Ben Hambly
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View article: Particle systems and the supercooled Stefan problem with non-integrable initial data
Particle systems and the supercooled Stefan problem with non-integrable initial data Open
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 f…
View article: Particle Systems and McKean--Vlasov Dynamics with Singular Interaction through Local Times
Particle Systems and McKean--Vlasov Dynamics with Singular Interaction through Local Times Open
We study a system of reflecting 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. We show that if this local time drift is too …
View article: Non-negative Martingale Solutions to the Stochastic Porous Medium Equation with Sticky Behavior
Non-negative Martingale Solutions to the Stochastic Porous Medium Equation with Sticky Behavior Open
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 F…
View article: Optimal Control of the Nonlinear Stochastic Fokker--Planck Equation
Optimal Control of the Nonlinear Stochastic Fokker--Planck Equation Open
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 …
View article: Stochastic PDEs for large portfolios with general mean-reverting volatility processes
Stochastic PDEs for large portfolios with general mean-reverting volatility processes Open
In this article we consider a large structural market model of defaultable assets, where the asset value processes are modelled by using stochastic volatility models with default upon hitting a lower boundary. The volatility processes are …
View article: Semilinear BSPDEs and Applications to McKean-Vlasov Control with Killing
Semilinear BSPDEs and Applications to McKean-Vlasov Control with Killing Open
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 condit…
View article: Control of McKean--Vlasov SDEs with Contagion Through Killing at a State-Dependent Intensity
Control of McKean--Vlasov SDEs with Contagion Through Killing at a State-Dependent Intensity Open
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 locat…
View article: Linear-quadratic Gaussian Games with Asymmetric Information: Belief Corrections Using the Opponents Actions
Linear-quadratic Gaussian Games with Asymmetric Information: Belief Corrections Using the Opponents Actions Open
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 ob…
View article: Contagious McKean--Vlasov problems with common noise: from smooth to singular feedback through hitting times
Contagious McKean--Vlasov problems with common noise: from smooth to singular feedback through hitting times Open
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 systemi…
View article: Recent advances in reinforcement learning in finance
Recent advances in reinforcement learning in finance Open
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 stoch…
View article: Dimension results and local times for superdiffusions on fractals
Dimension results and local times for superdiffusions on fractals Open
We consider the Dawson–Watanabe superprocess obtained from a spatial motion with sub-Gaussian transition densities on a metric measure space with finite Hausdorff dimension, and examine the dimensions of the range and the set of times when…
View article: An SPDE with Robin-type boundary for a system of elastically killed diffusions on the positive half-line
An SPDE with Robin-type boundary for a system of elastically killed diffusions on the positive half-line Open
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…
View article: Modelling Spikes and Pricing Swing Options in Electricity Markets
Modelling Spikes and Pricing Swing Options in Electricity Markets Open
Most electricity markets exhibit high volatilities and occasional distinctive price spikes, which result in demand for derivative products which protect the holder against high prices. In this paper we examine a simple spot price model tha…
View article: A probabilistic model for interfaces in a martensitic phase transition
A probabilistic model for interfaces in a martensitic phase transition Open
We analyse features of the patterns formed from a simple model for a martensitic phase transition that fragments the unit square into rectangles. This is a fragmentation model that can be encoded by a general branching random walk. An impo…
View article: McKean-Vlasov Equations with Positive Feedback through Elastic Stopping Times
McKean-Vlasov Equations with Positive Feedback through Elastic Stopping Times Open
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 absorbi…
View article: A stochastic model of chemorepulsion with additive noise and nonlinear sensitivity
A stochastic model of chemorepulsion with additive noise and nonlinear sensitivity Open
We consider a stochastic partial differential equation (SPDE) model for chemorepulsion, with non-linear sensitivity on the one-dimensional torus. By establishing an a priori estimate independent of the initial data, we show that there exis…
View article: McKean-Vlasov equations with positive feedback through elastic stopping times
McKean-Vlasov equations with positive feedback through elastic stopping times Open
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 absorbi…
View article: Recent Advances in Reinforcement Learning in Finance
Recent Advances in Reinforcement Learning in Finance Open
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 stoch…
View article: Policy Gradient Methods Find the Nash Equilibrium in N-player General-sum Linear-quadratic Games
Policy Gradient Methods Find the Nash Equilibrium in N-player General-sum Linear-quadratic Games Open
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 th…
View article: Policy Gradient Methods for the Noisy Linear Quadratic Regulator over a Finite Horizon
Policy Gradient Methods for the Noisy Linear Quadratic Regulator over a Finite Horizon Open
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.…
View article: An approximation of solutions to heat equations defined by generalized measure theoretic Laplacians
An approximation of solutions to heat equations defined by generalized measure theoretic Laplacians Open
We consider the heat equation defined by a generalized measure theoretic Laplacian on [0, 1]. This equation describes heat diffusion in a bar such that the mass distribution of the bar is given by a non-atomic Borel probabiliy measure $$\m…
View article: An Approximation of Solutions to Heat Equations defined by Generalized Measure Theoretic Laplacians
An Approximation of Solutions to Heat Equations defined by Generalized Measure Theoretic Laplacians Open
We consider the heat equation defined by a generalized measure theoretic Laplacian on $[0,1]$. This equation describes heat diffusion in a bar such that the mass distribution of the bar is given by a non-atomic Borel probabiliy measure $μ$…
View article: Stochastic PDEs for large portfolios with general mean-reverting volatility processes
Stochastic PDEs for large portfolios with general mean-reverting volatility processes Open
We consider a structural stochastic volatility model for the loss from a large portfolio of credit risky assets. Both the asset value and the volatility processes are correlated through systemic Brownian motions, with default determined by…