arXiv subjects

Luca Di Persio

Publications and source records attributed to Luca Di Persio.

At least 19 recordsLinked to original sources

arXiv2026

Trembling-Hand Perfection in Mean Field Games

We introduce an admissible trembling-hand refinement for stochastic mean field games formulated through relaxed controlled martingale problems. A relaxed MFG equilibrium is perfect if it is the common Wasserstein limit of two sequences of joint control-state laws. The first consists of full-support population perturbations admissible for the state laws they generate; the second consists of exact optimal responses to those perturbed state laws. This admissibility condition prevents the control and state coordinates of a perturbation from being varied independently. Under the standing continuity, growth, and coercivity assumptions, we prove existence without requiring compact controls or bounded coefficients. The proof combines a perturbed fixed-point argument in the compact-bounded case with truncation, uniform moment estimates, compactness, and stability of controlled martingale problems. A one-dimensional model with exactly two relaxed mean field game equilibria shows that the refinement is genuinely selective: only one equilibrium is trembling-hand perfect.

math.OC↗

arXiv2026

Shape analysis in Schauder spaces of the energy of heat problems in perturbed annular domains

This paper is devoted to the shape analysis of the energy of boundary value problems for the heat equation in a bounded perforated domain $\Omega^o \setminus \overline{\Omega^i[\phi]}$ of $\mathbb{R}^n$, where the outer boundary is fixed, and the inner boundary is given by a $C^{1,\alpha}$-perturbation $\phi$ of the boundary of a reference cavity. Under standard Dirichlet or Neumann boundary conditions, we prove that, in a suitable neighborhood of the identity $\phi_0$, the domain-to-energy map is of class $C^{\infty}$. The proof is based on the construction of a global diffeomorphism, smoothly depending on $\phi$, from the reference annulus onto the perturbed one, on a decomposition of the fixed domain into regions near, intermediate to, and far from the cavity, and on the smooth dependence of the layer heat potentials upon support perturbations.

math.AP↗

arXiv2026

Weibull-Stationary Stochastic Differential Equations for Conditional Long-Horizon Wind Power Forecasting

We present a one-month-ahead conditional probabilistic framework for wind-power forecasting at ten-minute resolution. Monthly Weibull shape and scale parameters are estimated from serially dependent SCADA wind-speed data, corrected through a Godambe covariance, and forecast by a heteroskedastic Kalman filter on a bivariate VAR(1) state-space model. Conditional on the MMSE forecasted Weibull invariant law, we construct and compare three positive wind-speed SDE models: an Ornstein-Uhlenbeck-Weibull transform, a Fokker-Planck drift-first diffusion, and a Fokker-Planck diffusion-first model. The simulated wind-speed ensembles are mapped to power through a calibrated XGBoost power curve. Applied to January 2021 data from a Senvion MM92 turbine at Kelmarsh Wind Farm, the three SDE formulations are statistically indistinguishable in probabilistic accuracy, with mean CRPS values between 1.569 and 1.575 m/s. The diffusion-first model is therefore preferred on computational grounds, reducing runtime by about a factor of seven relative to the OU-Weibull model. In the power domain, the Wasserstein distance between simulated and observed distributions is 26.1-27.6 kW, below $1.4\%$ of rated capacity, while the monthly energy-yield bias is about $-7.3\%$ for the examined month. Exceedance-probability errors remain below 1.6 percentage points over the 0-1500 kW range and about 2.2 percentage points near rated power. These quantities provide decision-relevant probabilistic inputs for downstream operational problems, rather than completed reserve, storage, market, or fatigue-optimization decisions. Full marginalisation over the Kalman predictive law of the Weibull parameters is left as a natural extension.

stat.AP↗

arXiv2026

Mean field games with terminal state constraints

We study a mean field game (MFG) of state and control with state dynamics described by stochastic differential equations driven by both idiosyncratic and common noise, and subject to the constraint that the terminal state variable belongs to a nonempty, convex, closed set. The mean-field interaction enters both the dynamics and the costs through state and control. We formulate an equivalent auxiliary MFG problem and derive the stochastic maximum principle for an auxiliary optimization problem under fixed flows. By means of a suitable forward-backward stochastic differential equation (FBSDE) of conditional McKean-Vlasov type, we establish a characterization of MFG solutions via such a system. Moreover, we prove the existence and uniqueness of solutions for the FBSDE system in the specific case where the running cost is zero and the state coefficients are linear. Additionally, we obtain uniqueness results in the sense of Lasry-Lions and we apply our findings to MFGs of optimal investment with a quadratic relative performance criterion.

math.OC↗

arXiv2026

A uniqueness result for finite-state mean field games with non-separable Hamiltonian

We study a class of continuous-time mean field games on a finite state space with transition rates depending on the population distribution, leading to a non-separable Hamiltonian. In this setting, classical Lasry--Lions monotonicity arguments do not apply directly. We establish a new uniqueness result on arbitrary time horizons under a combination of strong monotonicity assumptions on the costs and quantitative bounds on the interaction term in the dynamics.

math.OC↗

arXiv2026

Stochastic Port-Hamiltonian Neural Networks: Universal Approximation with Passivity Guarantees

Stochastic port-Hamiltonian systems represent open dynamical systems with dissipation, inputs, and stochastic forcing in an energy based form. We introduce stochastic port-Hamiltonian neural networks, SPH-NNs, which parameterize the Hamiltonian with a feedforward network and enforce skew symmetry of the interconnection matrix and positive semidefiniteness of the dissipation matrix. For It\^o dynamics we establish a weak passivity inequality in expectation under an explicit generator condition, stated for a stopped process on a compact set. We also prove a universal approximation result showing that, on any compact set and finite horizon, SPH-NNs approximate the coefficients of a target stochastic port-Hamiltonian system with $C^2$ accuracy of the Hamiltonian and yield coupled solutions that remain close in mean square up to the exit time. Experiments on noisy mass spring, Duffing, and Van der Pol oscillators show improved long horizon rollouts and reduced energy error relative to a multilayer perceptron baseline.

cs.LG↗

arXiv2026

Mild Solutions for Path-Dependent Parabolic PDEs with Neumann Boundary Conditions via Generalized BSDEs

We study a system of Forward-Backward Stochastic Differential Equations (FBSDEs) with time-delayed generators. The forward process includes a reflection component expressed via a Stieltjes integral, while the backward process takes the form of a Generalized BSDE. We establish the connection between this FBSDE system and non-linear path-dependent PDEs with Neumann boundary conditions by deriving a representation formula.

math.PR↗

arXiv2025

Integrating LSTM Networks with Neural Levy Processes for Financial Forecasting

This paper investigates an optimal integration of deep learning with financial models for robust asset price forecasting. Specifically, we developed a hybrid framework combining a Long Short-Term Memory (LSTM) network with the Merton-L\'evy jump-diffusion model. To optimise this framework, we employed the Grey Wolf Optimizer (GWO) for the LSTM hyperparameter tuning, and we explored three calibration methods for the Merton-Levy model parameters: Artificial Neural Networks (ANNs), the Marine Predators Algorithm (MPA), and the PyTorch-based TorchSDE library. To evaluate the predictive performance of our hybrid model, we compared it against several benchmark models, including a standard LSTM and an LSTM combined with the Fractional Heston model. This evaluation used three real-world financial datasets: Brent oil prices, the STOXX 600 index, and the IT40 index. Performance was assessed using standard metrics, including Mean Squared Error (MSE), Mean Absolute Error(MAE), Mean Squared Percentage Error (MSPE), and the coefficient of determination (R2). Our experimental results demonstrate that the hybrid model, combining a GWO-optimized LSTM network with the Levy-Merton Jump-Diffusion model calibrated using an ANN, outperformed the base LSTM model and all other models developed in this study.

q-fin.ST↗

arXiv2025

Port-Hamiltonian Neural Networks: From Theory to Simulation of Interconnected Stochastic Systems

This work introduces a new framework integrating port-Hamiltonian systems (PHS) and neural network architectures. This framework bridges the gap between deterministic and stochastic modeling of complex dynamical systems. We introduce new mathematical formulations and computational methods that expand the geometric structure of PHS to account for uncertainty, environmental noise, and random perturbations. Building on these advances, we introduce stochastic port-Hamiltonian neural networks (pHNNs), which facilitate the accurate learning and prediction of non-autonomous and interconnected stochastic systems. Our proposed framework generalizes passivity concepts to the stochastic regime, ensuring stability while maintaining the system's energy-consistent structure. Extensive simulations, including those involving damped mass-spring systems, Duffing oscillators, and robotic control tasks, demonstrate the capability of pHNNs to capture complex dynamics with high fidelity, even under noise and uncertainty. This unified approach establishes a foundation for the robust, data-driven modeling and control of nonlinear stochastic systems.

math-ph↗

arXiv2025

Gibbs measure over the cone of vector-valued discrete measures

We consider a gas whose each particle is characterised by a pair $(x,v_x)$ with the position $x\in \mathbb R^d$ and the velocity $v_x\in \mathbb R^d_0= \mathbb R^d\setminus \{0\}$. We define Gibbs measures on the cone of vector-valued measures and aim to prove their existence. We introduce the family of probability measures $\mu_\lambda$ on the cone $\mathbb K(\mathbb R^d)$. We define local Hamiltonian and partition functions for a positive, symmetric, bounded and measurable pair potential. Using those above, we define Gibbs's measure as a solution to the Dobrushin-Lanford-Ruelle equation. In particular, we focus on the subset of tempered Gibbs measures. To prove the existence of the Gibbs measure, we show that the subset of tempered Gibbs measures is non-empty and relatively compact.

math.PR↗

arXiv2025

Nonlocal Stochastic Optimal Control for Diffusion Processes: Existence, Maximum Principle and Financial Applications

This paper investigates the optimal control problem for a class of parabolic equations where the diffusion coefficient is influenced by a control function acting nonlocally. Specifically, we consider the optimization of a cost functional that incorporates a controlled probability density evolving under a Fokker-Planck equation with state-dependent drift and diffusion terms. The control variable is subject to spatial convolution through a kernel, inducing nonlocal interactions in both drift and diffusion terms. We establish the existence of optimal controls under appropriate convexity and regularity conditions, leveraging compactness arguments in function spaces. A maximum principle is derived to characterize the optimal control explicitly, revealing its dependence on the adjoint state and the nonlocal structure of the system. We further provide a rigorous financial application in the context of mean-variance portfolio optimization, where both the asset drift and volatility are controlled nonlocally, leading to an integral representation of the optimal investment strategy. The results offer a mathematically rigorous framework for optimizing diffusion-driven systems with spatially distributed control effects, broadening the applicability of nonlocal control methods to stochastic optimization and financial engineering.

math.OC↗

arXiv2024

Reinforcement Learning for Bidding Strategy Optimization in Day-Ahead Energy Market

In a day-ahead market, energy buyers and sellers submit their bids for a particular future time, including the amount of energy they wish to buy or sell and the price they are prepared to pay or receive. However, the dynamic for forming the Market Clearing Price (MCP) dictated by the bidding mechanism is frequently overlooked in the literature on energy market modelling. Forecasting models usually focus on predicting the MCP rather than trying to build the optimal supply and demand curves for a given price scenario. Following this approach, the article focuses on developing a bidding strategy for a seller in a continuous action space through a single agent Reinforcement Learning algorithm, specifically the Deep Deterministic Policy Gradient. The algorithm controls the offering curve (action) based on past data (state) to optimize future payoffs (rewards). The participant can access historical data on production costs, capacity, and prices for various sources, including renewable and fossil fuels. The participant gains the ability to operate in the market with greater efficiency over time to maximize individual payout.

math.OC↗

arXiv2024

Enhancing Multi-Step Brent Oil Price Forecasting with Ensemble Multi-Scenario Bi-GRU Networks

Despite numerous research efforts in applying deep learning to time series forecasting, achieving high accuracy in multi-step predictions for volatile time series like crude oil prices remains a significant challenge. Moreover, most existing approaches primarily focus on one-step forecasting, and the performance often varies depending on the dataset and specific case study. In this paper, we introduce an ensemble model to capture Brent oil price volatility and enhance the multi-step prediction. Our methodology employs a two-pronged approach. First, we assess popular deep-learning models and the impact of various external factors on forecasting accuracy. Then, we introduce an ensemble multi-step forecasting model for Brent oil prices. Our approach generates accurate forecasts by employing ensemble techniques across multiple forecasting scenarios using three BI-GRU networks.Extensive experiments were conducted on a dataset encompassing the COVID-19 pandemic period, which had a significant impact on energy markets. The proposed model's performance was evaluated using the standard evaluation metrics of MAE, MSE, and RMSE. The results demonstrate that the proposed model outperforms benchmark and established models.

cs.LG↗

arXiv2024

Multistep Brent Oil Price Forecasting with a Multi-Aspect Meta-heuristic Optimization and Ensemble Deep Learning Model

Accurate crude oil price forecasting is crucial for various economic activities, including energy trading, risk management, and investment planning. Although deep learning models have emerged as powerful tools for crude oil price forecasting, achieving accurate forecasts remains challenging. Deep learning models' performance is heavily influenced by hyperparameters tuning, and they are expected to perform differently under various circumstances. Furthermore, price volatility is also sensitive to external factors such as world events. To address these limitations, we propose a hybrid approach that integrates metaheuristic optimization with an ensemble of five widely used neural network architectures for time series forecasting. Unlike existing methods that apply metaheuristics to optimise hyperparameters within the neural network architecture, we exploit the GWO metaheuristic optimiser at four levels: feature selection, data preparation, model training, and forecast blending. The proposed approach has been evaluated for forecasting three-ahead days using real-world Brent crude oil price data, and the obtained results demonstrate that the proposed approach improves the forecasting performance measured using various benchmarks, achieving 0.000127 of MSE.

cs.LG↗

arXiv2024

Integrating Port-Hamiltonian Systems with Neural Networks: From Deterministic to Stochastic Frameworks

This article presents an innovative approach to integrating port-Hamiltonian systems with neural network architectures, transitioning from deterministic to stochastic models. The study presents novel mathematical formulations and computational models that extend the understanding of dynamical systems under uncertainty and complex interactions. It emphasizes the significant progress in learning and predicting the dynamics of non-autonomous systems using port-Hamiltonian neural networks (pHNNs). It also explores the implications of stochastic neural networks in various dynamical systems.

math.DS↗

arXiv2023

Explosion and non-explosion for the continuous-time frog model

We consider the continuous-time frog model on $\mathbb{Z}$. At time $t = 0$, there are $η(x)$ particles at $x\in \mathbb{Z}$, each of which is represented by a random variable. In particular, $(η(x))_{x \in \mathbb{Z} }$ is a collection of independent random variables with a common distribution $μ$, $μ(\mathbb{Z}_+) = 1$. The particles at the origin are active, all other ones being assumed as dormant, or sleeping. Active particles perform a simple symmetric continuous-time random walk in $\mathbb{Z} $ (that is, a random walk with $\exp(1)$-distributed jump times and jumps $-1$ and $1$, each with probability $1/2$), independently of all other particles. Sleeping particles stay still until the first arrival of an active particle to their location; upon arrival they become active and start their own simple random walks. Different sets of conditions are given ensuring explosion, respectively non-explosion, of the continuous-time frog model. Our results show in particular that if $μ$ is the distribution of $e^{Y \ln Y}$ with a non-negative random variable $Y$ satisfying $\mathbb{E} Y < \infty$, then a.s. no explosion occurs. On the other hand, if $a \in (0,1)$ and $μ$ is the distribution of $e^X$, where $\mathbb{P} \{X \geq t \} = t^{-a}$, $t \geq 1$, then explosion occurs a.s. The proof relies on a certain type of comparison to a percolation model which we call totally asymmetric discrete inhomogeneous Boolean percolation.

math.PR↗

arXiv2023

Interacting particle systems with continuous spins

We study a general class of interacting particle systems over a countable state space $V$ where on each site $x \in V$ the particle mass $η(x) \geq 0$ follows a stochastic differential equation. We construct the corresponding Markovian dynamics in terms of strong solutions to an infinite coupled system of stochastic differential equations and prove a comparison principle with respect to the initial configuration as well as the drift of the process. Using this comparison principle, we provide sufficient conditions for the existence and uniqueness of an invariant measure in the subcritical regime and prove convergence of the transition probabilities in the Wasserstein-1-distance. Finally, for sublinear drifts, we establish a linear growth theorem showing that the spatial spread is at most linear in time. Our results cover a large class of finite and infinite branching particle systems with interactions among different sites.

math.PR↗

arXiv2023

Minimal controllability time for systems with nonlinear drift under a compact convex state constraint

In this paper we estimate the minimal controllability time for a class of non-linear control systems with a bounded convex state constraint. An explicit expression is given for the controllability time if the image of the control matrix is of co-dimension one. A lower bound for the controllability time is given in the general case. The technique is based on finding a lower dimension system with the similar controllability properties as the original system. The controls corresponding to the minimal time, or time close to the minimal one, are discussed and computed analytically. The effectiveness of the proposed approach is illustrated by a few examples.

math.OC↗