random fields (in particular shot noise random fields), we compute their mean perimeter and total curvature integrals, and this provides new “explicit” computations of the mean perimeter and Euler characteristic densities of excursion sets, beyond the Gaussian framework: for piecewise constant shot noise random fields, we give some examples of completely explicit formulas, and for smooth shot noise random fields the provided examples are only partly explicit, since the formulas are given under the form of integrals of some special functions.
</p>projecteuclid.org/euclid.aoap/1591603216_20200608040026Mon, 08 Jun 2020 04:00 EDTBounds for the asymptotic distribution of the likelihood ratiohttps://projecteuclid.org/euclid.aoap/1591603217<strong>Andreas Anastasiou</strong>, <strong>Gesine Reinert</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 608--643.</p><p><strong>Abstract:</strong><br/>
In this paper, we give an explicit bound on the distance to chi-square for the likelihood ratio statistic when the data are realisations of independent and identically distributed random elements. To our knowledge, this is the first explicit bound which is available in the literature. The bound depends on the number of samples as well as on the dimension of the parameter space. We illustrate the bound with three examples: samples from an exponential distribution, samples from a normal distribution and logistic regression.
</p>projecteuclid.org/euclid.aoap/1591603217_20200608040026Mon, 08 Jun 2020 04:00 EDTOptimal position targeting via decoupling fieldshttps://projecteuclid.org/euclid.aoap/1591603218<strong>Stefan Ankirchner</strong>, <strong>Alexander Fromm</strong>, <strong>Thomas Kruse</strong>, <strong>Alexandre Popier</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 644--672.</p><p><strong>Abstract:</strong><br/>
We consider a variant of the basic problem of the calculus of variations, where the Lagrangian is convex and subject to randomness adapted to a Brownian filtration. We solve the problem by reducing it, via a limiting argument, to an unconstrained control problem that consists in finding an absolutely continuous process minimizing the expected sum of the Lagrangian and the deviation of the terminal state from a given target position. Using the Pontryagin maximum principle, we characterize a solution of the unconstrained control problem in terms of a fully coupled forward–backward stochastic differential equation (FBSDE). We use the method of decoupling fields for proving that the FBSDE has a unique solution. We exploit a monotonicity property of the decoupling field for solving the original constrained problem and characterize its solution in terms of an FBSDE with a free backward part.
</p>projecteuclid.org/euclid.aoap/1591603218_20200608040026Mon, 08 Jun 2020 04:00 EDTSemi-implicit Euler–Maruyama approximation for noncolliding particle systemshttps://projecteuclid.org/euclid.aoap/1591603219<strong>Hoang-Long Ngo</strong>, <strong>Dai Taguchi</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 673--705.</p><p><strong>Abstract:</strong><br/>
We introduce a semi-implicit Euler–Maruyama approximation which preserves the noncolliding property for some class of noncolliding particle systems such as Dyson–Brownian motions, Dyson–Ornstein–Uhlenbeck processes and Brownian particle systems with nearest neighbor repulsion, and study its rates of convergence in both $L^{p}$-norm and pathwise sense.
</p>projecteuclid.org/euclid.aoap/1591603219_20200608040026Mon, 08 Jun 2020 04:00 EDTTrading with small nonlinear price impacthttps://projecteuclid.org/euclid.aoap/1591603220<strong>Thomas Cayé</strong>, <strong>Martin Herdegen</strong>, <strong>Johannes Muhle-Karbe</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 706--746.</p><p><strong>Abstract:</strong><br/>
We study portfolio choice with small nonlinear price impact on general market dynamics. Using probabilistic techniques and convex duality, we show that the asymptotic optimum can be described explicitly up to the solution of a nonlinear ODE, which identifies the optimal trading speed and the performance loss due to the trading friction. Previous asymptotic results for proportional and quadratic trading costs are obtained as limiting cases. As an illustration, we discuss how nonlinear trading costs affect the pricing and hedging of derivative securities and active portfolio management.
</p>projecteuclid.org/euclid.aoap/1591603220_20200608040026Mon, 08 Jun 2020 04:00 EDTOptimal investment and consumption with labor income in incomplete marketshttps://projecteuclid.org/euclid.aoap/1591603221<strong>Oleksii Mostovyi</strong>, <strong>Mihai Sîrbu</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 747--787.</p><p><strong>Abstract:</strong><br/>
We consider the problem of optimal consumption from labor income and investment in a general incomplete semimartingale market. The economic agent cannot borrow against future income, so the total wealth is required to be positive at (all or some) previous times. Under very general conditions, we show that an optimal consumption and investment plan exists and is unique, and provide a dual characterization in terms of an optional strong supermartingale deflator and a decreasing part, which charges only the times when the no-borrowing constraint is binding. The analysis relies on the infinite-dimensional parametrization of the income/liability streams and, therefore, provides the first-order dependence of the optimal investment and consumption plans on future income/liabilities (as well as a pricing rule).
</p>projecteuclid.org/euclid.aoap/1591603221_20200608040026Mon, 08 Jun 2020 04:00 EDTOriented first passage percolation in the mean field limit, 2. The extremal processhttps://projecteuclid.org/euclid.aoap/1591603222<strong>Nicola Kistler</strong>, <strong>Adrien Schertzer</strong>, <strong>Marius A. Schmidt</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 788--811.</p><p><strong>Abstract:</strong><br/>
This is the second, and last paper in which we address the behavior of oriented first passage percolation on the hypercube in the limit of large dimensions. We prove here that the extremal process converges to a Cox process with exponential intensity. This entails, in particular, that the first passage time converges weakly to a random shift of the Gumbel distribution. The random shift, which has an explicit, universal distribution related to modified Bessel functions of the second kind, is the sole manifestation of correlations ensuing from the geometry of Euclidean space in infinite dimensions. The proof combines the multiscale refinement of the second moment method with a conditional version of the Chen–Stein bounds, and a contraction principle.
</p>projecteuclid.org/euclid.aoap/1591603222_20200608040026Mon, 08 Jun 2020 04:00 EDTNonlinear large deviations: Beyond the hypercubehttps://projecteuclid.org/euclid.aoap/1591603223<strong>Jun Yan</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 812--846.</p><p><strong>Abstract:</strong><br/>
By extending ( Adv. Math. 299 (2016) 396–450), we present a framework to calculate large deviations for nonlinear functions of independent random variables supported on compact sets in Banach spaces. Previous research on nonlinear large deviations has only focused on random variables supported on $\{-1,+1\}^{n}$, and accordingly we build theory for random variables with general distributions, increasing flexibility in the applications. As examples, we compute the large deviation rate functions for monochromatic subgraph counts in edge-colored complete graphs, and for triangle counts in dense random graphs with continuous edge weights. Moreover, we verify the mean field approximation for a class of vector spin models.
</p>projecteuclid.org/euclid.aoap/1591603223_20200608040026Mon, 08 Jun 2020 04:00 EDTA Berry–Esseen theorem for Pitman’s $\alpha $-diversityhttps://projecteuclid.org/euclid.aoap/1591603224<strong>Emanuele Dolera</strong>, <strong>Stefano Favaro</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 847--869.</p><p><strong>Abstract:</strong><br/>
This paper contributes to the study of the random number $K_{n}$ of blocks in the random partition of $\{1,\ldots,n\}$ induced by random sampling from the celebrated two parameter Poisson–Dirichlet process. For any $\alpha \in (0,1)$ and $\theta >-\alpha $ Pitman ( Combinatorial Stochastic Processes (2006) Springer, Berlin) showed that $n^{-\alpha }K_{n}\stackrel{\text{a.s.}}{\longrightarrow }S_{\alpha,\theta }$ as $n\rightarrow +\infty $, where the limiting random variable, referred to as Pitman’s $\alpha $-diversity, is distributed according to a polynomially scaled Mittag–Leffler distribution function. Our main result is a Berry–Esseen theorem for Pitman’s $\alpha $-diversity $S_{\alpha,\theta }$, namely we show that \[\mathop{\mathrm{sup}}_{x\geq 0}\biggl\vert \mathsf{P}\biggl[\frac{K_{n}}{n^{\alpha }}\leq x\biggr]-\mathsf{P}[S_{\alpha,\theta }\leq x]\biggr\vert \leq\frac{C(\alpha,\theta )}{n^{\alpha }}\] holds for every $n\in \mathbb{N}$ with an explicit constant term $C(\alpha,\theta )$, for $\alpha \in (0,1)$ and $\theta >0$. The proof relies on three intermediate novel results which are of independent interest: (i) a probabilistic representation of the distribution of $K_{n}$ in terms of a compound distribution; (ii) a quantitative version of the Laplace’s approximation method for integrals; (iii) a refined quantitative bound for Poisson approximation. An application of our Berry–Esseen theorem is presented in the context of Bayesian nonparametric inference for species sampling problems, quantifying the error of a posterior approximation that has been extensively applied to infer the number of unseen species in a population.
</p>projecteuclid.org/euclid.aoap/1591603224_20200608040026Mon, 08 Jun 2020 04:00 EDTA lower bound on the queueing delay in resource constrained load balancinghttps://projecteuclid.org/euclid.aoap/1591603225<strong>David Gamarnik</strong>, <strong>John N. Tsitsiklis</strong>, <strong>Martin Zubeldia</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 870--901.</p><p><strong>Abstract:</strong><br/>
We consider the following distributed service model: jobs with unit mean, general distribution, and independent processing times arrive as a renewal process of rate $\lambda n$, with $0<\lambda <1$, and are immediately dispatched to one of several queues associated with $n$ identical servers with unit processing rate. We assume that the dispatching decisions are made by a central dispatcher endowed with a finite memory, and with the ability to exchange messages with the servers.
We study the fundamental resource requirements (memory bits and message exchange rate), in order to drive the expected queueing delay in steady-state of a typical job to zero, as $n$ increases. We develop a novel approach to show that, within a certain broad class of “symmetric” policies, every dispatching policy with a message rate of the order of $n$, and with a memory of the order of $\log n$ bits, results in an expected queueing delay which is bounded away from zero, uniformly as $n\to \infty $. This complements existing results which show that, in the absence of such limitations on the memory or the message rate, there exist policies with vanishing queueing delay (at least with Poisson arrivals and exponential service times).
</p>projecteuclid.org/euclid.aoap/1591603225_20200608040026Mon, 08 Jun 2020 04:00 EDTThe social network model on infinite graphshttps://projecteuclid.org/euclid.aoap/1591603226<strong>Jonathan Hermon</strong>, <strong>Ben Morris</strong>, <strong>Chuan Qin</strong>, <strong>Allan Sly</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 902--935.</p><p><strong>Abstract:</strong><br/>
Given an infinite connected regular graph $G=(V,E)$, place at each vertex $\operatorname{Poisson}(\lambda)$ walkers performing independent lazy simple random walks on $G$ simultaneously. When two walkers visit the same vertex at the same time they are declared to be acquainted. We show that when $G$ is vertex-transitive and amenable, for all $\lambda>0$ a.s. any pair of walkers will eventually have a path of acquaintances between them. In contrast, we show that when $G$ is nonamenable (not necessarily transitive) there is always a phase transition at some $\lambda_{\mathrm{c}}(G)>0$. We give general bounds on $\lambda_{\mathrm{c}}(G)$ and study the case that $G$ is the $d$-regular tree in more detail. Finally, we show that in the nonamenable setup, for every $\lambda$ there exists a finite time $t_{\lambda}(G)$ such that a.s. there exists an infinite set of walkers having a path of acquaintances between them by time $t_{\lambda}(G)$.
</p>projecteuclid.org/euclid.aoap/1591603226_20200608040026Mon, 08 Jun 2020 04:00 EDTViscosity solutions to parabolic master equations and McKean–Vlasov SDEs with closed-loop controlshttps://projecteuclid.org/euclid.aoap/1591603227<strong>Cong Wu</strong>, <strong>Jianfeng Zhang</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 936--986.</p><p><strong>Abstract:</strong><br/>
The master equation is a type of PDE whose state variable involves the distribution of certain underlying state process. It is a powerful tool for studying the limit behavior of large interacting systems, including mean field games and systemic risk. It also appears naturally in stochastic control problems with partial information and in time inconsistent problems. In this paper we propose a novel notion of viscosity solution for parabolic master equations, arising mainly from control problems, and establish its wellposedness. Our main innovation is to restrict the involved measures to a certain set of semimartingale measures which satisfy the desired compactness. As an important example, we study the HJB master equation associated with the control problems for McKean–Vlasov SDEs. Due to practical considerations, we consider closed-loop controls. It turns out that the regularity of the value function becomes much more involved in this framework than the counterpart in the standard control problems. Finally, we build the whole theory in the path dependent setting, which is often seen in applications. The main result in this part is an extension of Dupire’s (2009) functional Itô formula. This Itô formula requires a special structure of the derivatives with respect to the measures, which was originally due to Lions in the state dependent case. We provided an elementary proof for this well known result in the short note (2017), and the same arguments work in the path dependent setting here.
</p>projecteuclid.org/euclid.aoap/1591603227_20200608040026Mon, 08 Jun 2020 04:00 EDTKinetically constrained models with random constraintshttps://projecteuclid.org/euclid.aoap/1591603228<strong>Assaf Shapira</strong>. <p><strong>Source: </strong>Annals of Applied Probability, Volume 30, Number 2, 987--1006.</p><p><strong>Abstract:</strong><br/>
We study two kinetically constrained models in a quenched random environment. The first model is a mixed threshold Fredrickson–Andersen model on $\mathbb{Z}^{2}$, where the update threshold is either $1$ or $2$. The second is a mixture of the Fredrickson–Andersen $1$-spin facilitated constraint and the North-East constraint in $\mathbb{Z}^{2}$. We compare three time scales related to these models—the bootstrap percolation time for emptying the origin, the relaxation time of the kinetically constrained model, and the time for emptying the origin of the kinetically constrained model—and understand the effect of the random environment on each of them.
</p>projecteuclid.org/euclid.aoap/1591603228_20200608040026Mon, 08 Jun 2020 04:00 EDT