arXiv subjects

Ben Hambly

Publications and source records attributed to Ben Hambly.

At least 19 recordsLinked to original sources

arXiv2026

A stochastic Fokker--Planck equation for the mean-field limit of a population of noisy integrate-and-fire neurons

We study a densely connected system of excitatory integrate-and-fire neurons which are subject to common noise. The system incorporates a gradual transmission of action potentials and captures the re- and hyperpolarization phases through a random refractory period followed by a reset to a randomized level below the rest potential. As the number of neurons tends to infinity, we show that there is weak convergence to a unique membrane potential density governed by a stochastic Fokker--Planck equation with a well-defined spike transmission rate. The latter is driven by the mean cumulative spike count, which is non-differentiable but shown to satisfy a generalized flux condition. We obtain the uniqueness of the Fokker--Planck equation from energy estimates in the dual of the first Sobolev space. Finally, we give a conditional McKean--Vlasov representation of the membrane potential density as the law of a representative neuron given the common noise.

math.AP↗

arXiv2026

Probabilistic estimates for a system of noisy integrate-and-fire neurons

In this note, we establish various probabilistic estimates for an interacting particle system that describes the evolution of the membrane potentials in a network of excitatory integrate-and-fire neurons, which are subject to both idiosyncratic and common noise. These estimates serve to support a separate work that studies the large population limit and the corresponding stochastic Fokker--Planck equation for the membrane potential density.

math.PR↗

arXiv2025

Scaling limit for Brownian motions on the $l$-level Sierpinski gaskets: The fractal to Euclidean crossover

In two dimensions, the $l$-level Sierpinski gasket $\mathrm{SG}(l)$ is obtained by splitting an equilateral triangle into a collection of $l^2$ equilateral triangles of equal size and with the same total area, retaining only the $l(l+1)/2$ triangles with the same orientation as the original triangle, and then iterating this procedure indefinitely. We show that the canonical diffusions on the spaces $\mathrm{SG}(l)$, $l\geq2$, can be rescaled to yield Brownian motion on the initial triangle. Our argument also applies to the analogous higher-dimensional Sierpinski gaskets. Moreover, we prove a local central limit theorem for the associated transition densities. Key to this is the derivation of a Poincar\'{e} inequality, in the proof of which we exploit the Euclidean-type mixing that occurs between the bottlenecks present at each scale of the fractal.

math.PR↗

arXiv2025

Particle systems and the supercooled Stefan problem with non-integrable initial data

We consider an infinite system of particles on the positive real line, initiated from a Poisson point process, which move according to Brownian motion up until the hitting time of a barrier. The barrier increases when it is hit, allowing for the possibility of sequences of successive jumps to occur instantaneously. Under certain conditions, the scaling limit gives a representation for the supercooled Stefan problem and its free boundary. This allows us to give a precise asymptotic limit for the barrier and determine the rate of convergence, resolving a conjecture of arXiv:1112.6257. From this representation, we also investigate properties of the supercooled Stefan problem for initial data not in $L^1(\mathbb{R}^+)$. In a critical case, where the jump size matches the density of the Poisson process and the Stefan problem has an instantaneous explosion, we instead recover the same scaling limit result as in arXiv:1705.10017.

math.PR↗

arXiv2025

Particle Systems and McKean--Vlasov Dynamics with Singular Interaction through Local Times

We study a system of reflected Brownian motions on the positive half-line in which each particle has a drift toward the origin determined by the local times at the origin of all the particles. If this local time drift is too strong, such systems exhibit a breakdown in their solutions in that there is a time beyond which the system cannot be extended. In the finite particle case we give a complete characterisation of this finite time breakdown, relying on a novel dynamic graph structure. We consider the mean-field limit of the system in the symmetric setting, which admits a McKean--Vlasov representation, and establish propagation of chaos. In the absence of breakdowns, the McKean--Vlasov equation exhibits multiple stationary and unique self-similar solutions and we prove convergence to these profiles. This work is motivated by models for liquidity in financial markets, the supercooled Stefan problem, and a toy model for cell polarisation.

math.PR↗

arXiv2024

Non-negative Martingale Solutions to the Stochastic Porous Medium Equation with Sticky Behavior

We construct non-negative martingale solutions to the stochastic porous medium equation in one dimension with homogeneous Dirichlet boundary conditions which exhibit a type of sticky behavior at zero. The construction uses the stochastic Faedo--Galerkin method via spatial semidiscretization, so that the pre-limiting system is given by a finite-dimensional diffusion with Wentzell boundary condition. We derive uniform moment estimates for the discrete systems by an Aubin-Lions-type interpolation argument, which enables us to implement a general weak convergence approach for the construction of martingale solutions of an SPDE using a Skorokhod representation-type result for non-metrizable spaces. We rely on a stochastic argument based on the occupation time formula for continuous semimartingales for the identification of the diffusion coefficient in the presence of an indicator function.

math.PR↗

arXiv2024

Optimal Control of the Nonlinear Stochastic Fokker--Planck Equation

We consider a control problem for the nonlinear stochastic Fokker--Planck equation. This equation describes the evolution of the distribution of nonlocally interacting particles affected by a common source of noise. The system is directed by a controller that acts on the drift term with the goal of minimising a cost functional. We establish the well-posedness of the state equation, prove the existence of optimal controls, and formulate a stochastic maximum principle (SMP) that provides necessary and sufficient optimality conditions for the control problem. The adjoint process arising in the SMP is characterised by a nonlocal (semi)linear backward SPDE for which we study existence and uniqueness. We also rigorously connect the control problem for the nonlinear stochastic Fokker--Planck equation to the control of the corresponding McKean--Vlasov SDE that describes the motion of a representative particle. Our work extends existing results for the control of the Fokker--Planck equation to nonlinear and stochastic dynamics. In particular, the sufficient SMP, which we obtain by exploiting the special structure of the Fokker--Planck equation, seems to be novel even in the linear deterministic setting. We illustrate our results with an application to a model of government interventions in financial systems, supplemented by numerical illustrations.

math.OC↗

arXiv2024

A McKean--Vlasov equation with positive feedback and blow-ups

We study a McKean--Vlasov equation arising from a mean-field model of a particle system with positive feedback. As particles hit a barrier they cause the other particles to jump in the direction of the barrier and this feedback mechanism leads to the possibility that the system can exhibit contagious blow-ups. Using a fixed-point argument we construct a differentiable solution up to a first explosion time. Our main contribution is a proof of uniqueness in the class of càdlàg functions, which confirms the validity of related propagation-of-chaos results in the literature. We extend the allowed initial conditions to include densities with any power law decay at the boundary, and connect the exponent of decay with the growth exponent of the solution in small time in a precise way. This takes us asymptotically close to the control on initial conditions required for a global solution theory. A novel minimality result and trapping technique are introduced to prove uniqueness.

math.PR↗

arXiv2023

Semilinear BSPDEs and Applications to McKean-Vlasov Control with Killing

We introduce a novel class of semilinear nonlocal backward stochastic partial differential equations (BSPDE) on half-spaces driven by an infinite-dimensional càdlàg martingale. The equations exhibit a degeneracy and have no explicit condition at the boundary of the half-space. To treat the existence and uniqueness of such BSPDEs we establish a generalisation of Itô's formula for infinite-dimensional càdlàg semimartingales, which addresses the occurrence of boundary terms. Next, we employ these BSPDEs to study the McKean--Vlasov control problem with killing and common noise proposed in [HJ23]. The particles in this control model live on the real line and are killed at a positive intensity whenever they are in the negative half-line. Accordingly, the interaction between particles occurs through the subprobability distribution of the living particles. We establish the existence of an optimal semiclosed-loop control that only depends on the particles' location and not their cumulative intensity. This problem cannot be addressed through classical mimicking arguments, because the particles' subprobability distribution cannot be reconstructed from their location alone. Instead, we represent optimal controls in terms of the solutions to BSPDEs of the above type and show those solutions do not depend on the intensity variable.

math.PR↗

arXiv2023

Control of McKean--Vlasov SDEs with Contagion Through Killing at a State-Dependent Intensity

We consider a novel McKean--Vlasov control problem with contagion through killing of particles and common noise. Each particle is killed at an exponential rate according to an intensity process that increases whenever the particle is located in a specific region. The removal of a particle pushes others towards the removal region, which can trigger cascades that see particles exiting the system in rapid succession. We study the control of such a system by a central agent who intends to preserve particles at minimal cost. Our theoretical contribution is twofold. Firstly, we rigorously justify the McKean--Vlasov control problem as the limit of a corresponding sequences of controlled finite particle systems. Our proof is based on a controlled martingale problem and tightness arguments. Secondly, we connect our framework with models in which particles are killed once they hit the boundary of the removal region. We show that these models appear in the limit as the exponential rate tends to infinity. As a corollary, we obtain new existence results for McKean--Vlasov SDEs with singular interaction through hitting times which extend those in the established literature. We conclude the paper with numerical investigations of our model applied to government control of systemic risk in financial systems.

math.PR↗

arXiv2023

Linear-quadratic Gaussian Games with Asymmetric Information: Belief Corrections Using the Opponents Actions

We consider two-player non-zero-sum linear-quadratic Gaussian games in which both players aim to minimize a quadratic cost function while controlling a linear and stochastic state process {using linear policies}. The system is partially observable with asymmetric information available to the players. In particular, each player has a private and noisy measurement of the state process but can see the history of their opponent's actions. The challenge of this asymmetry is that it introduces correlations into the players' belief processes for the state and leads to circularity in their beliefs about their opponents beliefs. We show that by leveraging the information available through their opponent's actions, both players can enhance their state estimates and improve their overall outcomes. In addition, we provide a closed-form solution for the Bayesian updating rule of their belief process. We show that there is a Nash equilibrium which is linear in the estimation of the state and with a value function incorporating terms that arise due to errors in the state estimation. We illustrate the results through an application to bargaining which demonstrates the value of these information corrections.

math.OC↗

arXiv2023

Contagious McKean--Vlasov problems with common noise: from smooth to singular feedback through hitting times

We consider a family of McKean--Vlasov equations arising as the large particle limit of a system of interacting particles on the positive half-line with common noise and feedback. Such systems are motivated by structural models for systemic risk with contagion. This contagious interaction is such that when a particle hits zero, the impact is to move all the others toward the origin through a kernel which smooths the impact over time. We study a rescaling of the impact kernel under which it converges to the Dirac delta function so that the interaction happens instantaneously and the limiting singular McKean--Vlasov equation can exhibit jumps. Our approach provides a novel method to construct solutions to such singular problems that allows for more general drift and diffusion coefficients and we establish weak convergence to relaxed solutions in this setting. With more restrictions on the coefficients we can establish an almost sure version showing convergence to strong solutions. Under some regularity conditions on the contagion, we also show a rate of convergence up to the time the regularity of the contagion breaks down. Lastly, we perform some numerical experiments to investigate the sharpness of our bounds for the rate of convergence.

math.PR↗

arXiv2023

Recent Advances in Reinforcement Learning in Finance

The rapid changes in the finance industry due to the increasing amount of data have revolutionized the techniques on data processing and data analysis and brought new theoretical and computational challenges. In contrast to classical stochastic control theory and other analytical approaches for solving financial decision-making problems that heavily reply on model assumptions, new developments from reinforcement learning (RL) are able to make full use of the large amount of financial data with fewer model assumptions and to improve decisions in complex financial environments. This survey paper aims to review the recent developments and use of RL approaches in finance. We give an introduction to Markov decision processes, which is the setting for many of the commonly used RL approaches. Various algorithms are then introduced with a focus on value and policy based methods that do not require any model assumptions. Connections are made with neural networks to extend the framework to encompass deep RL algorithms. Our survey concludes by discussing the application of these RL algorithms in a variety of decision-making problems in finance, including optimal execution, portfolio optimization, option pricing and hedging, market making, smart order routing, and robo-advising.

q-fin.MF↗

arXiv2022

An SPDE with Robin-type boundary for a system of elastically killed diffusions on the positive half-line

We consider a system of particles undergoing correlated diffusion with elastic boundary conditions on the half-line. By taking the large particle limit we establish existence and uniqueness for the limiting empirical measure valued process for the surviving particles. This process can be viewed as the weak form for an SPDE with a noisy Robin boundary condition satisfied by the particle density. We establish results on the $L^2$-regularity properties of this density process, showing that it is well behaved in the interior of the domain but may exhibit singularities on the boundary at a dense set of times. We also show existence of limit points for the empirical measure in the non-linear case where the particles have a measure dependent drift. We make connections for our linear problem to the corresponding absorbing and reflecting SPDEs, as the elastic parameter takes its extreme values.

math.PR↗

arXiv2022

Policy Gradient Methods Find the Nash Equilibrium in N-player General-sum Linear-quadratic Games

We consider a general-sum N-player linear-quadratic game with stochastic dynamics over a finite horizon and prove the global convergence of the natural policy gradient method to the Nash equilibrium. In order to prove the convergence of the method, we require a certain amount of noise in the system. We give a condition, essentially a lower bound on the covariance of the noise in terms of the model parameters, in order to guarantee convergence. We illustrate our results with numerical experiments to show that even in situations where the policy gradient method may not converge in the deterministic setting, the addition of noise leads to convergence.

math.OC↗

arXiv2022

McKean-Vlasov Equations with Positive Feedback through Elastic Stopping Times

We prove existence and uniqueness of physical and minimal solutions to McKean-Vlasov equations with positive feedback through elastic stopping times. We do this by establishing a relationship between this problem and a problem with absorbing stopping times. We show convergence of a particle system to the McKean-Vlasov equation. Moreover, we establish convergence of the elastic McKean-Vlasov problem to the problem with absorbing stopping times and to a reflecting Brownian motion as the elastic parameter goes to infinity or zero respectively.

math.PR↗

arXiv2022

A Stochastic Model of Chemorepulsion with Additive Noise and Nonlinear Sensitivity

We consider a stochastic partial differential equation (SPDE) model for chemorepulsion, with non-linear sensitivity on the one-dimensional torus. We show that for any suitable initial data there exists a pathwise unique, global solution to the SPDE. Furthermore we show that the associated semi-group is Markov and possesses a unique invariant measure, supported on a Hölder-Besov space of positive regularity, which the solution law converges to exponentially fast. We also establish tail bounds on the invariant measure that are heavier than Gaussian when measured using any $L^p$ norm.

math.PR↗

arXiv2021

Policy Gradient Methods for the Noisy Linear Quadratic Regulator over a Finite Horizon

We explore reinforcement learning methods for finding the optimal policy in the linear quadratic regulator (LQR) problem. In particular, we consider the convergence of policy gradient methods in the setting of known and unknown parameters. We are able to produce a global linear convergence guarantee for this approach in the setting of finite time horizon and stochastic state dynamics under weak assumptions. The convergence of a projected policy gradient method is also established in order to handle problems with constraints. We illustrate the performance of the algorithm with two examples. The first example is the optimal liquidation of a holding in an asset. We show results for the case where we assume a model for the underlying dynamics and where we apply the method to the data directly. The empirical evidence suggests that the policy gradient method can learn the global optimal solution for a larger class of stochastic systems containing the LQR framework and that it is more robust with respect to model mis-specification when compared to a model-based approach. The second example is an LQR system in a higher dimensional setting with synthetic data.

cs.LG↗