arXiv research

Let $X$be a complex hyperelliptic curve of genus two equipped with the canonical metric $ds^2$. We study mean field equations on complex hyperelliptic curves and show that the Gaussian curvature function of $(X,ds^2)$ determines an explicit solution to a mean field equation.

New inequality criterion for a mean field equation on spheres.

Mean-field approximations simplify insurance liability calculations.

The paper connects PSO and CBO methods using stochastic modeling and mean-field limits.

Study on the smoothness of solutions to a specific type of stochastic differential equation.

In his lectures at College de France, P.L. Lions introduced the concept of Master equation, see [5] for Mean Field Games. It is introduced in a heuristic fashion, from the system of partial differential equations, associated to a Nash equilibrium for a large, but finite, number of players. The method, also explained in…

Novel approach to Nash equilibrium in mean-field stochastic games with operator resolvents.

Revises mean-field theory of Santa Fe model using kinetic theory.

Recently, variational approximations such as the mean field approximation have received much interest. We extend the standard mean field method by using an approximating distribution that factorises into cluster potentials. This includes undirected graphs, directed acyclic graphs and junction trees. We derive generaliz…

Mean field game theory studies the behavior of a large number of interacting individuals in a game theoretic setting and has received a lot of attention in the past decade (Lasry and Lions, Japanese journal of mathematics, 2007). In this work, we derive mean field game partial differential equation systems from determi…

Let $Ω$ be an annulus. We prove that the mean field equation $-Δψ=\frac{e\sp{-βψ}}{\int\sbΩe\sp{-βψ}} $ admits a solution with zero boundary for $β\in (-16π,-8π)$. This is a supercritical case for the Moser-Trudinger inequality.

Develops a dynamic mean field theory for reinforcement learning.

Algorithm learns interaction kernels for particle systems from data.

We prove uniqueness of solutions to complex Monge-Ampère equations for small temperature.

Paper studies a generalized mean field equation on closed Riemann surfaces.

Study on Langevin dynamics convergence rates and their application to GAN training.

The paper analyzes the mean field Langevin dynamics and its convergence rate.

We study the mean field games equations, consisting of the coupled Kolmogorov-Fokker-Planck and Hamilton-Jacobi-Bellman equations. The equations are complemented by initial and terminal conditions. It is shown that with some specific choice of data, this problem can be reduced to solving a quadratically nonlinear syste…

Study on liquidation games with market drop-out, proving unique equilibria.

In this paper, we prove that the even solution of the mean field equation $Δu=λ(1-e^u)$ on $S^2$ must be axially symmetric when $4<λ\leq 8$. In particular, zero is the only even solution for $λ=6$. This implies the rigidity of Hawking mass for stable constant mean curvature(CMC) sphere with even symmetry.

Improved disability insurance model with collective health claims.

The paper proves existence of solutions for mean field equations on compact Riemann surfaces.

Proves equations for high-dimensional gradient-based methods from Gaussian data.

This paper extends transfer operator theory to McKean-Vlasov equations.

Modeling pollution from competing firms using mean-field games.

Given a regular bounded domain $Ω\subset\R{2m}$, we describe the limiting behavior of sequences of solutions to the mean field equation of order $2m$, $m\geq 1$, $(-Δ)^m u=ρ\frac{e^{2mu}}{\int_Ωe^{2mu}dx}\quad\text{in}Ω,$ under the Dirichlet boundary condition and the bound $0<ρ\leq C$. We emphasize the connection wi…

Mean field game with defaultable agents and systemic risk quantified.

Study on market entry timing in stock liquidation with trading constraints.

Study on mean field games with singular controls and their applications.

Study shows finite agent equilibrium converges to mean-field limit in asset pricing.

New neural networks learn mappings between probability measures and functions.

We review recent quantitative results on the approximation of mean field diffusion equations by large systems of interacting particles, obtained by optimal coupling methods. These results concern a larger range of models, more precise senses of convergence and links with the long time behaviour of the systems to be con…

Model analyzes competitive pricing strategies in large markets of perishable products.

Study optimizes portfolio liquidation strategies with complex market impacts.

Deep Galerkin Method estimates value function for mean-field control problem.

In this paper, we give an algebraic construction of the solution to the following mean field equation $Δψ+e^ψ=4π\sum_{i=1}^{2g+2}δ_{P_{i}},$ on a genus $g\geq 2$ hyperelliptic curve $(X,ds^{2})$ where $ds^{2}$ is a canonical metric on $X$ and $\{P_{1},\cdots,P_{2g+2}\}$ is the set of Weierstrass points on $X.$ Furt…

In this paper we study a continuous time equilibrium model of limit order book (LOB) in which the liquidity dynamics follows a non-local, reflected mean-field stochastic differential equation (SDE) with evolving intensity. Generalizing the basic idea of Ma et al. (2015), we argue that the frontier of the LOB (e.g., the…

The paper proves functions related to mean field equations on surfaces are Morse functions under certain conditions.

Study uses Mean Field Game to analyze Bitcoin mining hashpower dynamics.

Developed LQ MFG theory with common noise, proving existence and uniqueness.

Solves a game between brokers and informed traders using stochastic differential equations.

In this work, we systematically investigate mean field games and mean field type control problems with multiple populations using a coupled system of forward-backward stochastic differential equations of McKean-Vlasov type stemming from Pontryagin's stochastic maximum principle. Although the same cost functions as well…

New sampling method uses gradient-free IPS with RKHS velocity field.

The paper models asset pricing in a partially observed market using mean field game theory and exponential quadratic Gaussian framework.

Gradient descent finds global optima in ResNets with sufficient parameters.

Study on price formation among investors with exponential utility and liabilities.

Deep learning method proves convergence for high-dimensional PDEs.

Recent studies have suggested that the cognitive process of the human brain is realized as probabilistic inference and can be further modeled by probabilistic graphical models like Markov random fields. Nevertheless, it remains unclear how probabilistic inference can be implemented by a network of spiking neurons in th…