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作者：Nutz, Marcel; Wang, Ruodu; Zhang, Zhenyuan
作者单位：Columbia University; University of Waterloo; Stanford University
摘要：It is well known that martingale transport plans between marginals mu not equal nu are never given by Monge maps-with the understanding that the map is over the first marginal mu, or forward in time. Here, we change the perspective, with surprising results. We show that any distributions mu, nu in convex order with nu atomless admit a martingale coupling given by a Monge map over the second marginal nu. Namely, we construct a particular coupling called the barcode tingale transports are dense ...
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作者：Denisov, Denis; Wachtel, Vitali
作者单位：University of Manchester; University of Bielefeld
摘要：We consider a random walk in a truncated cone KN, which is obtained by slicing cone K by a hyperplane at a growing level of order N. We study the behaviour of the Green function in this truncated cone as N increases. Using these results we also obtain the asymptotic behaviour of the harmonic The obtained results are applied to a multidimensional gambler's problem studied by Diaconis and Ethier (Staist. Sci. 37 (2022) 289-305). In particular we confirm their conjecture that the probability of e...
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作者：Sethi, Deven; Siska, David
作者单位：University of Edinburgh
摘要：The modified method of successive approximations (MSA) is an iterative scheme for approximating solutions to stochastic control problems in continuous time based on Pontryagin optimality principle which, starting with an initial open loop control, solves the forward equation, the backward adjoint equation and then performs a static minimization step. We observe that this is an implicit Euler scheme for a gradient flow system. We prove that appropriate interpolations of the iterates of the modi...
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作者：Kazeykina, Anna; Ren, Zhenjie; Tan, Xiaolu; Yang, Junjian
作者单位：Universite Paris Saclay; Universite PSL; Universite Paris-Dauphine; Chinese University of Hong Kong; Technische Universitat Wien
摘要：We study the long time behavior of an underdamped mean -field Langevin (MFL) equation, and provide a general convergence as well as an exponential convergence rate result under different conditions. The results on the MFL equation can be applied to study the convergence of the Hamiltonian gradient descent algorithm for the overparametrized optimization. We then provide some numerical examples of the algorithm to train a generative adversarial network (GAN).
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作者：Biswas, Sani; Kumar, Chaman; Neelima; Dos Reis, Goncalo; Reisinger, Christoph
作者单位：Indian Institute of Technology System (IIT System); Indian Institute of Technology (IIT) - Roorkee; University of Delhi; University of Edinburgh; University of Oxford
摘要：We propose an explicit drift-randomised Milstein scheme for bothMcKean-Vlasov stochastic differential equations and associated high-dimen-sional interacting particle systems with common noise. By using a driftrandomisation step in space and measure, we establish the scheme's strongconvergence rate of 1 under reduced regularity assumptions on the drift co-efficient: no classical (Euclidean) derivatives in space or measure derivatives(e.g., Lions/Frechet) are required. The main result is establi...
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作者：Dhara, Souvik; Mukherjee, Debankur; Ramanan, Kavita
作者单位：University System of Georgia; Georgia Institute of Technology; Brown University
摘要：For an nxn matrix A(n), the r -> p operator norm is defined as parallel to An parallel to(r -> p):= sup(x is an element of R)(n):parallel to x parallel to r(<= 1)(n)(parallel to A)x parallel to(p) for r,p >= 1. For different choices of r and p, this norm corresponds to key quantities that arise in diverse applications including matrix condition number estimation, clustering of data, and construction of oblivious routing schemes in transportation networks. This article considers r -> p norms of...
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作者：Baker, Graeme; Shkolnikov, Mykhaylo
作者单位：Columbia University; Carnegie Mellon University
摘要：We consider a probabilistic formulation of a singular two-phase Stefan problem in one space dimension, which amounts to a coupled system of two McKean-Vlasov stochastic differential equations. In the financial context of systemic risk, this system models two competing regions with a large number of interconnected banks or firms at risk of default. Our main result shows the existence of a solution whose discontinuities obey the natural physicality condition for the problem at hand. Thus, this w...
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作者：Imkeller, Peter; Pellat, Rhoss likibi; Menoukeu-pamen, Olivier
作者单位：Humboldt University of Berlin; University of Liverpool
摘要：In this paper, we consider quadratic forward-backward SDEs (QFBSDEs), for which the drift in the forward equation does not satisfy the standard globally Lipschitz condition and the driver of the backward system possesses nonlinearity of type f (|y|)|z|2, where f is any locally integrable function. We prove both the Malliavin and classical differentiability of solutions to this type of QFBSDEs and provide representations of these derivatives processes. As a by-product, we derive a representatio...
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作者：Wang, Tianhao; Zhong, Xinyi; Fan, Zhou
作者单位：Yale University
摘要：Approximate message passing (AMP) algorithms provide a valuable tool for studying mean-field approximations and dynamics in a variety of applications. Although these algorithms are often first derived for matrices having independent Gaussian entries or satisfying rotational invariance in law, their state evolution characterizations are expected to hold over larger universality classes of random matrix ensembles. We develop several new results on AMP universality. For AMP algorithms tailored to...
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作者：Emilia Caballero, Maria; Gonzalez Casanov, Adrian; Perez, Jose-Luis
作者单位：Universidad Nacional Autonoma de Mexico
摘要：When two (possibly different in distribution) continuous-state branching processes with immigration are present, we study the relative frequency of one of them when the total mass is forced to be constant at a dense set of times. This leads to a SDE whose unique strong solution will be the definition of a Lambda-asymmetric frequency process (Lambda-AFP). We prove that it is a Feller process and we calculate a large population limit when the total mass tends to infinity. This allows us to study...
