arXiv subjects

Stéphane Villeneuve

Publications and source records attributed to Stéphane Villeneuve.

15 recordsLinked to original sources

arXiv2026

A continuous-time dynamic contracting problem with limited liability and finite horizon

We perform a detailed study of a principal--agent problem in a continuous time version of the celebrated Holmström--Milgrom model (Econometrica 55 (2), 1987) where we add limited liability for the Agent. We develop a probabilistic methodology to prove that the Principal's value function is the unique bounded classical solution to a fully nonlinear and fully degenerate partial differential equation (PDE) with Cauchy-Dirichlet boundary conditions on $[0,T]\times[0,\infty)$. Indeed, we also prove infinite continuous differentiability of the solution in the interior of the domain. The strength of our regularity result is such that we can ensure existence of optimal controls in strong form---a rare occurrence in dynamic contracting---and we obtain fine properties of the optimal control map, including a characterisation via a further nonlinear degenerate PDE.

math.OC↗

arXiv2026

Forward Hedging Reshapes Incentive Provision

We study how forward hedging reshapes incentive provision inside the firm. We consider a risk-averse producer facing demand and production risk that can either operate in-house or delegate production to a risk-averse agent under moral hazard, while hedging output in a competitive forward market with a rational market maker. Within a tractable continuous-time CARA framework, we jointly characterize optimal production, compensation, and static hedging in equilibrium. Delegation and external hedging are partial substitutes because both create value through risk sharing. Delegation can increase firm value even when the agent uses the same technology and is more risk averse than the principal, while access to forward hedging reduces the need to provide incentives through risk exposure. This mechanism delivers two main results. First, the principal hedges less under delegation than under in-house production. Second, this lower hedging demand under delegation raises the equilibrium forward price relative to the integrated benchmark. In the constant-demand case, we show that access to hedging lowers the agent's expected compensation under delegation. Numerical results indicate that this mechanism remains robust in the presence of demand uncertainty. More broadly, our results show that external risk transfer through financial markets feeds back into internal organizational design.

q-fin.MF↗

arXiv2026

A class of singular control problems with tipping points

Tipping points characterize situations where a regulated system may experience a sudden and irreversible change and are generally associated with a random state of the system below which the change materializes. In this paper, we study a singular stochastic control problem in which the performance criterion depends on the hitting time of a random state that is not a stopping time for the reference filtration. We establish a connection between the value of this problem and that of a singular control problem involving a diffusion and its running minimum. We provide a verification lemma that we apply to explicitly solve a resource-extraction problem with an ex-ante unknown tipping point.

math.OC↗

arXiv2025

Well-posedness of McKean-Vlasov SDEs with density-dependent drift

In this paper, we study well-posedness of McKean-Vlasov stochastic differential equations (SDE) whose drift depends pointwisely on marginal density and satisfies a local integrability condition in time-space variables. The drift and noise coefficients are assumed to be Lipschitz continuous in distribution variable with respect to Wasserstein metric $W_p$. Our approach is by approximation with mollifiers. We prove strong existence of a solution. Weak and strong uniqueness are obtained when $p=1$, the drift coefficient is bounded, and the diffusion coefficient is distribution free.

math.PR↗

arXiv2025

Nash equilibria for dividend distribution with competition

We construct Nash equilibria in feedback form for a class of two-person stochastic games of singular control with absorption, arising from a stylized model for corporate finance. More precisely, the paper focusses on a strategic dynamic game in which two financially-constrained firms operate in the same market. The firms distribute dividends and are faced with default risk. The strategic interaction arises from the fact that if one firm defaults, the other one becomes a monopolist and increases its profitability. The firms choose their dividend distribution policies from a class of randomised strategies and we identify two types of equilibria, depending on the firms' initial endowments. In both situations the optimal strategies and the equilibrium payoffs are found explicitly.

math.OC↗

arXiv2025

Freidlin-Wentzell type exit-time estimates for time-inhomogeneous diffusions and their applications

This paper investigates the exit-time problem for time-inhomogeneous diffusion processes. The focus is on the small-noise behavior of the exit time from a bounded positively invariant domain. We demonstrate that, when the drift and diffusion terms are uniformly close to some time-independent functions, the exit time grows exponentially both in probability and in $L_1$ as a parameter that controls the noise tends to zero. We also characterize the exit position of the time-inhomogeneous process. Additionally, we investigate the impact of relaxing the uniform closeness condition on the exit-time behavior. As an application, we extend these results to the McKean-Vlasov process. Our findings improve upon existing results in the literature for the exit-time problem for this class of processes.

math.PR↗

arXiv2024

A Stochastic Non-Zero-Sum Game of Controlling the Debt-to-GDP Ratio

We introduce a non-zero-sum game between a government and a legislative body to study the optimal level of debt. Each player, with different time preferences, can intervene on the stochastic dynamics of the debt-to-GDP ratio via singular stochastic controls, in view of minimizing non-continuously differentiable running costs. We completely characterise Nash equilibria in the class of Skorokhod-reflection-type policies. We highlight the importance of different time preferences resulting in qualitatively different type of equilibria. In particular, we show that, while it is always optimal for the government to devise an appropriate debt issuance policy, the legislator should optimally impose a debt ceiling only under relatively low discount rates and a laissez-faire policy can be optimal for high values of the legislator's discount rate.

math.OC↗

arXiv2022

Gaussian Agency problems with memory and Linear Contracts

Can a principal still offer optimal dynamic contracts that are linear in end-of-period outcomes when the agent controls a process that exhibits memory? We provide a positive answer by considering a general Gaussian setting where the output dynamics are not necessarily semi-martingales or Markov processes. We introduce a rich class of principal-agent models that encompasses dynamic agency models with memory. From the mathematical point of view, we develop a methodology to deal with the possible non-Markovianity and non-semimartingality of the control problem, which can no longer be directly solved by means of the usual Hamilton-Jacobi-Bellman equation. Our main contribution is to show that, for one-dimensional models, this setting always allows for optimal linear contracts in end-of-period observable outcomes with a deterministic optimal level of effort. In higher dimension, we show that linear contracts are still optimal when the effort cost function is radial and we quantify the gap between linear contracts and optimal contracts for more general quadratic costs of efforts.

math.OC↗

arXiv2022

Swarm gradient dynamics for global optimization: the density case

Using jointly geometric and stochastic reformulations of nonconvex problems and exploiting a Monge-Kantorovich gradient system formulation with vanishing forces, we formally extend the simulated annealing method to a wide class of global optimization methods. Due to an inbuilt combination of a gradient-like strategy and particles interactions, we call them swarm gradient dynamics. As in the original paper of Holley-Kusuoka-Stroock, the key to the existence of a schedule ensuring convergence to a global minimizer is a functional inequality. One of our central theoretical contributions is the proof of such an inequality for one-dimensional compact manifolds. We conjecture the inequality to be true in a much wider setting. We also describe a general method allowing for global optimization and evidencing the crucial role of functional inequalities {à} la Łojasiewicz.

math.AP↗

arXiv2021

A Class of Explicit optimal contracts in the face of shutdown

What type of delegation contract should be offered when facing a risk of the magnitude of the pandemic we are currently experiencing and how does the likelihood of an exogenous early termination of the relationship modify the terms of a full-commitment contract? We study these questions by considering a dynamic principal-agent model that naturally extends the classical Holmstr{ö}m-Milgrom setting to include a risk of default whose origin is independent of the inherent agency problem. We obtain an explicit characterization of the optimal wage along with the optimal action provided by the agent. The optimal contract is linear by offering both a fixed share of the output which is similar to the standard shutdown-free Holmstr{ö}m-Milgrom model and a linear prevention mechanism that is proportional to the random lifetime of the contract. We then tweak the model to add a possibility for risk mitigation through investment and study its optimality.

math.OC↗

arXiv2019

A Dynkin game on assets with incomplete information on the return

This paper studies a 2-players zero-sum Dynkin game arising from pricing an option on an asset whose rate of return is unknown to both players. Using filtering techniques we first reduce the problem to a zero-sum Dynkin game on a bi-dimensional diffusion $(X,Y)$. Then we characterize the existence of a Nash equilibrium in pure strategies in which each player stops at the hitting time of $(X,Y)$ to a set with moving boundary. A detailed description of the stopping sets for the two players is provided along with global $C^1$ regularity of the value function.

math.PR↗

arXiv2019

On a Monotone Dynamic Approach to Optimal Stopping Problems for Continuous-Time Markov Chains

This paper is concerned with the solution of the optimal stopping problem associated to the valuation of Perpetual American options driven by continuous time Markov chains. We introduce a new dynamic approach for the numerical pricing of this type of American options where the main idea is to build a monotone sequence of almost excessive functions that are associated to hitting times of explicit sets. Under minimal assumptions about the payoff and the Markov chain, we prove that the value function of an American option is characterized by the limit of this monotone sequence.

math.PR↗

arXiv2016

Numerical approximation of a cash-constrained firm value with investment opportunities

We consider a singular control problem with regime switching that arises in problems of optimal investment decisions of cash-constrained firms. The value function is proved to be the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation. Moreover, we give regularity properties of the value function as well as a description of the shape of the control regions. Based on these theoretical results, a numerical deterministic approximation of the related HJB variational inequality is provided. We finally show that this numerical approximation converges to the value function. This allows us to describe the investment and dividend optimal policies.

q-fin.CP↗

arXiv2015

Liquidity Management with Decreasing-returns-to-scale and Secured Credit Line

This paper examines the dividend and investment policies of a cash constrained firm that has access to costly external funding. We depart from the literature by allowing the firm to issue collateralized debt to increase its investment in productive assets resulting in a performance sensitive interest rate on debt. We formulate this problem as a bi-dimensional singular control problem and use both a viscosity solution approch and a verification technique to get qualitative properties of the value function. We further solve quasi-explicitly the control problem in two special cases.

q-fin.PM↗

arXiv2008

A mixed singular/switching control problem for a dividend policy with reversible technology investment

We consider a mixed stochastic control problem that arises in Mathematical Finance literature with the study of interactions between dividend policy and investment. This problem combines features of both optimal switching and singular control. We prove that our mixed problem can be decoupled in two pure optimal stopping and singular control problems. Furthermore, we describe the form of the optimal strategy by means of viscosity solution techniques and smooth-fit properties on the corresponding system of variational inequalities. Our results are of a quasi-explicit nature. From a financial viewpoint, we characterize situations where a firm manager decides optimally to postpone dividend distribution in order to invest in a reversible growth opportunity corresponding to a modern technology. In this paper a reversible opportunity means that the firm may disinvest from the modern technology and return back to its old technology by receiving some gain compensation. The results of our analysis take qualitatively different forms depending on the parameters values.

math.PR↗