arXiv subjects

Emma Hubert

Publications and source records attributed to Emma Hubert.

10 recordsLinked to original sources

arXiv2026

Forcing and duality-corrected contracts for volatility control

In this paper, we revisit the construction of optimal incentives in continuous-time principal-agent problems with drift and volatility control. Originally, a general approach relying on dynamic programming and second-order backward stochastic differential equations (2BSDEs) was developed by Cvitani\'c, Possama\"i, and Touzi (2018) [8] to determine the optimal form of contracts in this setting. More recently, Chiusolo and Hubert (2026) [5] proposed a BSDE-based approach by introducing an alternative `contractible-volatility' problem for the principal. In addition to the proposed new method, this work highlights that the optimality result of [8] actually hinges on an assumption, stated below as Assumption 2.3, which may not hold in general. Motivated by this, we introduce in this paper a more general class of contracts, parametrised by a function $\psi$ subject to conditions that make the contract revealing for the agent and without loss of generality for the principal. We further provide two natural specifications of $\psi$: one, inspired by the BSDE approach, yielding a forcing-type contract; the other, motivated by the 2BSDE approach, correcting the duality gap when Assumption 2.3 is not satisfied.

math.OC↗

arXiv2026

Trading Electrons: Predicting DART Spread Spikes in ISO Electricity Markets

We study the problem of forecasting and optimally trading day-ahead versus real-time (DART) price spreads in U.S. wholesale electricity markets. Building on the framework of Galarneau-Vincent et al., we extend spike prediction from a single zone to a multi-zone setting and treat both positive and negative DART spikes within a unified statistical model. To translate directional signals into economically meaningful positions, we develop a structural and market-consistent price impact model based on day-ahead bid stacks. This yields closed-form expressions for the optimal vector of zonal INC/DEC quantities, capturing asymmetric buy/sell impacts and cross-zone congestion effects. When applied to NYISO, the resulting impact-aware strategy significantly improves the risk-return profile relative to unit-size trading and highlights substantial heterogeneity across markets and seasons.

q-fin.TR↗

arXiv2025

A Mean Field Game for Capacity Expansion Modeling

This paper studies the optimal investment behavior of renewable electricity producers in a competitive market, where both prices and installation costs are influenced by aggregate industry activity. We model the resulting crowding effects using a mean field game framework, capturing the strategic interactions among a continuum of heterogeneous producers. The equilibrium dynamics are characterized via a coupled system of Hamilton-Jacobi-Bellman and Fokker-Planck equations, which describe the value function of a representative producer and the evolution of the distribution of installed capacities over time. We analyze both deterministic and stochastic versions of the model, providing analytical insights in tractable cases and developing numerical methods to approximate the general solution. Simulation results illustrate how aggregate investment responds to changing market conditions, cost structures, and exogenous productivity shocks.

math.OC↗

arXiv2024

A new approach to principal-agent problems with volatility control

The recent work by Cvitani\'c, Possama\"i, and Touzi (2018) [9] presents a general approach for continuous-time principal-agent problems, through dynamic programming and second-order backward stochastic differential equations (BSDEs). In this paper, we provide an alternative formulation of the principal-agent problem, which can be solved simply by relying on the theory of BSDEs. This reformulation is strongly inspired by an important remark in [9], namely that if the principal observes the output process in continuous-time, she can compute its quadratic variation pathwise. While in [9], this information is used in the contract, our reformulation consists in assuming that the principal could directly control this process, in a `first-best' fashion. The resolution approach for this alternative problem actually follows the line of the so-called `Sannikov's trick' in the literature on continuous-time principal-agent problems, as originally introduced by Sannikov (2008) [28]. We then show that the solution to this `first-best' formulation is identical to the solution of the original problem. More precisely, using the contract form introduced in [9] as `penalisation contracts', we highlight that this `first-best' scenario can be achieved even if the principal cannot directly control the quadratic variation. Nevertheless, we do not have to rely on the theory of 2BSDEs to prove that such contracts are optimal, as their optimality is ensured by showing that the `first-best' scenario is achieved. We believe that this more straightforward approach to solve continuous-time principal-agent problems with volatility control will facilitate the dissemination of these problems across many fields, and its extension to even more intricate problems.

math.OC↗

arXiv2024

Closed-loop equilibria for Stackelberg games: a story about stochastic targets

We provide a general approach to reformulating any continuous-time stochastic Stackelberg differential game under closed-loop strategies as a single-level optimisation problem with target constraints. More precisely, we consider a Stackelberg game in which the leader and the follower can both control the drift and the volatility of a stochastic output process, in order to maximise their respective expected utility. The aim is to characterise the Stackelberg equilibrium when the players adopt 'closed-loop strategies', i.e. their decisions are based solely on the historical information of the output process, excluding especially any direct dependence on the underlying driving noise, often unobservable in real-world applications. We first show that, by considering the second-order backward stochastic differential equation associated with the continuation utility of the follower as a controlled state variable for the leader, the latter's unconventional optimisation problem can be reformulated as a more standard stochastic control problem with target constraints. Thereafter, adapting the methodology developed by Soner and Touzi (2002a) or Bouchard, Elie and Imbert (2010), the optimal strategies, as well as the corresponding value of the Stackelberg equilibrium, can be characterised through the solution of a well-specified system of Hamilton- Jacobi-Bellman equations. For a more comprehensive insight, we illustrate our approach through a simple example, facilitating both theoretical and numerical detailed comparisons with the solutions under different information structures studied in the literature.

math.OC↗

arXiv2020

Incentives, lockdown, and testing: from Thucydides's analysis to the COVID-19 pandemic

In this work, we provide a general mathematical formalism to study the optimal control of an epidemic, such as the COVID-19 pandemic, via incentives to lockdown and testing. In particular, we model the interplay between the government and the population as a principal-agent problem with moral hazard, \`a la Cvitani\'c, Possama\"i, and Touzi [27], while an epidemic is spreading according to dynamics given by compartmental stochastic SIS or SIR models, as proposed respectively by Gray, Greenhalgh, Hu, Mao, and Pan [45] and Tornatore, Buccellato, and Vetro [88]. More precisely, to limit the spread of a virus, the population can decrease the transmission rate of the disease by reducing interactions between individuals. However, this effort, which cannot be perfectly monitored by the government, comes at social and monetary cost for the population. To mitigate this cost, and thus encourage the lockdown of the population, the government can put in place an incentive policy, in the form of a tax or subsidy. In addition, the government may also implement a testing policy in order to know more precisely the spread of the epidemic within the country, and to isolate infected individuals. In terms of technical results, we demonstrate the optimal form of the tax, indexed on the proportion of infected individuals, as well as the optimal effort of the population, namely the transmission rate chosen in response to this tax. The government's optimisation problem then boils down to solving an Hamilton-Jacobi-Bellman equation. Numerical results confirm that if a tax policy is implemented, the population is encouraged to significantly reduce its interactions. If the government also adjusts its testing policy, less effort is required on the population side, individuals can interact almost as usual, and the epidemic is largely contained by the targeted isolation of positively-tested individuals.

q-bio.PE↗

arXiv2020

Continuous-time incentives in hierarchies

This paper studies continuous-time optimal contracting in a hierarchy problem which generalises the model of Sung (2015). The hierarchy is modeled by a series of interlinked principal-agent problems, leading to a sequence of Stackelberg equilibria. More precisely, the principal can contract with the managers to incentivise them to act in her best interest, despite only observing the net benefits of the total hierarchy. Managers in turn subcontract with the agents below them. Both agents and managers independently control in continuous time a stochastic process representing their outcome. First, we show through a continuous-time adaptation of Sung's model that, even if the agents only control the drift of their outcome, their manager controls the volatility of their continuation utility. This first simple example justifies the use of recent results on optimal contracting for drift and volatility control, and therefore the theory of second-order backward stochastic differential equations, developed in the theoretical part of this paper, dedicated to a more general model. The comprehensive approach we outline highlights the benefits of considering a continuous-time model and opens the way to obtain comparative statics. We also explain how the model can be extended to a large-scale principal-agent hierarchy. Since the principal's problem can be reduced to only an $m$-dimensional state space and a $2m$-dimensional control set, where $m$ is the number of managers immediately below her, and is therefore independent of the size of the hierarchy below these managers, the dimension of the problem does not explode.

math.OC↗

arXiv2020

Contact rate epidemic control of COVID-19: an equilibrium view

We consider the control of the COVID-19 pandemic through a standard SIR compartmental model. This control is induced by the aggregation of individuals' decisions to limit their social interactions: when the epidemic is ongoing, an individual can diminish his/her contact rate in order to avoid getting infected, but this effort comes at a social cost. If each individual lowers his/her contact rate, the epidemic vanishes faster, but the effort cost may be high. A Mean Field Nash equilibrium at the population level is formed, resulting in a lower effective transmission rate of the virus. We prove theoretically that equilibrium exists and compute it numerically. However, this equilibrium selects a sub-optimal solution in comparison to the societal optimum (a centralized decision respected fully by all individuals), meaning that the cost of anarchy is strictly positive. We provide numerical examples and a sensitivity analysis, as well as an extension to a SEIR compartmental model to account for the relatively long latent phase of the COVID-19 disease. In all the scenarii considered, the divergence between the individual and societal strategies happens both before the peak of the epidemic, due to individuals' fears, and after, when a significant propagation is still underway.

physics.soc-ph↗

arXiv2020

Optimal contracts under adverse selection for staple goods: efficiency of in-kind insurance

An income loss can have a negative impact on households, forcing them to reduce their consumption of some staple goods. This can lead to health issues and, consequently, generate significant costs for society. We suggest that consumers can, to prevent these negative consequences, buy insurance to secure sufficient consumption of a staple good if they lose part of their income. We develop a two-period/two-good principal-agent problem with adverse selection and endogenous reservation utility to model insurance with in-kind benefits. This model allows us to obtain semi-explicit solutions for the insurance contract and is applied to the context of fuel poverty. For this application, our model allows to conclude that, even in the least efficient scenario from the households point of view, i.e., when the insurance is provided by a monopoly, this mechanism decreases significantly the risk of fuel poverty of households by ensuring them a sufficient consumption of energy. The effectiveness of in-kind insurance is highlighted through a comparison with income insurance, but our results nevertheless underline the need to regulate such insurance market.

econ.GN↗

arXiv2019

Mean-field moral hazard for optimal energy demand response management

We study the problem of demand response contracts in electricity markets by quantifying the impact of considering a mean-field of consumers, whose consumption is impacted by a common noise. We formulate the problem as a Principal-Agent problem with moral hazard in which the Principal - she - is an electricity producer who observes continuously the consumption of a continuum of risk-averse consumers, and designs contracts in order to reduce her production costs. More precisely, the producer incentivises the consumers to reduce the average and the volatility of their consumption in different usages, without observing the efforts they make. We prove that the producer can benefit from considering the mean-field of consumers by indexing contracts on the consumption of one Agent and aggregate consumption statistics from the distribution of the entire population of consumers. In the case of linear energy valuation, we provide closed-form expression for this new type of optimal contracts that maximises the utility of the producer. In most cases, we show that this new type of contracts allows the Principal to choose the risks she wants to bear, and to reduce the problem at hand to an uncorrelated one.

math.PR↗