mean field game for linear quadratic stochastic recursive systems abstract this paper focuses on linear quadratic lq for short mean field games described by forward backward stochastic differential equations fbsdes for short in which the individual control region is postulated to be convex the decentralized strategies and consistency condition are represented by a kind of coupled mean field fbsdes with projection operators the well posedness of consistency condition system is obtained using the monotonicity condition method the nash equilibrium property is discussed as well

equations differentielles sur l espace de wiener et formules de cameron martin non lineaires proprietes algebriques des approximations sur r n une formule de perturbation des flots et convergence des approximations formule de changement de variable sur x et equation differentielle

on the invariant measures for the two dimensional euler flow it is proven that the canonical gibbs measure associated with a gas of vortices of intensity converges in the limitn 0 n const to a gaussian measure which is invariant for the two dimensional euler equation

on tests against one sided hypotheses in some generalized linear models one sided hypotheses arise naturally in many situations when testing against such hypotheses it is desirable to take the available one sided information into account rather than simply applying a two sided test what we expect to gain by applying a one sided test instead of a two sided test is an increase in the power of the test we consider various tests of one sided hypotheses in a class of models that includes generalized linear and cox regression models the tests are likelihood ratio wald score generalized distance and a pearson chi square it is shown that these test statistics are asymptomatically equivalent in terms of local power this is a generalization of the well known corresponding result for two sided alternatives two examples are also discussed they are on 1 testing for interaction in binomial response models and 2 comparison of treatments with ordinal categorical responses

global flows with invariant gibbs measures for euler and navier stokes two dimensional fluids we construct a family of probability spaces p 0 associated with the euler equation for a two dimensional inviscid incompressible fluid which carries a pointwise flow t time evolution leavingp globally invariant t is obtained as the limit of galerkin approximations associated with euler equations p is also in invariant measure for a stochastic process associated with a navier stokes equation with viscosity stochastically perturbed by a white noise force

large deviations for stationary measures of stochastic nonlinear wave equation with smooth white noise we prove the freidlin wentzell type large deviations principle for the family of stationary measures of stochastic nonlinear wave nlw equation with white noise we do not assume that the limiting equation possesses a unique equilibrium and do not impose roughness on the noise this allows to provide the first such result in the pde setting

inference and testing on the boundary in extended constant conditional correlation garch models we consider inference and testing in extended constant conditional correlation garch models in the case where the true parameter vector is a boundary point of the parameter space this is of particular importance when testing for volatility spillovers in the model the large sample properties of the qmle are derived together with the limiting distributions of the related lr wald and lm statistics due to the boundary problem these large sample properties become nonstandard the size and power properties of the tests are investigated in a simulation study as an empirical illustration we test for no volatility spillovers between foreign exchange rates

linear quadratic mean field games with a major player the multi scale approach this paper considers linear quadratic lq mean field games with a major player and analyzes an asymptotic solvability problem it starts with a large scale system of coupled dynamic programming equations and applies a re scaling technique introduced in huang and zhou 2018 and huang and zhou 2020 to derive a set of riccati equations in lower dimensions the solvability of which determines the necessary and sufficient condition for asymptotic solvability we next derive the mean field limit of the strategies and the value functions finally we show that the two decentralized strategies can be interpreted as the best responses of a major player and a representative minor player embedded in an infinite population which have the property of consistent mean field approximations

linear quadratic n person and mean field games with ergodic cost we consider stochastic differential games with n players linear gaussian dynamics in arbitrary state space dimension and long time average cost with quadratic running cost admissible controls

bootstrap inference on the boundary of the parameter space with application to conditional volatility models abstract it is a well established fact that with an unknown number of nuisance parameters at the boundary testing a null hypothesis on the boundary of the parameter space is infeasible in practice as the limiting distributions of standard test statistics are non pivotal in particular likelihood ratio statistics have limiting distributions which can be characterized in terms of quadratic forms minimized over cones where the shape of the cones depends on the unknown location of the possibly multiple model parameters not restricted by the null hypothesis we propose to solve this inference problem by a novel bootstrap which we show to be valid under general conditions irrespective of the presence of unknown nuisance parameters on the boundary that is the new bootstrap replicates the unknown limiting distribution of the likelihood ratio statistic under the null hypothesis and is bounded in probability under the alternative the new bootstrap approach which is very simple to implement is based on shrinkage of the parameter estimates used to generate the bootstrap sample toward the boundary of the parameter space at an appropriate rate as an application of our general theory we treat the problem of inference in finite order arch models with coefficients subject to inequality constraints extensive monte carlo simulations illustrate that the proposed bootstrap has attractive finite sample properties both under the null and under the alternative hypothesis

nonlinear filtering theory for mckean vlasov type stochastic differential equations we consider estimation theory for partially observed stochastic dynamical systems whose state equations are given by mckean vlasov type stochastic differential equations and hence contain a measure term corresponding to the distribution of the solution of the state process nonlinear filtering equations are derived in this framework based upon the classification that the measure term is either stochastic or deterministic and that either the state or the measure term is estimated when the measure term is deterministic the standard theory holds further when the measure is random the induced functions in the dynamics of the state become random and a similar recursion for the optimal filter is obtained the joint estimation of state and the measure term is next considered the extended state in this setup is shown to be a banach space valued stochastic process with random functions in its state dynamics and a nonlinear filtering equation for this setup is provided the first step of such an analysis requ

stochastic volterra equations in banach spaces and stochastic partial differential equation abstract in this paper we study the existence uniqueness and large deviation estimate for stochastic volterra integral equations with singular kernels in 2 smooth banach spaces then we apply them to a large class of semilinear stochastic partial differential equations spde and obtain the existence of unique maximal strong solutions in the sense of sde and pde under local lipschitz conditions moreover stochastic navier stokes equations are also investigated

on the asymptotics of constrained m estimation limit theorems for an m estimate constrained to lie in a closed subset of r d are given under two different sets of regularity conditions a consistent sequence of global optimizers converges under chernoff regularity of the parameter set a n consistent sequence of local optimizers converges under clarke regularity of the parameter set in either case the asymptotic distribution is a projection of a normal random vector on the tangent cone of the parameter set at the true parameter value limit theorems for the optimal value are also obtained agreeing with chernoff s result in the case of maximum likelihood with global optimizers

robust mean field games for coupled markov jump linear systems abstractwe consider robust stochastic large population games for coupled markov jump linear systems mjlss the n agents individual mjlss are governed by different infinitesimal generators and are affected not only by the control input but also by an individual disturbance or adversarial input the mean field term representing the average behaviour of n agents is included in the individual worst case cost function to capture coupling effects among agents to circumvent the computational complexity and analyse the worst case effect of the disturbance we use robust mean field game theory to design low complexity robust decentralised controllers and to characterise the associated worst case disturbance we show that with the individual robust decentralised controller and the corresponding worst case disturbance which constitute a saddle point solution to a generic stochastic differential game for mjlss the actual mean field behaviour can be approximated by a deterministic function which is a fixed

asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions abstract large sample properties of the likelihood function when the true parameter value may be on the boundary of the parameter space are described specifically the asymptotic distribution of maximum likelihood estimators and likelihood ratio statistics are derived these results generalize the work of moran 1971 chant 1974 and chernoff 1954 some of chant s results are shown to be incorrect the approach used in deriving these results follows from comments made by moran and chant the problem is shown to be asymptotically equivalent to the problem of estimating the restricted mean of a multivariate gaussian distribution from a sample of size 1 in this representation the gaussian random variable corresponds to the limit of the normalized score statistic and the estimate of the mean corresponds to the limit of the normalized maximum likelihood estimator thus the limiting distribution of the maximum likelihood estimator is the same as the distribution of the projection of the gaussian random v

non standard inference for augmented double autoregressive models with null volatility coefficients abstract this paper considers an augmented double autoregressive dar model which allows null volatility coefficients to circumvent the over parameterization problem in the dar model since the volatility coefficients might be on the boundary the statistical inference methods based on the gaussian quasi maximum likelihood estimation gqmle become non standard and their asymptotics require the data to have a finite sixth moment which narrows the applicable scope in studying heavy tailed data to overcome this deficiency this paper develops a systematic statistical inference procedure based on the self weighted gqmle for the augmented dar model except for the lagrange multiplier test statistic the wald quasi likelihood ratio and portmanteau test statistics are all shown to have non standard asymptotics the entire procedure is valid as long as the data are stationary and its usefulness is illustrated by simulation studies and one real example

rare events in the boussinesq system with fluctuating dynamical boundary conditions abstract the boussinesq system models various phenomena in geophysical and climate dynamics it is a coupled system of the navier stokes equations and the salinity transport equation due to uncertainty in salinity flux on fluid boundary this system is subject to random fluctuations on the boundary this stochastic boussinesq system can be transformed into a random dynamical system rare events or small probability events are investigated in the context of large deviations a large deviations principle is established via a weak convergence approach based on a recently developed variational representation of functionals of infinite dimensional brownian motion

convex analysis for lqg systems with applications to major minor lqg mean field game systems abstract we develop a convex analysis approach for solving lqg optimal control problems and apply it to major minor mm lqg mean field game mfg systems the approach retrieves the best response strategies for the major agent and all minor agents that attain an nash equilibrium an important and distinctive advantage to this approach is that unlike the classical approach in the literature we are able to avoid imposing assumptions on the evolution of the mean field in particular this provides a tool for dealing with complex and non standard systems

ϵ nash mean field games for general linear quadratic systems with applications this paper is concerned with general mean field mf linear quadratic lq games of stochastic large population system where the individual diffusion coefficient can depend on both the state and the control of the agent moreover the control weight in the cost functional could be indefinite the asymptotic suboptimality property namely nash equilibrium of the decentralized strategies for the lq games is derived through the consistency condition the impact of the population s collective behaviors and the consistency of the mean field estimation are illustrated by the numerical results a pricing problem is also studied for which the decentralized suboptimal price is obtained

testing when a parameter is on the boundary of the maintained hypothesis this paper considers testing problems where several of the standard regularity conditions fail to hold we consider the case where i parameter vectors in the null hypothesis may lie on the boundary of the maintained hypothesis and ii there may be a nuisance parameter that appears under the alternative hypothesis but not under the null the paper establishes the asymptotic null and local alternative distributions of quasi likelihood ratio rescaled quasi likelihood ratio wald and score tests in this case the results apply to tests based on a wide variety of extremum estimators and apply to a wide variety of models examples treated in the paper are i tests of the null hypothesis of no conditional heteroskedasticity in a garch 1 1 regression model and ii tests of the null hypothesis that some random coefficients have variances equal to zero in a random coefficients regression model with possibly correlated random coefficients

on the asymptotics of constrained local m estimators we discuss in this paper asymptotics of locally optimal solutions of maximum likelihood and more generally m estimation procedures in cases where the true value of the parameter vector lies on the boundary of the parameter set s we give a counterexample showing that regularity of s in the sense of clarke is not sufficient for asymptotic equivalence of n consistent locally optimal m estimators we argue further that stronger properties such as so called near convexity or prox regularity of s are required in order to ensure that any two n consistent locally optimal m estimators have the same asymptotics

asymptotics of cholesky garch models and time varying conditional betas this paper proposes a new model with time varying slope coefficients our model called char is a cholesky garch model based on the cholesky decomposition of the conditional variance matrix introduced by pourahmadi 1999 in the context of longitudinal data we derive stationarity and invertibility conditions and prove consistency and asymptotic normality of the full and equation by equation qml estimators of this model we then show that this class of models is useful to estimate conditional betas and compare it to the approach proposed by engle 2016 finally we use real data in a portfolio and risk management exercise we find that the char model outperforms a model with constant betas as well as the dynamic conditional beta model of engle 2016

ϵ nash mean field games for linear quadratic systems with random jumps and applications this paper studies linear quadratic gaussian lqg games of stochastic large population system with jump diffusion processes the most distinguishing feature compared with the well studied mean fi

large deviation for navier stokes equations with small stochastic perturbation we consider a navier stokes evolution equation with a small stochastic perturbation the unperturbed navier stokes equation approximates incompressible viscous fluid flows in a two dimensional bounded region with an adhesive boundary condition the small stochastic perturbation accounts for often ignored quantum effects and other sources of fluctuations the weak solution process for the stochastic navier stokes equation is constructed via a girsanov type transformation large deviation results in terms of wentzel freidlin estimates are obtained for both the infinite dimensional brownian motion and the weak solution process for the stochastic navier stokes equation

adaptive inference for a semiparametric generalized autoregressive conditional heteroskedasticity model abstract this paper considers a semiparametric generalized autoregressive conditional heteroskedasticity s garch model for this model we first estimate the time varying long run component for unconditional variance by the kernel estimator and then estimate the non time varying parameters in garch type short run component by the quasi maximum likelihood estimator qmle we show that the qmle is asymptotically normal with the parametric convergence rate next we construct a lagrange multiplier test for linear parameter constraint and a portmanteau test for model checking and obtain their asymptotic null distributions our entire statistical inference procedure works for the non stationary data with two important features first our qmle and two tests are adaptive to the unknown form of the long run component second our qmle and two tests share the same efficiency and testing power as those in variance targeting method when the s garch model is stationary

epsilon nash equilibria for partially observed lqg mean field games with a major player huang 2010 and nguyen and huang 2012 solved the linear quadratic mean field systems and control problem in the case where there is a major agent i e non asymptotically vanishing as the population size goes to infinity together with a population of minor agents i e individually asymptotically negligible the new feature in this case is that the mean field becomes stochastic and then by minor agent state extension the existence of epsilon nash equilibria together with the individual agents control laws that yield the equilibria may be established this paper presents results initially announced by caines and kizilkale 2013 2014 where it is shown that if the major agent s state is partially observed by the minor agents and if the major agent completely observes its own state all agents can recursively generate estimates in general individually distinct of the major agent s state and the mean field and thence generate feedback controls yielding epsilon nash equilibria

backward mean field linear quadratic gaussian lqg games full and partial information this paper introduces the backward mean field mf linear quadratic gaussian lqg games for short bmflqg of weakly coupled stochastic large population system in contrast to the well studied forward mean field lqg games the individual state in our large population system follows the backward stochastic differential equation bsde whose terminal instead initial condition should be prescribed two classes of bmflqg games are discussed here and their decentralized strategies are derived through the consistency condition in the first class the individual agents of large population system are weakly coupled in their state dynamics and the full information can be accessible to all agents in the second class the coupling structure lies in the cost functional with only partial information structure in both classes the asymptotic near optimality property namely epsilon nash equilibrium of decentralized strategies are verified to this end some estimates to bsde are presented in the large population setting

a score test against one sided alternatives abstract a score type statistic t s is introduced for testing h 0 against k 0 and more general one sided hypotheses when nuisance parameters may be present is a vector parameter the main advantages of t s are that it requires estimation of the model only under the null hypothesis and that it is asymptotically equivalent to the likelihood ratio statistic these are precisely the reasons for the popularity of the score tests for testing against two sided alternatives in this sense t s preserves the main attractive features of the classical two sided score test the theoretical results are presented in a general framework where the likelihood based score function is replaced by an estimating function so that the test is applicable even if the exact population distribution is unknown computation of t s is simplified by the fact that it can be computed easily once the corresponding two sided statistic has been computed the relevance and simplicity of t s are illustrated by discussing

mean field stackelberg games aggregation of delayed instructions in this paper we consider an n player interacting strategic game in the presence of a endogenous dominating player who gives direct influence on individual agents through its impact on their control in the sense of stackelberg game and then on the whole community each individual agent is subject to a delay effect on collecting information specifically at a delay time from the dominating player the size of his delay is completely known by the agent while to others including the dominating player his delay plays as a hidden random variable coming from a common fixed distribution by invoking a noncanonical fixed point property we show that for a general class of finite n player games each of them converges to the mean field counterpart which may possess an optimal solution that can serve as an epsilon nash equilibrium for the corresponding finite n player game second we provide with explicit solutions a comprehensive study on the corresponding linear quadratic mean field games of

epsilon nash mean field game theory for nonlinear stochastic dynamical systems with major and minor agents this paper studies large population dynamic games involving nonlinear stochastic dynamical systems with agents of the following mixed types i a major agent and ii a population of n minor agents where n is very large the major and minor agents are coupled via both i their individual nonlinear stochastic dynamics and ii their individual finite time horizon nonlinear cost functions this problem is analyzed by the so called epsilon nash mean field game theory a distinct feature of the mixed agent mean field game problem is that even asymptotically as the population size n approaches infinity the noise process of the major agent causes random fluctuation of the mean field behavior of the minor agents to deal with this the overall asymptotic n rightarrow infty mean field game problem is decomposed into i two nonstandard stochastic optimal control problems with random coefficient processes which yield forward adapted stochastic best response control processes determined from the

statistical behavior and coherent structures in two dimensional inviscid turbulence the ergodic and mixing questions are studied for the dynamical systems obtained by truncation of the fourier transformed two dimensional euler equations with periodic boundary conditions such statistical properties are important because they bear a close relationship to the existence of coherent structures in turbulent flow fields these theoretical relationships are derived in particular for the case of macroscopic vortex structures the main emphasis is on the numerical analysis of these statistical properties several different systems with various a priori statistical properties ergodic nonergodic unknown have been computed in this manner comparison standards are established for systems with unknown statistical properties a careful error analysis is presented for the results in particular it is found that final integration times for the modal truncations must be kept small relative to a relaxation time in order to insure numerical accuracy the consequences of this fact are important for th

the evolutionary game of pressure or interference resistance and collaboration in this paper we extend the framework of the evolutionary inspection game put forward recently by the author and coworkers to a large class of conflict interactions to address the pressure executed by the major player or principal on the large group of small players who can resist this pressure or collaborate with the major player we prove rigorous results on the convergence of various markov decision models of interacting small agents including evolutionary growth i e pairwise in groups and by coalition formation to a deterministic evolution on the distributions of the state spaces of small players paying main attention to situations with an infinite state space of small players we supply precise rates of convergence the theoretical results of the paper are applied to the analysis of the processes of inspection corruption cyber security counter terrorism banks and firms merging strategically enhanced preferential attachment and many other

open loop nash equilibrium in polynomial differential games via state dependent riccati equation this paper studies finite as well as infinite time horizon nonzero sum polynomial differential games in both cases we explore the so called state dependent riccati equations to find a set of strategies that guarantee an open loop nash equilibrium for this particular class of nonlinear games such a method presents advantages in simplicity of the design of equilibrium strategies and yields computationally effective solution algorithms we demonstrate that this solution leads the game to an e or quasi equilibrium and provide an upper bound for this e quantity the proposed solution is given as a set of n coupled polynomial riccati like state dependent differential equations where each equation includes a p linear form tensor representation for its polynomial part we provide an algorithm for finding the solution of the state dependent algebraic equation in the infinite time case based on a hamiltonian approach and give conditions on the existence of such stabilizing solutions for a third order polynomial a numerical example is presented to illustrate effectiveness of the approach

testing the nullity of garch coefficients correction of the standard tests and relative efficiency comparisons this article is concerned with testing the nullity of coefficients in generalized autoregressive conditionally heteroscedastic garch models the problem is nonstandard because the quasi maximum likelihood estimator is subject to positivity constraints this article establishes the asymptotic null and local alternative distributions of wald score and quasi likelihood ratio tests efficiency comparisons under fixed alternatives are considered two cases of special interest are tests of the null hypothesis of one coefficient equal to zero and tests of the null hypothesis of no conditional heteroscedasticity finally the proposed approach is used in the analysis of financial data and suggests reconsidering the preeminence of garch 1 1 among garch models

self similar universal homogeneous statistical solutions of the navier stokes equations in this note we consider a family of statistical solutions of the navier stokes equations i e time dependent solutions of the hopf equation which seem to constitute the rigorous mathematical framework for the theory of homogeneous turbulence 1 13 the main feature of these solutions is that they are the transforms under suitable scalings of thestationary statistical solutions of a new system of equations the eq 2 below