Directed chain stochastic differential equations

Detering, Nils; Fouque, Jean‐Pierre; Ichiba, Tomoyuki

doi:10.1016/j.spa.2019.07.009

Cited by 10 publications

(13 citation statements)

References 23 publications

(20 reference statements)

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“…Along with the definition of γ t (see (4.13) and the subsequent line), this verifies both properties (7) and (10) of Definition 3.8, with Y = X.…”

Section: Proof Of Theorem 311supporting

confidence: 65%

“…It remains to check that we have all of the ingredients required by Definition 3.8 for a solution of the local equation. We begin with the integrability condition stated as property (10) in Definition 3.8. Note that Lemma 4.1 and the linear growth of b from Assumption (A.1) ensure that, by Jensen's inequality and (4.13),…”

Section: Proof Of Theorem 311mentioning

confidence: 99%

“…The only other result that we are aware of that provides an autonomous characterization of marginals of an infinite system of interacting diffusions on a sparse graph was obtained recently in [10], which treats a Markovian interacting diffusive particle system with identity diffusion coefficient in the special case where the interaction graph is a directed line, without any feedback of the interactions. In this specific setting, a coupling argument is used to obtain an autonomous characterization of the law of the trajectories of any contiguous set of particles in terms of a non-linear diffusion process.…”

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confidence: 99%

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Marginal dynamics of interacting diffusions on unimodular Galton-Watson trees

Lacker¹,

Ramanan²,

Wu³

2020

Preprint

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Consider a system of homogeneous interacting diffusive particles labeled by the nodes of a unimodular Galton-Watson tree, where the state of each node evolves infinitesimally like a d-dimensional diffusion whose drift coefficient depends on (the histories of) its own state and the states of neighboring nodes, and whose diffusion coefficient depends only on its own state. Under suitable regularity assumptions on the coefficients, an autonomous characterization is obtained for the marginal distribution of the dynamics of the neighborhood of a typical node in terms of a certain local equation, which is a new kind of stochastic differential equation that is nonlinear in the sense of McKean. This equation describes a finite-dimensional non-Markovian stochastic process whose infinitesimal evolution at any time depends not only on the structure and current state of the neighborhood, but also on the conditional law of the current state given the past of the neighborhood process until that time. Such marginal distributions are of interest because they arise as weak limits of both marginal distributions and empirical measures of interacting diffusions on many sequences of sparse random graphs, including the configuration model and Erdös-Rényi graphs whose average degrees converge to a finite non-zero limit. The results obtained complement classical results in the mean-field regime, which characterize the limiting dynamics of homogeneous interacting diffusions on complete graphs, as the number of nodes goes to infinity, in terms of a corresponding nonlinear Markov process. However, in the sparse graph setting, the topology of the graph matters, and the analysis requires a completely different approach. Important ingredients of the proofs include conditional independence and symmetry properties for particle trajectories on unimodular Galton-Watson trees, a stochastic analytic result on projections of Itô processes, and a recursive construction for establishing well-posedness of the local equation.

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“…Along with the definition of γ t (see (4.13) and the subsequent line), this verifies both properties (7) and (10) of Definition 3.8, with Y = X.…”

Section: Proof Of Theorem 311supporting

confidence: 65%

Section: Proof Of Theorem 311mentioning

confidence: 99%

mentioning

confidence: 99%

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Marginal dynamics of interacting diffusions on unimodular Galton-Watson trees

Lacker¹,

Ramanan²,

Wu³

2020

Preprint

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“…The combination of network, dynamic, and game-theoretic effects is essential for many recent models of large economic and financial systems, and several recent studies have attacked specific models combining some of these features; see [6,10,16,33] and references therein. Even without game-theoretic (strategic) features, incorporating network effects into large-scale dynamic models already presents many mathematical challenges, which very recent work has begun to address; see [2,12,32] and [14,28,34] for studies of dense and sparse graph regimes, respectively. Notably, prior work studied dense graph regimes, and most questions in the sparse regime (roughly defined as finite limiting neighborhood sizes) remain open, as was highlighted in particular in the recent paper [5].…”

Section: Introductionmentioning

confidence: 99%

A case study on stochastic games on large graphs in mean field and sparse regimes

Lacker¹,

Soret²

2020

Preprint

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We study a class of linear-quadratic stochastic differential games in which each player interacts directly only with its nearest neighbors in a given graph. We find a semiexplicit Markovian equilibrium for any transitive graph, in terms of the empirical eigenvalue distribution of the graph's normalized Laplacian matrix. This facilitates large-population asymptotics for various graph sequences, with several sparse and dense examples discussed in detail. In particular, the mean field game is the correct limit only in the dense graph case, i.e., when the degrees diverge in a suitable sense. Even though equilibrium strategies are nonlocal, depending on the behavior of all players, we use a correlation decay estimate to prove a propagation of chaos result in both the dense and sparse regimes, with the sparse case owing to the large distances between typical vertices. Without assuming the graphs are transitive, we show also that the mean field game solution can be used to construct decentralized approximate equilibria on any sufficiently dense graph sequence.

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“…3, pp. 315-317, © 2019 The last problem, which is related to the stochastic block model and community detection, is partially motivated by Detering et al (2019), in which a system with both mean-field and sparse local interactions on a circle is studied and a detection problem is discussed.…”

mentioning

confidence: 99%

Open Problem—Weakly Interacting Particle Systems on Dense Random Graphs

2019

Stochastic Systems

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Please scroll down for article-it is on subsequent pages With 12,500 members from nearly 90 countries, INFORMS is the largest international association of operations research (O.R.) and analytics professionals and students. INFORMS provides unique networking and learning opportunities for individual professionals, and organizations of all types and sizes, to better understand and use O.R. and analytics tools and methods to transform strategic visions and achieve better outcomes. For more information on INFORMS, its publications, membership, or meetings visit http://www.informs.org

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Directed chain stochastic differential equations

Cited by 10 publications

References 23 publications

Marginal dynamics of interacting diffusions on unimodular Galton-Watson trees

Marginal dynamics of interacting diffusions on unimodular Galton-Watson trees

A case study on stochastic games on large graphs in mean field and sparse regimes

Open Problem—Weakly Interacting Particle Systems on Dense Random Graphs

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