Marginal Laws Research Articles

Does Government Digital Transformation Drive High-Quality Urban Economic Development? Evidence from E-Government Platform Construction

Digitalization represents a pivotal global development trend and serves as a significant force propelling economic and social transformation. This manuscript uses the global Malmquist–Luenberger (GML) model to estimate green total factor productivity (GTFP) across 284 Chinese cities from 2003 to 2018, taking the pilot policy of “construction and application of e-government public platforms based on cloud computing” as an example to assess the impact of government digital transformation on the qualitative development of the economy by using a difference-in-differences model to explore the path of its role and driving mechanism. The results reveal that government digital transformation promotes the qualitative improvement of the city’s economic development, and its driving effect shows a marginal incremental law. Moreover, government digital transformation can contribute to the formation of a “latecomer advantage” in the lagging regions, which creates a “catch-up effect” on the regions with favorable development foundations, excellent geographical conditions, high urban ranking, and high education quality. Additionally, government digital transformation boosts economic and social development quality through both innovation spillover and structural optimization.

On a multi-dimensional McKean-Vlasov SDE with memorial and singular interaction associated to the parabolic-parabolic Keller-Segel model

In this work, we first prove the well-posedness of the non-linear martingale problem related to a McKean-Vlasov stochastic differential equation with singular interaction kernel in ℝ d for d≥3. The particularity of our setting is that the McKean-Vlasov process we study interacts at each time with all its past time marginal laws by means of a singular space-time kernel. Second, we prove that our stochastic process is a probabilistic interpretation for the parabolic-parabolic Keller-Segel system in ℝ d . We thus obtain a well-posedness result to the latter under explicit smallness condition on the parameters of the model.

Strong solutions to McKean–Vlasov SDEs with coefficients of Nemytskii type: the time-dependent case

We consider a large class of nonlinear FPKEs with coefficients of Nemytskii type depending explicitly on time and space, for which it is known that there exists a sufficiently Sobolev-regular Schwartz-distributional solution u∈L1∩L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u\\in L^1\\cap L^\\infty $$\\end{document}. We show that there exists a unique strong solution to the associated McKean–Vlasov SDE with time marginal law densities u. In particular, every weak solution of this equation with time marginal law densities u can be written as a functional of the driving Brownian motion. Moreover, plugging any Brownian motion into this very functional produces a weak solution with time marginal law densities u.

Density analysis for coupled forward–backward SDEs with non-Lipschitz drifts and applications

We explore the existence of a continuous marginal law with respect to the Lebesgue measure for each component (X,Y,Z) of the solution to coupled quadratic forward–backward stochastic differential equations (QFBSDEs) for which the drift coefficient of the forward component is either bounded and measurable or Hölder continuous. Our approach relies on a combination of the existence of a weak decoupling field (see Delarue and Guatteri, 2006), the integration with respect to space time local time (see Eisenbaum, 2006), the analysis of the backward Kolmogorov equation associated to the forward component along with an Itô-Tanaka trick (see Flandoli et al., 2009). The framework of this paper is beyond all existing papers on density analysis for Markovian BSDEs and constitutes a major refinement of the existing results. We also derive a comonotonicity theorem for the control variable in this frame and thus extending the works (Chen et al., 2005; Dos Rei and Dos Rei 2013).As applications of our results, we first analyse the regularity of densities of solution to coupled FBSDEs. In the second example, we consider a regime switching term structure interest rate models (see for e.g., Ma et al. (2017)) for which the corresponding FBSDE has discontinuous drift. Our results enables us to: firstly study classical and Malliavin differentiability of the solutions for such models, secondly the existence of density of such solutions. Lastly we consider a pricing and hedging problem of contingent claims on non-tradable underlying, when the dynamic of the latter is given by a regime switching SDE (i.e., the drift coefficient is allowed to be discontinuous). We obtain a representation of the derivative hedge as the weak derivative of the indifference price function, thus extending the result in Ankirchner et al. (2010).

Central limit theorem in uniform metrics for generalized Kac equations

The aim of this paper is to give explicit rates for the speed of convergence to equilibrium of the solution of the generalized Kac equation in two strong metrics: the total variation distance (TV) and the uniform metric between characteristic functions (χ0). A fundamental role in our study is played by the probabilistic representation of the solution of the generalized Kac equation as marginal law of a stochastic process which is a weighted random sum of i.i.d. random variables, where the weights are positive and dependent. Exponential bounds for the total variation distance between the solution and the gaussian stationary state of the Kac equation have been proved by Dolera, Gabetta and Regazzini (2009). In our more general setting the equilibrium states are scale mixtures of stable distributions and hence not necessarily gaussian. Therefore we develop new tools based on ideal metrics that are used in the literature for quantitative central limit theorems for i.i.d. random variables in the domain of attraction of a stable distribution. We obtain first exponential bounds in the so-called ”r-smoothed total variation” and in the weighted χr-metric for a suitable r, then we deduce rates of convergence with respect to the “corresponding” uniform metrics TV and χ0.

Generalized gamma ARMA process for synthetic aperture radar amplitude and intensity data

AbstractWe propose a new autoregressive moving average (ARMA) process with generalized gamma (G) marginal law, called G‐ARMA. We derive some of its mathematical properties: moment‐based closed‐form expressions, score function, and Fisher information matrix. We provide a procedure for obtaining maximum likelihood estimates for the G‐ARMA parameters. Its performance is quantified and discussed using Monte Carlo experiments, considering (among others) various link functions. Finally, our proposal is applied to solve remote sensing problems using synthetic aperture radar (SAR) imagery. In particular, the G‐ARMA process is applied to real data from images taken in the Munich and San Francisco regions. The results show that G‐ARMA describes the neighborhoods of SAR features better than the gamma‐ARMA process (a reference for asymmetric positive data). For pixel ray modeling, our proposal outperforms and gamma‐ARMA.

Nonlinear Fokker–Planck Equations with Time-Dependent Coefficients

An operator-based approach is used here to prove the existence and uniqueness of a strong solution to the time-varying nonlinear Fokker–Planck equation , in the Sobolev space , under appropriate conditions on and It is proved also that if is a density of a probability measure, so is for all . Moreover, we construct a weak solution to the McKean–Vlasov SDE associated with the Fokker–Planck equation such that is the density of its time marginal law.

On solutions of the distributional Bellman equation

<abstract><p>In distributional reinforcement learning (RL), not only expected returns but the complete return distributions of a policy are taken into account. The return distribution for a fixed policy is given as the solution of an associated distributional Bellman equation. In this note, we consider general distributional Bellman equations and study the existence and uniqueness of their solutions, as well as tail properties of return distributions. We give necessary and sufficient conditions for the existence and uniqueness of return distributions and identify cases of regular variation.</p> <p>We link distributional Bellman equations to multivariate affine distributional equations. We show that any solution of a distributional Bellman equation can be obtained as the vector of marginal laws of a solution to a multivariate affine distributional equation. This makes the general theory of such equations applicable to the distributional reinforcement learning setting.</p></abstract>

Strong solutions to McKean–Vlasov SDEs with coefficients of Nemytskii-type

We study a large class of McKean–Vlasov SDEs with drift and diffusion coefficient depending on the density of the solution’s time marginal laws in a Nemytskii-type of way. A McKean–Vlasov SDE of this kind arises from the study of the associated nonlinear FPKE, for which is known that there exists a bounded Sobolev-regular Schwartz-distributional solution u. Via the superposition principle, it is already known that there exists a weak solution to the McKean–Vlasov SDE with time marginal densities u. We show that there exists a strong solution the McKean–Vlasov SDE, which is unique among weak solutions with time marginal densities u. The main tool is a restricted Yamada–Watanabe theorem for SDEs, which is obtained by an observation in the proof of the classical Yamada–Watanabe theorem.

Variable elimination, graph reduction and the efficient g-formula

SummaryWe study efficient estimation of an interventional mean associated with a point exposure treatment under a causal graphical model represented by a directed acyclic graph without hidden variables. Under such a model, a subset of the variables may be uninformative, in that failure to measure them neither precludes identification of the interventional mean nor changes the semiparametric variance bound for regular estimators of it. We develop a set of graphical criteria that are sound and complete for eliminating all the uninformative variables, so that the cost of measuring them can be saved without sacrificing estimation efficiency, which could be useful when designing a planned observational or randomized study. Further, we construct a reduced directed acyclic graph on the set of informative variables only. We show that the interventional mean is identified from the marginal law by the g-formula (Robins, 1986) associated with the reduced graph, and the semiparametric variance bounds for estimating the interventional mean under the original and the reduced graphical model agree. The g-formula is an irreducible, efficient identifying formula in the sense that the nonparametric estimator of the formula, under regularity conditions, is asymptotically efficient under the original causal graphical model, and no formula with this property exists that depends only on a strict subset of the variables.

Business-transactions and Accumulation of Economic Resources

Operational management implies not only production management but logistics, innovative management, transaction and quality management and other components of organization of business as well. Operational managers bear the responsibility that company should possess advanced production and organization methods, implement continued modernization and remain the leading positions. In order to beat the competitors, together with care on price and quality - the firm should try to implement an innovative technology and in such manner satisfy the customers` requirements in a better way. The Indicator of enterprise`s backwardness is relatively high cost price, low-grade production or both at a time, which presents the ineffective combination of resources. Division of costs into fixed and variable types are being performed according to the short-term periods. As for long-term periods, according them all the costs are variable. Short-time period is the time interval, during of which it is possible to increase the production capacity only by improvement of variable costs. If the part of resources applied in the manufacture are left unchanged and they are added with some other resource, then production volume might be increased in the beginning, but since then, whilst the optimal correlation of resources will be violated, the moment will rise, when an additional resources involved in production will gain ever less and less return. In such case, the law of decreased return (or the decreased marginal production capacity law) is active. The least is nothing more than a grow of variable expenses per production unit under the conditions of unchanged costs. Marginal return - means the surplus of production capacity, which is conditioned by application of additional unit of resource. Key words: business, marketing, agreement, transactions, enterprise, resources, production, operation, goods, automatization, firm, cost, raw materials, materials, calculation.

Weak quantitative propagation of chaos via differential calculus on the space of measures

Consider the metric space (P2(Rd),W2) of square integrable laws on Rd with the topology induced by the 2-Wasserstein distance W2. Let Φ:P2(Rd)→R and μ∈P2(Rd). In this work, we consider (a) μN being the empirical measure of N-samples from μ, and the other case in which (b) μN is the empirical measure of marginal laws of the particle system of a McKean–Vlasov PDE (μt)t. The main result of this paper is to show that under suitable regularity conditions, we have |Φ(μ)−EΦ(μN)|=∑j=1k−1Cj Nj+O(1 Nk), for some positive constants C1,…,Ck−1 that do not depend on N, where k corresponds to the degree of smoothness. The case where the samples are i.i.d. is studied using functional derivatives on the space of measures. The case of particle systems relies on an Itô-type formula for the flow of probability measures and is intimately connected to PDEs on the space of measures, called the master equation in the literature of mean-field games. We state general regularity conditions required for each case and analyze the regularity in the case of functionals of the laws of McKean–Vlasov PDEs. Ultimately, this work reveals quantitative estimates of propagation of chaos for interacting particle systems. Furthermore, we are able to provide weak propagation of chaos estimates for ensembles of interacting particles and show that these may have some remarkable properties.

Game on Random Environment, Mean-Field Langevin System, and Neural Networks

In this paper, we study a class of games regularized by relative entropy where the players’ strategies are coupled through a random environment. Besides existence and uniqueness of equilibria for such games, we prove, under different sets of hypotheses that the marginal laws of the corresponding mean-field Langevin systems can converge toward the games’ equilibria. As an application, we show that dynamic games fall in this framework by considering the time horizon as environment. Concerning applications, our results allow analysis of stochastic gradient descent algorithms for deep neural networks in the context of supervised learning and for generative adversarial networks.

Constrained overdamped Langevin dynamics for symmetric multimarginal optimal transportation

The Strictly Correlated Electrons (SCE) limit of the Levy–Lieb functional in Density Functional Theory (DFT) gives rise to a symmetric multi-marginal optimal transport problem with Coulomb cost, where the number of marginal laws is equal to the number of electrons in the system, which can be very large in relevant applications. In this work, we design a numerical method, built upon constrained overdamped Langevin processes to solve Moment Constrained Optimal Transport (MCOT) relaxations (introduced in [A. Alfonsi, R. Coyaud, V. Ehrlacher and D. Lombardi, Approximation of optimal transport problems with marginal moments constraints, Math. Comp. 90 (2021) 689–737; C. Villani, Optimal Transport: Old and New (Springer Science & Business Media, 2008)]) of symmetric multi-marginal optimal transport problems with Coulomb cost. Some minimizers of such relaxations can be written as discrete measures charging a low number of points belonging to a space whose dimension, in the symmetrical case, scales linearly with the number of marginal laws. We leverage the sparsity of those minimizers in the design of the numerical method and prove that there is no strict local minimizer to the resulting problem. We illustrate the performance of the proposed method by numerical examples which solves MCOT relaxations of 3D systems with up to 100 electrons.

Log-unimodality for free positive multiplicative Brownian motion

We prove that the marginal law $\sigma _{t}\mathrel {\scriptstyle {\boxtimes }}\nu $ of free positive multiplicative Brow\-nian motion is log-unimodal for all $t \gt 0$ if $\nu $ is a multiplicatively symmetric log-unimodal distribution, and that $\sigma

Quasi-stationary distribution for the Langevin process in cylindrical domains, part II: overdamped limit

Consider the Langevin process, described by a vector (positions and momenta) in Rd×Rd. Let O be a C2 open bounded and connected set of Rd. Recent works showed the existence of a unique quasi-stationary distribution (QSD) of the Langevin process on the domain D:=O×Rd. In this article, we study the overdamped limit of this QSD, i.e. when the friction coefficient goes to infinity. In particular, we show that the marginal law in position of the overdamped limit is the QSD of the overdamped Langevin process on the domain O.

Quantitative approximate independence for continuous mean field Gibbs measures

Many Gibbs measures with mean field interactions are known to be chaotic, in the sense that any collection of k particles in the n-particle system are asymptotically independent, as n→∞ with k fixed or perhaps k=o(n). This paper quantifies this notion for a class of continuous Gibbs measures on Euclidean space with pairwise interactions, with main examples being systems governed by convex interactions and uniformly convex confinement potentials. The distance between the marginal law of k particles and its limiting product measure is shown to be O((k∕n)c∧2), with c proportional to the squared temperature. In the high temperature case, this improves upon prior results based on subadditivity of entropy, which yield O(k∕n) at best. The bound O((k∕n)2) cannot be improved, as a Gaussian example demonstrates. The results are non-asymptotic, and distance is quantified via relative Fisher information, relative entropy, or the squared quadratic Wasserstein metric. The method relies on an a priori functional inequality for the limiting measure, used to derive an estimate for the k-particle distance in terms of the (k+1)-particle distance.

Probabilistic representation of integration by parts formulae for some stochastic volatility models with unbounded drift

In this paper, we establish a probabilistic representation as well as some integration by parts formulae for the marginal law at a given time maturity of some stochastic volatility model with unbounded drift. Relying on a perturbation technique for Markov semigroups, our formulae are based on a simple Markov chain evolving on a random time grid for which we develop a tailor-made Malliavin calculus. Among other applications, an unbiased Monte Carlo path simulation method stems from our formulas so that it can be used in order to numerically compute with optimal complexity option prices as well as their sensitivities with respect to the initial values or Greeks in finance, namely the Delta and Vega, for a large class of non-smooth European payoff. Numerical results are proposed to illustrate the efficiency of the method.

On the Impact of Asymmetric Dependence in the Actuarial Pricing of Joint Life Insurance Policies

Multipopulation mortality modeling is a significant research problem in actuarial science. Mortality functions involving multiple lives are also essential to determine the pricing of premiums. Moreover, the lifetime models based on dependence and asymmetry are more realistic. Hence, this paper applies an asymmetric copula model, Generalized FGM (GFGM) to model the bivariate joint distribution of future lifetimes. Premiums of first-death life insurance products are calculated based on the proposed model and compared with independent and symmetrical models. The results display that asymmetry has a significant effect on premium calculations. Also, it is concluded that the lowest premiums are generally in asymmetric lifetime models. This paper also provides analytical examples for the proposed model with Gompertz’s marginal law.

The Bivariate Extended Poisson Distribution of Type 1

In this paper, we will construct the bivariate extended Poisson distribution whichgeneralizes the univariate extended Poisson distribution. This law will be obtained by the method of the product of its marginal laws by a factor. This method was demonstrated in [7]. Thus we call the bivariate extended Poisson distribution of type 1 the bivariate extended Poisson distribution obtained by the method of the product of its marginal distributions by a factor. We will show that this distribution belongs to the family of bivariate Poisson distributions and and will highlight the conditions relating to the independence of the marginal variables. A simulation study was realised.