arXiv subjects

Giorgio Ferrari

Publications and source records attributed to Giorgio Ferrari.

At least 37 records · Page 2Linked to original sources

arXiv2025

Entropy Regularization in Mean-Field Games of Optimal Stopping

We study mean-field games of optimal stopping (OS-MFGs) and introduce an entropy-regularized framework to enable learning-based solution methods. By utilizing randomized stopping times, we reformulate the OS-MFG as a mean-field game of singular stochastic controls (SC-MFG) with entropy regularization. We establish the existence of equilibria and prove their stability as the entropy parameter vanishes. Fictitious play algorithms tailored for the regularized setting are introduced, and we show their convergence under both Lasry-Lions monotonicity and supermodular assumptions on the reward functional. Our work lays the theoretical foundation for model-free learning approaches to OS-MFGs.

math.OC↗

arXiv2025

Singular Control in a Cash Management Model with Ambiguity

We consider a singular control model of cash reserve management, driven by a diffusion under ambiguity. The manager is assumed to have maxmin preferences over a set of priors characterized by $κ$-ignorance. A verification theorem is established to determine the firm's cost function and the optimal cash policy; the latter taking the form of a control barrier policy. In a model driven by arithmetic Brownian motion, we use Dynkin games to show that an increase in ambiguity leads to higher expected costs under the worst-case prior and a narrower inaction region. The latter effect can be used to provide an ambiguity-driven explanation for observed cash management behavior. Our findings can be applied to broader applications of singular control in managing inventories under ambiguity.

q-fin.RM↗

arXiv2025

Existence of Strong Randomized Equilibria in Mean-Field Games of Optimal Stopping with Common Noise

We study a mean-field game of optimal stopping and investigate the existence of strong solutions via a connection with the Bank-El Karoui's representation problem. Under certain continuity assumptions, where the common noise is generated by a countable partition, we show that a strong randomized mean-field equilibrium exists, in which the mean-field interaction term is adapted to the common noise and the stopping time is randomized. Furthermore, under suitable monotonicity assumptions and for a general common noise, we provide a comparative statics analysis of the set of strong mean-field equilibria with strict equilibrium stopping times.

math.OC↗

arXiv2025

Striking the Balance: Life Insurance Timing and Asset Allocation in Financial Planning

This paper investigates the consumption and investment decisions of an individual facing uncertain lifespan and stochastic labor income within a Black-Scholes market framework. A key aspect of our study involves the agent's option to choose when to acquire life insurance for bequest purposes. We examine two scenarios: one with a fixed bequest amount and another with a controlled bequest amount. Applying duality theory and addressing free-boundary problems, we analytically solve both cases, and provide explicit expressions for value functions and optimal strategies in both cases. In the first scenario, where the bequest amount is fixed, distinct outcomes emerge based on different levels of risk aversion parameter $γ$: (i) the optimal time for life insurance purchase occurs when the agent's wealth surpasses a critical threshold if $γ\in (0,1)$, or (ii) life insurance should be acquired immediately if $γ>1$. In contrast, in the second scenario with a controlled bequest amount, regardless of $γ$ values, immediate life insurance purchase proves to be optimal. Finally, we extend the analysis to consider a scenario in which the individual earmarks part of her initial wealth for inheritance, where a critical wealth threshold consistently emerges.

q-fin.PM↗

arXiv2025

Cooperation, Correlation and Competition in Ergodic N-player Games and Mean-field Games of Singular Controls: A Case Study

We consider a class of $N$-player games and mean-field games of singular controls with ergodic performance criterion, providing a benchmark case for irreversible investment games featuring mean-field interaction and strategic complementarities. The state of each player follows a geometric Brownian motion, controlled additively through a nondecreasing process, while agents seek to maximize a long-term average reward functional with a power-type instantaneous profit, under strategic complementarity. We explore three different notions of optimality, which, in the mean-field limit, correspond to the mean-field control solution, mean-field coarse correlated equilibria, and mean-field Nash equilibria. We explicitly compute equilibria in the three cases and compare them numerically, in terms of yielded payoffs and existence conditions. Finally, we show that the mean-field control and mean-field equilibria can approximate the cooperative and competitive equilibria, respectively, in the corresponding $N$-player game when $N$ is sufficiently large. Our analysis of the mean-field control problem features a novel Lagrange multiplier approach, which proves crucial in establishing the approximation result, while the treatment of mean-field coarse correlated equilibria necessitates a new, specifically tailored definition for the stationary setting.

math.OC↗

arXiv2025

Integrated electronic controller for dynamic self-configuration of photonic circuits

Reconfigurable photonic integrated circuits (PICs) can implement arbitrary operations and signal processing functionalities directly in the optical domain. Run-time configuration of these circuits requires an electronic control layer to adjust the working point of their building elements and make them compensate for thermal drifts or degradations of the input signal. As the advancement of photonic foundries enables the fabrication of chips of increasing complexity, developing scalable electronic controllers becomes crucial for the operation of complex PICs. In this paper, we present an electronic application-specific integrated circuit (ASIC) designed for reconfiguration of PICs featuring numerous tuneable elements. Each channel of the ASIC controller independently addresses one optical component of the PIC, as multiple parallel local feedback loops are operated to achieve full control. The proposed design is validated through real-time reconfiguration of a 16-channel silicon photonics adaptive beam coupler. Results demonstrate automatic coupling of an arbitrary input beam to a single-mode waveguide, dynamic compensation of beam wavefront distortions and successful transmission of a 50 Gbit/s signal through an optical free-space link. The low power consumption and compactness of the electronic chip provide a scalable paradigm that can be seamlessly extended to larger photonic architectures.

physics.optics↗

arXiv2024

Ergodic Mean-Field Games of Singular Control with Regime-Switching (Extended Version)

This paper studies a class of stationary mean-field games of singular stochastic control with regime-switching. The representative agent adjusts the dynamics of a Markov-modulated Itô-diffusion via a two-sided singular stochastic control and faces a long-time-average expected profit criterion. The mean-field interaction is of scalar type and it is given through the stationary distribution of the population. Via a constructive approach, we prove the existence and uniqueness of the stationary mean-field equilibrium. Furthermore, we show that this realizes a symmetric $\varepsilon_N$-Nash equilibrium for a suitable ergodic $N$-player game with singular controls. The proof hinges on the characterization of the optimal solution to the representative player's ergodic singular stochastic control problem with regime switching in terms of an auxiliary Dynkin game, which is of independent interest and appears here for the first time.

math.OC↗

arXiv2024

Numerical approximation of Dynkin games with asymmetric information

We propose an implementable, neural network-based structure preserving probabilistic numerical approximation for a generalized obstacle problem describing the value of a zero-sum differential game of optimal stopping with asymmetric information. The target solution depends on three variables: the time, the spatial (or state) variable, and a variable from a standard $(I-1)$-simplex which represents the probabilities with which the $I$ possible configurations of the game are played. The proposed numerical approximation preserves the convexity of the continuous solution as well as the lower and upper obstacle bounds. We show convergence of the fully-discrete scheme to the unique viscosity solution of the continuous problem and present a range of numerical studies to demonstrate its applicability.

math.NA↗

arXiv2024

Uncertainty over Uncertainty in Environmental Policy Adoption: Bayesian Learning of Unpredictable Socioeconomic Costs

The socioeconomic impact of pollution naturally comes with uncertainty due to, e.g., current new technological developments in emissions' abatement or demographic changes. On top of that, the trend of the future costs of the environmental damage is unknown: Will global warming dominate or technological advancements prevail? The truth is that we do not know which scenario will be realised and the scientific debate is still open. This paper captures those two layers of uncertainty by developing a real-options-like model in which a decision maker aims at adopting a once-and-for-all costly reduction in the current emissions rate, when the stochastic dynamics of the socioeconomic costs of pollution are subject to Brownian shocks and the drift is an unobservable random variable. By keeping track of the actual evolution of the costs, the decision maker is able to learn the unknown drift and to form a posterior dynamic belief of its true value. The resulting decision maker's timing problem boils down to a truly two-dimensional optimal stopping problem which we address via probabilistic free-boundary methods and a state-space transformation. We completely characterise the solution by showing that the optimal timing for implementing the emissions reduction policy is the first time that the learning process has become ``decisive'' enough; that is, when it exceeds a time-dependent percentage. This is given in terms of an endogenously determined threshold function, which solves uniquely a nonlinear integral equation. We numerically illustrate our results, discuss the implications of the optimal policy and also perform comparative statics to understand the role of the relevant model's parameters in the optimal policy.

math.OC↗

arXiv2024

A Stationary Equilibrium Model of Green Technology Adoption with Endogenous Carbon Price

This paper proposes and analyzes a stationary equilibrium model for a competitive industry which endogenously determines the carbon price necessary to achieve a given emission target. In the model, firms are identified by their level of technology and make production, entry, and abatement decisions. Polluting firms are subject to a carbon price and abatement is formulated as an irreversible investment, which entails a sunk cost and results in the firms switching to a carbon neutral technology. In equilibrium, we identify a carbon price and a stationary distribution of incumbent, polluting firms, that guarantee the compliance with a certain emission target. Our general theoretical framework is complemented with a case study with Brownian technology shocks, in which we discuss some implications of our model. We observe that a carbon pricing system alongside installation subsidies and tax benefits for green firms trigger earlier investment, while higher income taxes for polluting firms may be distorting. Moreover, we discuss the role of a welfare maximizing regulator, who, by optimally setting the emission target, may mitigate or revert some parameters' effects observed in the model with fixed limit.

q-fin.MF↗

arXiv2024

Multiple equilibria in mean-field game models for large oligopolies with strategic complementarities

We consider continuous-time mean-field stochastic games with strategic complementarities. The interaction between the representative productive firm and the population of rivals comes through the price at which the produced good is sold and the intensity of interaction is measured by a so-called "strenght parameter" $ξ$. Via lattice-theoretic arguments we first prove existence of equilibria and provide comparative statics results when varying $ξ$. A careful numerical study based on iterative schemes converging to suitable maximal and minimal equilibria allows then to study in relevant financial examples how the emergence of multiple equilibria is related to the strenght of the strategic interaction.

math.OC↗

arXiv2024

Irreversible reinsurance: Minimization of Capital Injections in Presence of a Fixed Cost

We propose a model in which, in exchange to the payment of a fixed transaction cost, an insurance company can choose the retention level as well as the time at which subscribing a perpetual reinsurance contract. The surplus process of the insurance company evolves according to the diffusive approximation of the Cramér-Lundberg model, claims arrive at a fixed constant rate, and the distribution of their sizes is general. Furthermore, we do not specify any specific functional form of the retention level. The aim of the company is to take actions in order to minimize the sum of the expected value of the total discounted flow of capital injections needed to avoid bankruptcy and of the fixed activation cost of the reinsurance contract. We provide an explicit solution to this problem, which involves the resolution of a static nonlinear optimization problem and of an optimal stopping problem for a reflected diffusion. We then illustrate the theoretical results in the case of proportional and excess-of-loss reinsurance, by providing a numerical study of the dependency of the optimal solution with respect to the model's parameters.

math.OC↗

arXiv2023

Multidimensional singular control and related Skorokhod problem: sufficient conditions for the characterization of optimal controls

We characterize the optimal control for a class of singular stochastic control problems as the unique solution to a related Skorokhod reflection problem. The considered optimization problems concern the minimization of a discounted cost functional over an infinite time-horizon through a process of bounded variation affecting an Itô-diffusion. The setting is multidimensional, the drift of the state equation and the costs are convex, the volatility matrix can be constant or linear in the state. We prove that the optimal control acts only when the underlying diffusion attempts to exit the so-called waiting region, and that the direction of this action is prescribed by the derivative of the value function. Our approach is based on the study of a suitable monotonicity property of the derivative of the value function through its interpretation as the value of an optimal stopping game. Such a monotonicity allows to construct nearly optimal policies which reflect the underlying diffusion at the boundary of approximating waiting regions. The limit of this approximation scheme then provides the desired characterization. Our result applies to a relevant class of linear-quadratic models, among others. Furthermore, it allows to construct the optimal control in degenerate and non degenerate settings considered in the literature, where this important aspect was only partially addressed.

math.OC↗

arXiv2023

A Stationary Mean-Field Equilibrium Model of Irreversible Investment in a Two-Regime Economy

We consider a mean-field model of firms competing à la Cournot on a commodity market, where the commodity price is given in terms of a power inverse demand function of the industry-aggregate production. Investment is irreversible and production capacity depreciates at a constant rate. Production is subject to Gaussian productivity shocks, while large non-anticipated macroeconomic events driven by a two-state continuous-time Markov chain can change the volatility of the shocks, as well as the price function. Firms wish to maximize expected discounted revenues of production, net of investment and operational costs. Investment decisions are based on the long-run stationary price of the commodity. We prove existence, uniqueness and characterization of the stationary mean-field equilibrium of the model. The equilibrium investment strategy is of barrier-type and it is triggered by a couple of endogenously determined investment thresholds, one per state of the economy. We provide a quasi-closed form expression of the stationary density of the state and we show that our model can produce Pareto distribution of firms' size. This is a feature that is consistent both with observations at the aggregate level of industries and at the level of a particular industry. We establish a relation between economic instability and market concentration and we show how macroeconomic instability can harm firms' profitability more than productivity fluctuations.

math.OC↗

arXiv2022

Two-sided Singular Control of an Inventory with Unknown Demand Trend (Extended Version)

We study the problem of optimally managing an inventory with unknown demand trend. Our formulation leads to a stochastic control problem under partial observation, in which a Brownian motion with non-observable drift can be singularly controlled in both an upward and downward direction. We first derive the equivalent separated problem under full information, with state-space components given by the Brownian motion and the filtering estimate of its unknown drift, and we then completely solve this latter problem. Our approach uses the transition amongst three different but equivalent problem formulations, links between two-dimensional bounded-variation stochastic control problems and games of optimal stopping, and probabilistic methods in combination with refined viscosity theory arguments. We show substantial regularity of (a transformed version of) the value function, we construct an optimal control rule, and we show that the free boundaries delineating (transformed) action and inaction regions are bounded globally Lipschitz continuous functions. To our knowledge this is the first time that such a problem has been solved in the literature.

math.OC↗

arXiv2022

Optimal Execution with Multiplicative Price Impact and Incomplete Information on the Return

We study an optimal liquidation problem with multiplicative price impact in which the trend of the asset's price is an unobservable Bernoulli random variable. The investor aims at selling over an infinite time-horizon a fixed amount of assets in order to maximize a net expected profit functional, and lump-sum as well as singularly continuous actions are allowed. Our mathematical modelling leads to a singular stochastic control problem featuring a finite-fuel constraint and partial observation. We provide the complete analysis of an equivalent three-dimensional degenerate problem under full information, whose state process is composed of the asset's price dynamics, the amount of available assets in the portfolio, and the investor's belief about the true value of the asset's trend. The optimal execution rule and the problem's value function are expressed in terms of the solution to a truly two-dimensional optimal stopping problem, whose associated belief-dependent free boundary $b$ triggers the investor's optimal selling rule. The curve $b$ is uniquely determined through a nonlinear integral equation, for which we derive a numerical solution through an application of the Monte-Carlo method. This allows us to understand the sensitivity of the problem's solution with respect to the relevant model's parameters as well as the value of information in our model.

q-fin.MF↗

arXiv2022

Optimal Dividends under Markov-Modulated Bankruptcy Level

This paper proposes and studies an optimal dividend problem in which a two-state regime-switching environment affects the dynamics of the company's cash surplus and, as a novel feature, also the bankruptcy level. The aim is to maximize the total expected profits from dividends until bankruptcy. The company's optimal dividend payout is therefore influenced by four factors simultaneously: Brownian fluctuations in the cash surplus, as well as regime changes in drift, volatility and bankruptcy levels. In particular, the average profitability can assume different signs in the two regimes. We find a rich structure of the optimal strategy, which, depending on the interaction of the model's parameters, can be either of barrier-type or of liquidation-barrier type. Furthermore, we provide explicit expressions of the optimal policies and value functions. Finally, we complement our theoretical results by a detailed numerical study, where also a thorough analysis of the sensitivities of the optimal dividend policy with respect to the problem's parameters is performed.

math.OC↗

arXiv2022

Optimal Vaccination in a SIRS Epidemic Model

We propose and solve an optimal vaccination problem within a deterministic compartmental model of SIRS type: the immunized population can become susceptible again, e.g.\ because of a not complete immunization power of the vaccine. A social planner thus aims at reducing the number of susceptible individuals via a vaccination campaign, while minimizing the social and economic costs related to the infectious disease. As a theoretical contribution, we provide a technical non-smooth verification theorem, guaranteeing that a semiconcave viscosity solution to the Hamilton-Jacobi-Bellman equation identifies with the minimal cost function, provided that the closed-loop equation admits a solution. Conditions under which the closed-loop equation is well-posed are then derived by borrowing results from the theory of \emph{Regular Lagrangian Flows}. From the applied point of view, we provide a numerical implementation of the model in a case study with quadratic instantaneous costs. Amongst other conclusions, we observe that in the long-run the optimal vaccination policy is able to keep the percentage of infected to zero, at least when the natural reproduction number and the reinfection rate are small.

math.OC↗