arXiv subjects

Athena Picarelli

Publications and source records attributed to Athena Picarelli.

17 recordsLinked to original sources

arXiv2026

An Optimal Energy Production Problem with Energy Source Switching and Load Following Nuclear Power Plants

The integration of weather-dependent renewable generation increases the volatility of residual demand and raises the value of dispatchable low-carbon flexibility. This paper studies the optimal operation of a load-following nuclear power plant owned by a producer that must balance stochastic residual demand while accounting for ramping limits and costly changes in operating regimes. Nuclear output can be increased, decreased, or kept constant, and the production decision is formulated as a finite-horizon optimal switching problem. We analyze both a closed-economy benchmark, where excess production cannot be sold and shortages require costly back-up generation, and an open-economy setting, where the producer can trade electricity at prices driven by aggregate market residual demand. The value functions are characterized as viscosity solutions of a system of Hamilton-Jacobi-Bellman quasi-variational inequalities, and optimal policies are computed using a monotone semi-Lagrangian scheme. The numerical results show how shortage costs, switching costs, ramping capability, and market access shape optimal nuclear load following. The analysis highlights the economic value of controllable low-carbon capacity in renewable-intensive systems and provides implications for flexibility remuneration, balancing-market design, and interconnection policy.

math.OC↗

arXiv2025

A Semi-Lagrangian scheme for Hamilton-Jacobi-Bellman equations with Dirichlet boundary conditions

We study the numerical approximation of time-dependent, possibly degenerate, second-order Hamilton-Jacobi-Bellman equations in bounded domains with nonhomogeneous Dirichlet boundary conditions. It is well known that convergence towards the exact solution of the equation, considered here in the viscosity sense, holds if the scheme is monotone, consistent, and stable. While standard finite difference schemes are, in general, not monotone, the so-called semi-Lagrangian schemes are monotone by construction. On the other hand, these schemes make use of a wide stencil and, when the equation is set in a bounded domain, this typically causes an overstepping of the boundary and hence the loss of consistency. We propose here a semi-Lagrangian scheme defined on an unstructured mesh, with a suitable treatment at grid points near the boundary to preserve consistency, and show its convergence for problems where the viscosity solution can even be discontinuous. We illustrate the numerical convergence in several tests, including degenerate and first-order equations.

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arXiv2025

Extended mean field control: a global numerical solution via finite-dimensional approximation

We investigate the global numerical approximation of a class of extended mean field control problems (MFC), where the dynamics and costs depend on the joint distribution of the state and the control. We propose a framework to approximate the value function globally over the Wasserstein space, moving beyond the restriction of fixed initial conditions. Our approach exploits the propagation of chaos by approximating the infinite-dimensional MFC problem by an $N$-player cooperative game, together with the usage of finite-dimensional solvers. This method avoids the need to parametrise functions on an infinite-dimensional space, offering a balance between probabilistic rigor and computational efficiency.

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arXiv2025

Convergence of a Deep BSDE solver with jumps

We study the error arising in the numerical approximation of FBSDEs and related PIDEs by means of a deep learning-based method. Our results focus on decoupled FBSDEs with jumps and extend the seminal work of HAn and Long (2020) analyzing the numerical error of the deep BSDE solver proposed in E et al. (2017). We provide a priori and a posteriori error estimates for the finite and infinite activity case.

math.PR↗

arXiv2022

Deep Quadratic Hedging

We propose a novel computational procedure for quadratic hedging in high-dimensional incomplete markets, covering mean-variance hedging and local risk minimization. Starting from the observation that both quadratic approaches can be treated from the point of view of backward stochastic differential equations (BSDEs), we (recursively) apply a deep learning-based BSDE solver to compute the entire optimal hedging strategies paths. This allows us to overcome the curse of dimensionality, extending the scope of applicability of quadratic hedging in high dimension. We test our approach with a classic Heston model and with a multiasset and multifactor generalization thereof, showing that this leads to high levels of accuracy.

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arXiv2022

Deep xVA solver -- A neural network based counterparty credit risk management framework

In this paper, we present a novel computational framework for portfolio-wide risk management problems, where the presence of a potentially large number of risk factors makes traditional numerical techniques ineffective. The new method utilises a coupled system of BSDEs for the valuation adjustments (xVA) and solves these by a recursive application of a neural network based BSDE solver. This not only makes the computation of xVA for high-dimensional problems feasible, but also produces hedge ratios and dynamic risk measures for xVA, and allows simulations of the collateral account.

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arXiv2022

A deep solver for BSDEs with jumps

The aim of this work is to propose an extension of the deep solver by Han, Jentzen, E (2018) to the case of forward backward stochastic differential equations (FBSDEs) with jumps. As in the aforementioned solver, starting from a discretized version of the FBSDE and parametrizing the (high dimensional) control processes by means of a family of artificial neural networks (ANNs), the FBSDE is viewed as a model-based reinforcement learning problem and the ANN parameters are fitted so as to minimize a prescribed loss function. We take into account both finite and infinite jump activity by introducing, in the latter case, an approximation with finitely many jumps of the forward process. We successfully apply our algorithm to option pricing problems in low and high dimension and discuss the applicability in the context of counterparty credit risk.

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arXiv2021

On the set of robust sustainable thresholds for uncertain control systems

In natural resource management, or more generally in the study of sustainability issues, often the objective is to maintain the state of a given system within a desirable configuration, typically established in terms of standards or thresholds. For instance, in fisheries management, the procedure of designing policies may include keeping the spawning stock biomass over a critical threshold and also ensuring minimal catches. Given a controlled dynamical system in discrete-time, representing the evolution of some natural resources under the action of controls and uncertainties, and an initial endowment of the resources, the aim of this paper is to characterize the set of robust sustainable thresholds, that is, the thresholds for which there exists some control path, along with its corresponding state trajectory, satisfying for all possible uncertainty scenarios, prescribed mixed constraints parametrized by such thresholds. This set provides useful information to users and decision-makers, illustrating the trade-offs between constraints and it is strongly related to the robust viability, one of the key concepts in viability theory, discipline that study the consistency between a controlled dynamical system and given constraints. Specifically, we are concerned with characterizing the weak and strong Pareto fronts of the set of robust sustainable thresholds, providing a practical method for computing such objects based on optimal control theory and a level-set approach. A numerical example, relying on renewable resource management, is shown to demonstrate the proposed method.

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arXiv2020

A level-set approach for stochastic optimal control problems under controlled-loss constraints

We study a family of optimal control problems under a set of controlled-loss constraints holding at different deterministic dates. The characterization of the associated value function by a Hamilton-Jacobi-Bellman equation usually calls for additional strong assumptions on the dynamics of the processes involved and the set of constraints. To treat this problem in absence of those assumptions, we first convert it into a state-constrained stochastic target problem and then apply a level-set approach. With this approach, the state constraints can be managed through an exact penalization technique.

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arXiv2020

Probabilistic error analysis for some approximation schemes to optimal control problems

We introduce a class of numerical schemes for optimal control problems based on a novel Markov chain approximation, which uses, in turn, a piecewise constant policy approximation, Euler-Maruyama time stepping, and a Gauss-Hermite approximation of the Gaussian increments. We provide lower error bounds of order arbitrarily close to 1/2 in time and 1/3 in space for Lipschitz viscosity solutions, coupling probabilistic arguments with regularization techniques as introduced by Krylov. The corresponding order of the upper bounds is 1/4 in time and 1/5 in space. For sufficiently regular solutions, the order is 1 in both time and space for both bounds. Finally, we propose techniques for further improving the accuracy of the individual components of the approximation.

math.OC↗

arXiv2020

Duality-based a posteriori error estimates for some approximation schemes for optimal investment problems

We consider a Markov chain approximation scheme for utility maximization problems in continuous time, which uses, in turn, a piecewise constant policy approximation, Euler-Maruyama time stepping, and a Gauss-Hermite approximation of the Gaussian increments. The error estimates previously derived in Picarelli and Reisinger (2019) are asymmetric between lower and upper bounds due to the control approximation and improve on known results in the literature in the lower case only. In the present paper, we use duality results to obtain a posteriori upper error bounds which are empirically of the same order as the lower bounds. The theoretical results are confirmed by our numerical tests.

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arXiv2019

Optimal management of pumped hydroelectric production with state constrained optimal control

We present a novel technique to solve the problem of managing optimally a pumped hydroelectric storage system. This technique relies on representing the system as a stochastic optimal control problem with state constraints, these latter corresponding to the finite volume of the reservoirs. Following the recent level-set approach presented in O. Bokanowski, A. Picarelli, H. Zidani, "State-constrained stochastic optimal control problems via reachability approach", SIAM J. Control and Optim. 54 (5) (2016), we transform the original constrained problem in an auxiliary unconstrained one in augmented state and control spaces, obtained by introducing an exact penalization of the original state constraints. The latter problem is fully treatable by classical dynamic programming arguments.

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arXiv2019

Some regularity and convergence results for parabolic Hamilton-Jacobi-Bellman equations in bounded domains

We study the approximation of parabolic Hamilton-Jacobi-Bellman (HJB) equations in bounded domains with strong Dirichlet boundary conditions. We work under the assumption of the existence of a sufficiently regular barrier function for the problem to obtain well-posedness and regularity of a related switching system and the convergence of its components to the HJB equation. In particular, we show existence of a viscosity solution to the switching system by a novel construction of sub- and supersolutions and application of Perron's method. Error bounds for monotone schemes for the HJB equation are then derived from estimates near the boundary, where the standard regularisation procedure for viscosity solutions is not applicable, and are found to be of the same order as known results for the whole space. We deduce error bounds for some common finite difference and truncated semi-Lagrangian schemes.

math.NA↗

arXiv2019

Improved order 1/4 convergence for piecewise constant policy approximation of stochastic control problems

In N.V. Krylov, Approximating value functions for controlled degenerate diffusion processes by using piece-wise constant policies, Electron. J. Probab., 4(2), 1999, it is proved under standard assumptions that the value functions of controlled diffusion processes can be approximated with order 1/6 error by those with controls which are constant on uniform time intervals. In this note we refine the proof and show that the provable rate can be improved to 1/4, which is optimal in our setting. Moreover, we demonstrate the improvements this implies for error estimates derived by similar techniques for approximation schemes, bringing these in line with the best available results from the PDE literature.

math.PR↗

arXiv2018

Stability and convergence of second order backward differentiation schemes for parabolic Hamilton-Jacobi-Bellman equations

We study a second order BDF (Backward Differentiation Formula) scheme for the numerical approximation of parabolic HJB (Hamilton-Jacobi-Bellman) equations. The scheme under consideration is implicit, non-monotone, and second order accurate in time and space. The lack of monotonicity prevents the use of well-known convergence results for solutions in the viscosity sense. In this work, we establish rigorous stability results in a general nonlinear setting as well as convergence results for some particular cases with additional regularity assumptions. While most results are presented for one-dimensional, linear parabolic and non-linear HJB equations, some results are also extended to multiple dimensions and to Isaacs equations. Numerical tests are included to validate the method.

math.NA↗

arXiv2016

High-order filtered schemes for time-dependent second order HJB equations

In this paper, we present and analyse a class of "filtered" numerical schemes for second order Hamilton-Jacobi-Bellman equations. Our approach follows the ideas introduced in B.D. Froese and A.M. Oberman, Convergent filtered schemes for the Monge-Amp\`ere partial differential equation, SIAM J. Numer. Anal., 51(1):423--444, 2013, and more recently applied by other authors to stationary or time-dependent first order Hamilton-Jacobi equations. For high order approximation schemes (where "high" stands for greater than one), the inevitable loss of monotonicity prevents the use of the classical theoretical results for convergence to viscosity solutions. The work introduces a suitable local modification of these schemes by "filtering" them with a monotone scheme, such that they can be proven convergent and still show an overall high order behaviour for smooth enough solutions. We give theoretical proofs of these claims and illustrate the behaviour with numerical tests from mathematical finance, focussing also on the use of backward difference formulae (BDF) for constructing the high order schemes.

math.NA↗

arXiv2011

A patchy Dynamic Programming scheme for a class of Hamilton-Jacobi-Bellman equations

In this paper we present a new algorithm for the solution of Hamilton-Jacobi-Bellman equations related to optimal control problems. The key idea is to divide the domain of computation into subdomains which are shaped by the optimal dynamics of the underlying control problem. This can result in a rather complex geometrical subdivision, but it has the advantage that every subdomain is invariant with respect to the optimal dynamics, and then the solution can be computed independently in each subdomain. The features of this dynamics-dependent domain decomposition can be exploited to speed up the computation and for an efficient parallelization, since the classical transmission conditions at the boundaries of the subdomains can be avoided. For their properties, the subdomains are patches in the sense introduced by Ancona and Bressan [ESAIM Control Optim. Calc. Var., 4 (1999), pp. 445-471]. Several examples in two and three dimensions illustrate the properties of the new method.

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