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作者：Hu, Yaozhong; Liu, Yanghui; Nualart, David
作者单位：University of Alberta; City University of New York (CUNY) System; Baruch College (CUNY); University of Kansas
摘要：We study the Crank-Nicolson scheme for stochastic differential equations (SDEs) driven by a multidimensional fractional Brownian motion with Hurst parameterH > 1/2. It is well known that for ordinary differential equations with proper conditions on the regularity of the coefficients, the Crank-Nicolson scheme achieves a convergence rate of n(-2), regardless of the dimension. In this paper we show that, due to the interactions between the driving processes, the corresponding Crank-Nicolson sche...
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作者：Shao, Qi-Man; Zhang, Mengchen; Zhang, Zhuo-Song
作者单位：Southern University of Science & Technology; Chinese University of Hong Kong; Hong Kong University of Science & Technology; National University of Singapore
摘要：A Cramer-type moderate deviation theorem quantifies the relative error of the tail probability approximation. It provides a criterion whether the limiting tail probability can be used to estimate the tail probability under study. Chen, Fang and Shao (2013) obtained a general Cramer-type moderate result using Stein's method when the limiting was a normal distribution. In this paper, Cramer-type moderate deviation theorems are established for nonnormal approximation under a general Stein identit...
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作者：Henry-Labordere, Pierre; Touzi, Nizar
作者单位：Universite Paris Cite; Institut Polytechnique de Paris; ENSTA Paris
摘要：We provide probabilistic representations of the solution of some semi-linear hyperbolic and high-order PDEs based on branching diffusions. These representations pave the way for an approximation of the solution by the standard Monte Carlo method, whose error estimate is controlled by the standard central limit theorem, thus partly bypassing the curse of dimensionality. We illustrate the numerical implications in the context of some popular PDEs in physics such as nonlinear Klein-Gordon equatio...
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作者：Jaber, Eduardo Abi; Miller, Enzo; Pham, Huyen
作者单位：Universite Paris Cite
摘要：We provide an exhaustive treatment of linear-quadratic control problems for a class of stochastic Volterra equations of convolution type, whose kernels are Laplace transforms of certain signed matrix measures which are not necessarily finite. These equations are in general neither Markovian nor semimartingales, and include the fractional Brownian motion with Hurst index smaller than 1/2 as a special case. We establish the correspondence of the initial problem with a possibly infinite dimension...
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作者：Vogrinc, Jure; Kendall, Wilfrid S.
作者单位：University of Warwick
摘要：For sufficiently smooth targets of product form it is known that the variance of a single coordinate of the proposal in RWM (random walk Metropolis) and MALA (Metropolis adjusted Langevin algorithm) should optimally scale as n(-1) and as n(-1/3) with dimension n, and that the acceptance rates should be tuned to 0.234 and 0.574. We establish counterexamples to demonstrate that smoothness assumptions of the order of C-1(R) for RWM and C-3(R) for MALA are indeed required if these scaling rates ar...
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作者：Hermon, Jonathan; Salez, Justin
作者单位：University of British Columbia; Universite PSL; Universite Paris-Dauphine; Universite PSL
摘要：We resolve a long-standing conjecture of Wilson (Ann. Appl. Probab. 14 (2004) 274-325), reiterated by Oliveira (2016), asserting that the mixing time of the interchange process with unit edge rates on the n-dimensional hyper-cube is of order n. This follows from a sharp inequality established at the level of Dirichlet forms, from which we also deduce that macroscopic cycles emerge in constant time, and that the log-Sobolev constant of the exclusion process is of order 1. Beyond the hypercube, ...
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作者：Dianetti, Jodi; Ferrari, Giorgio; Fischer, Markus; Nendel, Max
作者单位：University of Bielefeld; University of Padua
摘要：We study mean field games with scalar Ito-type dynamics and costs that are submodular with respect to a suitable order relation on the state and measure space. The submodularity assumption has a number of interesting consequences. First, it allows us to prove existence of solutions via an application of Tarski's fixed point theorem, covering cases with discontinuous dependence on the measure variable. Second, it ensures that the set of solutions enjoys a lattice structure: in particular, there...
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作者：Ernst, Philip A.; Franceschi, Sandro
作者单位：Rice University; Universite Paris Saclay
摘要：Let pi be the occupancy density of an obliquely reflected Brownian motion in the half plane and let (rho, alpha) be the polar coordinates of a point in the upper half plane. This work determines the exact asymptotic behavior of pi(rho, alpha) as rho -> infinity with alpha epsilon (0, pi). We find explicit functions a, b, c such that pi(rho, alpha) similar to(rho -> infinity) a(alpha) rho(b(alpha)) e(-c(alpha)rho). This closes an open problem first stated by Professor J. Michael Harrison in Aug...
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作者：Foutel-Rodier, Felix; Lambert, Amaury; Schertzer, Emmanuel
作者单位：Universite Paris Cite; Sorbonne Universite; Universite PSL; College de France; Institut National de la Sante et de la Recherche Medicale (Inserm)
摘要：Kingman's (1978) representation theorem (J. Lond. Math. Soc. (2) 18 (1978) 374-380) states that any exchangeable partition of N can be represented as a paintbox based on a random mass-partition. Similarly, any exchangeable composition (i.e., ordered partition of N) can be represented as a paintbox based on an interval-partition (Gnedin (1997) Ann. Probab. 25 (1997) 1437-1450). Our first main result is that any exchangeable coalescent process (not necessarily Markovian) can be represented as a ...
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作者：Cao, Sky
作者单位：Stanford University
摘要：For random combinatorial optimization problems, there has been much progress in establishing laws of large numbers and computing limiting constants for the optimal values of various problems. However, there has not been as much success in proving central limit theorems. This paper introduces a method for establishing central limit theorems in the sparse graph setting. It works for problems that display a key property which has been variously called endogeny, long-range independence and replica...
