arXiv subjects

Dylan Possamaï

Publications and source records attributed to Dylan Possamaï.

At least 37 records · Page 2Linked to original sources

arXiv2023

Randomness and early termination: what makes a game exciting?

In this paper we revisit an open problem posed by Aldous on the max-entropy win-probability martingale: given two players of equal strength, such that the win-probability is a martingale diffusion, which of these processes has maximum entropy and hence gives the most excitement for the spectators? We study a terminal-boundary value problem for the nonlinear parabolic PDE $2\partial_te(t,x)=\log(-\partial_{xx}e(t,x))$ derived by Aldous and prove its wellposedness and regularity of its solution by combining PDE analysis and probabilistic tools, in particular the reformulation as a stochastic control problem with restricted control set, which allows us to deduce strict ellipticity. We establish key qualitative properties of the solution including concavity, monotonicity, convergence to a steady state for long remaining time and the asymptotic behaviour shortly before the terminal time. Moreover, we construct convergent numerical approximations. The analytical and numerical results allow us to highlight the behaviour of the win-probability process in the present case where the match may end early, in contrast to recent work by Backhoff-Veraguas and Beiglböck where the match always runs the full length.

math.PR↗

arXiv2023

Stability of backward stochastic differential equations: the general case

In this paper, we obtain stability results for backward stochastic differential equations with jumps (BSDEs) in a very general framework. More specifically, we consider a convergent sequence of standard data, each associated to their own filtration, and we prove that the associated sequence of (unique) solutions is also convergent. The current result extends earlier contributions in the literature of stability of BSDEs and unifies several frameworks for numerical approximations of BSDEs and their implementations.

math.PR↗

arXiv2023

Time-inconsistent contract theory

This paper investigates the moral hazard problem in finite horizon with both continuous and lump-sum payments, involving a time-inconsistent sophisticated agent and a standard utility maximiser principal. Building upon the so-called dynamic programming approach in Cvitanić, Possamaï, and Touzi [18] and the recently available results in Hernández and Possamaï [43], we present a methodology that covers the previous contracting problem. Our main contribution consists in a characterisation of the moral hazard problem faced by the principal. In particular, it shows that under relatively mild technical conditions on the data of the problem, the supremum of the principal's expected utility over a smaller restricted family of contracts is equal to the supremum over all feasible contracts. Nevertheless, this characterisation yields, as far as we know, a novel class of control problems that involve the control of a forward Volterra equation via Volterra-type controls, and infinite-dimensional stochastic target constraints. Despite the inherent challenges associated to such a problem, we study the solution under three different specifications of utility functions for both the agent and the principal, and draw qualitative implications from the form of the optimal contract. The general case remains the subject of future research.

econ.TH↗

arXiv2022

Is there a Golden Parachute in Sannikov's principal-agent problem?

This paper provides a complete review of the continuous-time optimal contracting problem introduced by Sannikov, in the extended context allowing for possibly different discount rates for both parties. The agent's problem is to seek for optimal effort, given the compensation scheme proposed by the principal over a random horizon. Then, given the optimal agent's response, the principal determines the best compensation scheme in terms of running payment, retirement, and lump-sum payment at retirement. A Golden Parachute is a situation where the agent ceases any effort at some positive stopping time, and receives a payment afterwards, possibly under the form of a lump sum payment, or of a continuous stream of payments. We show that a Golden Parachute only exists in certain specific circumstances. This is in contrast with the results claimed by Sannikov, where the only requirement is a positive agent's marginal cost of effort at zero. Namely, we show that there is no Golden Parachute if this parameter is too large. Similarly, in the context of a concave marginal utility, there is no Golden Parachute if the agent's utility function has a too negative curvature at zero. In the general case, we prove that an agent with positive reservation utility is either never retired by the principal, or retired above some given threshold (as in Sannikov's solution). We show that different discount factors induce a face-lifted utility function, which allows to reduce the analysis to a setting similar to the equal discount rates one. Finally, we also confirm that an agent with small reservation utility may have an informational rent, meaning that the principal optimally offers him a contract with strictly higher utility than his participation value.

econ.TH↗

arXiv2022

A unified approach to a priori estimates for supersolutions of BSDEs in general filtrations

We provide a unified approach to a priori estimates for supersolutions of BSDEs in general filtrations, which may not be quasi left-continuous. Unlike the previous related approaches in simpler settings, our results do not only rely on a simple application of Itô's formula and classical estimates, but use crucially appropriate generalizations of deep estimates for supermartingales obtained by Meyer. As an example of application, we prove that reflected BSDEs are well-posed in a general framework which has not been covered so far in the existing literature.

math.PR↗

arXiv2022

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, à la Cvitanić, Possamaï, 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↗

arXiv2022

A mean-field game of market-making against strategic traders

We design a market-making model à la Avellaneda-Stoikov in which the market-takers act strategically, in the sense that they design their trading strategy based on an exogenous trading signal. The market-maker chooses her quotes based on the average market-takers' behaviour, modelled through a mean-field interaction. We derive, up to the resolution of a coupled HJB--Fokker--Planck system, the optimal controls of the market-maker and the representative market-taker. This approach is flexible enough to incorporate different behaviours for the market-takers and takes into account the impact of their strategies on the price process.

q-fin.MF↗

arXiv2021

Pollution regulation for electricity generators in a transmission network

In this paper we study a pollution regulation problem in an electricity market with a network structure. The market is ruled by an independent system operator (ISO for short) who has the goal of reducing the pollutant emissions of the providers in the network, by encouraging the use of cleaner technologies. The problem of the ISO formulates as a contracting problem with each one of the providers, who interact among themselves by playing a stochastic differential game. The actions of the providers are not observable by the ISO which faces moral hazard. By using the dynamic programming approach, we represent the value function of the ISO as the unique viscosity solution to the corresponding Hamilton-Jacobi-Bellman equation. We prove that this solution is smooth and characterise the optimal controls for the ISO. Numerical solutions to the problem are presented and discussed. We consider also a simpler problem for the ISO, with constant production levels, that can be solved explicitly in a particular setting.

math.OC↗

arXiv2021

Me, myself and I: a general theory of non-Markovian time-inconsistent stochastic control for sophisticated agents

We develop a theory for continuous-time non-Markovian stochastic control problems which are inherently time-inconsistent. Their distinguishing feature is that the classical Bellman optimality principle no longer holds. Our formulation is cast within the framework of a controlled non-Markovian forward stochastic differential equation, and a general objective functional setting. We adopt a game-theoretic approach to study such problems, meaning that we seek for sub-game perfect Nash equilibrium points. As a first novelty of this work, we introduce and motivate a refinement of the definition of equilibrium that allows us to establish a direct and rigorous proof of an extended dynamic programming principle, in the same spirit as in the classical theory. This in turn allows us to introduce a system consisting of an infinite family of backward stochastic differential equations analogous to the classical HJB equation. We prove that this system is fundamental, in the sense that its well-posedness is both necessary and sufficient to characterise the value function and equilibria. As a final step we provide an existence and uniqueness result. Some examples and extensions of our results are also presented.

math.OC↗

arXiv2021

Optimal make take fees in a multi market maker environment

Following the recent literature on make take fees policies, we consider an exchange wishing to set a suitable contract with several market makers in order to improve trading quality on its platform. To do so, we use a principal-agent approach, where the agents (the market makers) optimise their quotes in a Nash equilibrium fashion, providing best response to the contract proposed by the principal (the exchange). This contract aims at attracting liquidity on the platform. This is because the wealth of the exchange depends on the arrival of market orders, which is driven by the spread of market makers. We compute the optimal contract in quasi explicit form and also derive the optimal spread policies for the market makers. Several new phenomena appears in this multi market maker setting. In particular we show that it is not necessarily optimal to have a large number of market makers in the presence of a contracting scheme.

q-fin.TR↗

arXiv2021

Governmental incentives for green bonds investment

Motivated by the recent studies on the green bond market, we build a model in which an investor trades on a portfolio of green and conventional bonds, both issued by the same governmental entity. The government provides incentives to the bondholder in order to increase the amount invested in green bonds. These incentives are, optimally, indexed on the prices of the bonds, their quadratic variation and covariation. We show numerically on a set of French governmental bonds that our methodology outperforms the current tax-incentives systems in terms of green investments. Moreover, it is robust to model specification for bond prices and can be applied to a large portfolio of bonds using classical optimisation methods.

q-fin.MF↗

arXiv2020

Equilibrium Asset Pricing with Transaction Costs

We study risk-sharing economies where heterogenous agents trade subject to quadratic transaction costs. The corresponding equilibrium asset prices and trading strategies are characterised by a system of nonlinear, fully-coupled forward-backward stochastic differential equations. We show that a unique solution generally exists provided that the agents' preferences are sufficiently similar. In a benchmark specification with linear state dynamics, the illiquidity discounts and liquidity premia observed empirically correspond to a positive relationship between transaction costs and volatility.

q-fin.PM↗

arXiv2020

Corrigendum for "Second-order reflected backward stochastic differential equations" and "Second-order BSDEs with general reflection and game options under uncertainty"

The aim of this short note is to fill in a gap in our earlier paper [16] on 2BSDEs with reflections, and to explain how to correct the subsequent results in the second paper [15]. We also provide more insight on the properties of 2RBSDEs, in the light of the recent contributions [13, 23] in the so--called $G-$framework.

math.PR↗

arXiv2020

A unified approach to well-posedness of type-I backward stochastic Volterra integral equations

We study a novel general class of multidimensional type-I backward stochastic Volterra integral equations. Toward this goal, we introduce an infinite dimensional system of standard backward SDEs and establish its well-posedness, and we show that it is equivalent to that of a type-I backward stochastic Volterra integral equation. We also establish a representation formula in terms of non-linear semilinear partial differential equation of Hamilton-Jacobi-Bellman type. As an application, we consider the study of time-inconsistent stochastic control from a game-theoretic point of view. We show the equivalence of two current approaches to this problem from both a probabilistic and an analytic point of view.

math.PR↗

arXiv2020

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↗

arXiv2020

McKean-Vlasov optimal control: the dynamic programming principle

We study the McKean-Vlasov optimal control problem with common noise in various formulations, namely the strong and weak formulation, as well as the Markovian and non-Markovian formulations, and allowing for the law of the control process to appear in the state dynamics. By interpreting the controls as probability measures on an appropriate canonical space with two filtrations, we then develop the classical measurable selection, conditioning and concatenation arguments in this new context, and establish the dynamic programming principle under general conditions.

math.OC↗

arXiv2020

McKean-Vlasov optimal control: limit theory and equivalence between different formulations

We study a McKean-Vlasov optimal control problem with common noise, in order to establish the corresponding limit theory, as well as the equivalence between different formulations, including the strong, weak and relaxed formulation. In contrast to the strong formulation, where the problem is formulated on a fixed probability space equipped with two Brownian filtrations, the weak formulation is obtained by considering a more general probability space with two filtrations satisfying an $(H)$-hypothesis type condition from the theory of enlargement of filtrations. When the common noise is uncontrolled, our relaxed formulation is obtained by considering a suitable controlled martingale problem. As for classical optimal control problems, we prove that the set of all relaxed controls is the closure of the set of all strong controls, when considered as probability measures on the canonical space. Consequently, we obtain the equivalence of the different formulations of the control problem, under additional mild regularity conditions on the reward functions. This is also a crucial technical step to prove the limit theory of the McKean-Vlasov control problem, that is to say proving that it consists in the limit of a large population control problem with common noise.

math.OC↗

arXiv2019

An adverse selection approach to power pricing

We study the optimal design of electricity contracts among a population of consumers with different needs. This question is tackled within the framework of Principal-Agent problems in presence of adverse selection. The particular features of electricity induce an unusual structure on the production cost, with no decreasing return to scale. We are nevertheless able to provide an explicit solution for the problem at hand. The optimal contracts are either linear or polynomial with respect to the consumption. Whenever the outside options offered by competitors are not uniform among the different type of consumers, we exhibit situations where the electricity provider should contract with consumers with either low or high appetite for electricity.

math.OC↗