Study on the smoothness of solutions to a specific type of stochastic differential equation.
problem Regularity of solutions to mean-field G-SDEs. method Analysis of first and second order Fréchet differentiability in the random initial condition.
result Established the Fréchet differentiability of the solution and specified the corresponding equations.
The paper connects PSO and CBO methods using stochastic modeling and mean-field limits.
problem Global optimization problems with particle swarm optimization and consensus based optimization.
method Stochastic differential equations and mean-field approximation to derive macroscopic hydrodynamic equations.
result Derives mean-field approximation for PSO and links it to CBO methods.
This paper extends transfer operator theory to McKean-Vlasov equations.
problem Analyzing the behavior of complex dynamical systems using transfer operators.
method Extended dynamic mode decomposition and Galerkin projection.
result Finite-dimensional approximations of transfer operators computed.
Solves a game between brokers and informed traders using stochastic differential equations.
problem Optimizing wealth in a game between brokers and informed traders with private signals.
method Closed-form solutions to a mean-field game using forward-backward SDEs.
result Optimal trading strategies for both brokers and informed traders are found.
Study improves L∞ estimates and extreme value behavior in stochastic differential games.
problem Analyzing the mean-field limit of diffusive games through master equation.
method Using the Master Equation to approximate state processes and establishing L∞ estimates for the total error. result Established No∞ asymptotic behavior of upper order statistics of Nash states, initiating Extreme Value Theory for stochastic differential games. Study shows finite agent equilibrium converges to mean-field limit in asset pricing.
problem Asset pricing equilibrium in markets with finite vs infinite agents.
method Existence of finite agent equilibrium and strong convergence to mean-field limit.
result Finite agent equilibrium converges to mean-field limit under suitable conditions.
Conservative SPDEs emerge from fluctuating SGD dynamics in neural networks.
problem Understanding the convergence of stochastic gradient descent to SPDEs.
method Mean-field analysis and central limit theorem for SPDEs.
result Optimal convergence rates for SPDEs derived from SGD.
Study on LOB dynamics using mean-field game theory.
problem Modeling liquidity dynamics in limit order books.
method Mean-field stochastic differential equation and control problem formulation.
result Equilibrium density function of LOB can be derived.
Analyzes high-dimensional SGD dynamics using DMFT.
problem Understanding the high-dimensional behavior of multi-pass SGD with small batch sizes.
method Derives DMFT equations for high-dimensional SGD dynamics.
result Proves DMFT equations characterize the asymptotic distribution of SGF parameters.
Two neural network methods solve the master equation for MFGs.
problem Approximating Nash equilibria in stochastic, finite-agent games.
method Backward induction and direct PDE tackling neural networks.
result Neural networks can approximate the master equation's solution.
The paper extends mean field results to three-layer neural networks using SGD.
problem Understanding the dynamics of training three-layer neural networks with SGD.
method Extending mean field results from two-layer networks to three-layer networks with two hidden layers, using non-linear partial differential equations.
result The distributions of weights in the two hidden layers are independent.
This paper studies insurers' robust strategies in a stochastic game with model uncertainty and volatility risk.
problem Model uncertainty and volatility risk in insurers' surplus processes.
method Formulates robust mean-field games with insurers competing based on mean-variance criterion under worst-case scenario.
result Derives semi-closed forms of equilibrium strategies for insurers and mean-field equilibrium, ensuring existence and uniqueness.
Study on price formation among investors with exponential utility and liabilities.
problem Equilibrium price formation among investors with heterogeneous risk-averseness and liabilities.
method Mean-field game theory and mean-field backward stochastic differential equations (BSDE).
result Existence of equilibrium risk-premium process and market clearing in the large population limit.
Paper analyzes SHB method for neural networks, proving stability, connectivity, and global convergence.
problem Theoretical understanding of SHB method for neural networks.
method Mean-field analysis of SHB dynamics related to a partial differential equation.
result SHB method converges to global optimum and exhibits stability and connectivity.
In this paper we study mean-field type control problems with risk-sensitive performance functionals. We establish a stochastic maximum principle (SMP) for optimal control of stochastic differential equations (SDEs) of mean-field type, in which the drift and the diffusion coefficients as well as the performance function…
Novel approach to Nash equilibrium in mean-field stochastic games with operator resolvents.
problem Finding Nash equilibrium in mean-field stochastic games with mean-field interaction.
method Proposed a novel approach to derive Nash equilibrium semi-explicitly using operator resolvents and stochastic Fredholm equations.
result Equilibrium of the N-player game converges to mean-field equilibrium, and ε-Nash equilibrium derived as a by-product. We rigorously prove a central limit theorem for neural network models with a single hidden layer. The central limit theorem is proven in the asymptotic regime of simultaneously (A) large numbers of hidden units and (B) large numbers of stochastic gradient descent training iterations. Our result describes the neural net…
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…
Stochastic mirror descent improves performance on ensemble models.
problem Improving performance of ensemble models using stochastic mirror descent.
method Utilizes mirror potential to influence training algorithm's implicit bias, mapping evolution to continuous time process.
result Converges to a nonlinear PDE in asymptotic regime of large networks, with mirror potential affecting gradient flow.
Method identifies IPS governing equations from particle data efficiently.
problem Identify governing equations of interacting particle systems efficiently.
method Combines mean-field theory and WSINDy for large N and M. result Converges with rate O(N−1/2) for N≥100. Mean-field approximations simplify insurance liability calculations.
problem High-dimensional system of equations makes insurance liability calculation infeasible.
method Use mean-field model to replace high-dimensional system with a low-dimensional non-linear system.
result Insurance liability converges to mean-field approximation as cohort size increases.
Model asset pricing with habit formation in a large market.
problem Understanding asset pricing in large heterogeneous markets with habit formation.
method Mean field game theory and quadratic-growth mean field BSDEs.
result Derives a semi-analytic solution for asset pricing model.
Study shows rate of convergence for particle approximation of PDEs in Wasserstein space.
problem Analyzing convergence rates for particle approximations of PDEs in Wasserstein space.
method Backward stochastic differential equations techniques.
result Proved a rate of convergence of order 1/N for pathwise error and 1/sqrt(N) for L2-error on the derivative.
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…
The paper models asset pricing in a partially observed market using mean field game theory and exponential quadratic Gaussian framework.
problem Asset pricing in a market with partial observation and heterogeneous agents.
method Mean field game theory, exponential quadratic Gaussian framework, Kalman-Bucy filtering theory.
result Characterization of equilibrium risk premium through mean field BSDE and construction of unobservable risk premium process.
DeepGSB solves MFGs with non-differentiable preferences.
problem Solving MFGs with non-differentiable preferences and exact population convergence.
method Generalized Schrödinger Bridge via Forward-Backward SDEs and Temporal Difference learning.
result DeepGSB provides necessary and sufficient conditions for mean-field problems.
The paper studies stability of mean-field variational inference for log-concave distributions.
problem Stability of mean-field variational inference for log-concave distributions.
method Novel approach via linearized optimal transport, lifting non-convex problem to convex optimization over transport maps.
result Dimension-free Lipschitz continuity of the MFVI optimizer with respect to the target distribution, measured in 2-Wasserstein distance.
Develops asset pricing models with mean field game theory for heterogeneous agents.
problem Tackles equilibrium asset pricing in incomplete markets with heterogeneous agents.
method Uses mean field game theory and mean field backward stochastic differential equations (BSDEs).
result Derives equilibrium risk premium and shows market clearing in the large population limit.
Neural networks with a large number of parameters admit a mean-field description, which has recently served as a theoretical explanation for the favorable training properties of "overparameterized" models. In this regime, gradient descent obeys a deterministic partial differential equation (PDE) that converges to a glo…
Proves equations for high-dimensional gradient-based methods from Gaussian data.
problem High-dimensional asymptotics of gradient-based learning algorithms.
method Closed-form equations derived from dynamical mean-field theory.
result Equations match those from discretized DMFT for gradient flow.
Framework detects tipping points in complex systems using ML.
problem Detecting tipping points in complex, emergent systems.
method Combining manifold learning, neural networks, and Gaussian processes.
result Reduced-order models for mesoscopic and mean-field dynamics.
Parameter inference for stochastic differential equations is challenging due to the presence of a latent diffusion process. Working with an Euler-Maruyama discretisation for the diffusion, we use variational inference to jointly learn the parameters and the diffusion paths. We use a standard mean-field variational appr…
Method combines deep learning and elicitability for solving complex stochastic equations.
problem Solving McKean-Vlasov FBSDEs with common noise.
method Combines Picard iterations, elicitability, and deep learning.
result Validated on systemic-risk model and extended to quantile-mediated interactions.
We extend the Deep Galerkin Method (DGM) introduced in Sirignano and Spiliopoulos (2018)} to solve a number of partial differential equations (PDEs) that arise in the context of optimal stochastic control and mean field games. First, we consider PDEs where the function is constrained to be positive and integrate to uni…
We consider learning two layer neural networks using stochastic gradient descent. The mean-field description of this learning dynamics approximates the evolution of the network weights by an evolution in the space of probability distributions in RD (where D is the number of parameters associated to each neuron). T…
The asymptotic pseudo-trajectory approach to stochastic approximation of Benaim, Hofbauer and Sorin is extended for asynchronous stochastic approximations with a set-valued mean field. The asynchronicity of the process is incorporated into the mean field to produce convergence results which remain similar to those of a…
A stochastic theory for the toppling activity in sandpile models is developed, based on a simple mean-field assumption about the toppling process. The theory describes the process as an anti-persistent Gaussian walk, where the diffusion coefficient is proportional to the activity. It is formulated as a generalization o…
Algorithm generates private continuous-time data for sensitive domains.
problem Private generation of continuous-time data for sensitive domains.
method Mean-field Langevin dynamics and noisy particle gradient descent.
result Strong privacy guarantees for one-time data contributions.
Financial markets are often driven by latent factors which traders cannot observe. Here, we address an algorithmic trading problem with collections of heterogeneous agents who aim to perform optimal execution or statistical arbitrage, where all agents filter the latent states of the world, and their trading actions hav…
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…
Paper studies systemic robustness in financial networks using particle systems.
problem Budget control and default risk in regional financial networks.
method Mean-field particle system approach, McKean-Vlasov equations, asymptotic analysis.
result Systemic robustness measured by the proportion of surviving entities in large particle systems.
Study on price formation in financial markets with a single default event.
problem Equilibrium price formation in financial markets with a single default risk.
method Characterized optimal strategies using quadratic-growth BSDEs, derived market-clearing condition, and established mean-field BSDE solvability.
result Characterized equilibrium risk premium and its dependence on default risk factors.
Even when confronted with the same data, agents often disagree on a model of the real-world. Here, we address the question of how interacting heterogenous agents, who disagree on what model the real-world follows, optimize their trading actions. The market has latent factors that drive prices, and agents account for th…
New MFG model for MV portfolio management with peer-based risk aversion.
problem Time-inconsistent mean-variance portfolio management with peer-based risk aversion.
method Mean-field game, smooth regularization, fixed-point arguments, convergence analysis.
result Existence of mean-field equilibrium in time-inconsistent MFG.
The paper analyzes arbitrage opportunities in a large investor market with common stock noises.
problem Identifying arbitrage opportunities in a market with many competitive investors.
method Stochastic differential games and mean-field systems to study market dynamics and optimal arbitrage.
result Optimal arbitrage is characterized by a solution to a Cauchy PDE involving volatility terms.
Study uses Mean Field Game to analyze Bitcoin mining hashpower dynamics.
problem Analyzing the hashpower distribution in Bitcoin mining.
method Mean Field Game framework and master equation approach.
result Hashpower reaches steady state or increases with demand.
Gradient descent finds global optima in ResNets with sufficient parameters.
problem Finding optimal parameters in ResNet models.
method Mean-field analysis and gradient-flow PDE to study convergence of first-order optimization methods.
result First-order methods can find global minimizers in overparameterized ResNets.
Developed LQ MFG theory with common noise, proving existence and uniqueness.
problem Linear-quadratic mean field games with common noise.
method Coupled forward-backward stochastic evolution equations (FBSEEs) in Hilbert spaces.
result Existence and uniqueness of solutions for small and arbitrary finite time horizons.