VILLENEUVE Stephane

In the presence of agency friction, incentive contracts are designed to align the manager's goals with those of the firm's owner. However, the contractual environment is subject to shocks beyond the manager's control that impact the firm's future profitability. These shocks may be due, for example, to increased regulation, changes in the market or the emergence of a new alternative to the manager. The question then arises as to how contracts are designed when such shocks are anticipated at the time the contract is signed. To understand this effect, we conduct three studies. In the first paper, we explore how an incentive contract evolves upon the emergence of automation technologies that can replace the manager in the asset management context. We study a continuous-time principal-agent problem where the performance of an asset is determined by the unobserved effort of the manager, and the automation technology emerges in an uncertain future. Our model suggests that the empirically observed layoffs that accompany the emergence of automation technology may have a contractual basis. In the second study, we explore how changes in the agent's ability to divert cash flows impact the design of an optimal contract. We construct a continuous-time principal-agent model where the agent can divert cash flows out of the owner's view. While it is clear that mitigating agency friction is valuable to the business owner, its effect on incentive provision throughout the contractual relationship is unclear. First, our result suggests that bonus compression at the time of the shock: the reduction (respectively, increase) in bonuses expected by good (respectively, bad) managers. Second, our analysis also predicts that this type of regulation leads to retention of bad managers, defined as keeping a manager in place when his or her poor performance would have induced his or her dismissal in the absence of the cash flow detour profit shock. In the third study, we continue the previous study with an empirical approach. We analyze the Compensation Discussion and Analysis (CD&A) introduced in the US starting in 2007. We focus on the impact of this reform on the decision to lay off employees in non-financial companies of the S&P 500. We find that the introduction of the CD&A law significantly reduced the probability of CEO layoffs in non-financial firms. While the literature has shown that exogenous industry-level shocks have an impact on the layoff decision, we document that changes in the regulatory environment also matter.

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 C1 regularity of the value function.

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 Holmström-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 Holmström-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.

This thesis is composed of two independent parts, the first one grouping two distinct problems. In the first part, we first study the Principal-Agent problem in degenerate systems, which arise naturally in partially observed environments where the Agent and the Principal observe only a part of the system. We present an approach based on the stochastic maximum principle, which aims to extend existing work that uses the principle of dynamic programming in non-degenerate systems. First, we solve the Principal problem in an extended contract set given by the first-order condition of the Agent problem in the form of a path-dependent stochastic differential equation (EDSPR). Then we use the sufficient condition of the Agent problem to verify that the obtained optimal contract is implementable. A parallel study is devoted to the existence and uniqueness of the solution of path-dependent EDSPRs in Chapter IV. We extend the decoupling field method to cases where the coefficients of the equations can depend on the trajectory of the forward process. We also prove a stability property for such EDSPRs. Finally, we study the moral hazard problem with several Principals. The Agent can only work for one Principal at a time and thus faces an optimal switching problem. Using the randomization method we show that the Agent's value function and its optimal effort are given by an Itô process. This representation helps us to solve the Principal problem when there are infinitely many Principals in equilibrium according to a mean-field game. We justify the mean-field formulation by a chaos propagation argument.The second part of this thesis consists of chapters V and VI. The motivation of this work is to give a rigorous theoretical foundation for the convergence of gradient descent type algorithms which are often used in the solution of non-convex problems such as the calibration of a neural network. For non-convex problems of the hidden layer neural network type, the key idea is to transform the problem into a convex problem by raising it in the space of measurements. We show that the corresponding energy function admits a unique minimizer which can be characterized by a first order condition using the derivation in the space of measures in the sense of Lions. We then present an analysis of the long term behavior of the Langevin mean-field dynamics, which has a gradient flow structure in the 2-Wasserstein metric. We show that the marginal law flow induced by the mean-field Langevin dynamics converges to a stationary law using La Salle's invariance principle, which is the minimizer of the energy function.In the case of deep neural networks, we model them using a continuous-time optimal control problem. We first give the first order condition using Pontryagin's principle, which will then help us to introduce the system of mean-field Langevin equations, whose invariant measure corresponds to the minimizer of the optimal control problem. Finally, with the reflection coupling method we show that the marginal law of the mean-field Langevin system converges to the invariant measure with an exponential speed.

In this thesis, we are mainly interested in three research topics, relatively independent, but nevertheless related through the thread of interactions and incentives, as highlighted in the introduction constituting the first chapter.In the first part, we present extensions of contract theory, allowing in particular to consider a multitude of players in principal-agent models, with drift and volatility control, in the presence of moral hazard. In particular, Chapter 2 presents a continuous-time optimal incentive problem within a hierarchy, inspired by the one-period model of Sung (2015) and enlightening in two respects: on the one hand, it presents a framework where volatility control occurs in a perfectly natural way, and, on the other hand, it highlights the importance of considering continuous-time models. In this sense, this example motivates the comprehensive and general study of hierarchical models carried out in the third chapter, which goes hand in hand with the recent theory of second-order stochastic differential equations (2EDSR). Finally, in Chapter 4, we propose an extension of the principal-agent model developed by Aïd, Possamaï, and Touzi (2019) to a continuum of agents, whose performances are in particular impacted by a common hazard. In particular, these studies guide us towards a generalization of the so-called revealing contracts, initially proposed by Cvitanić, Possamaï and Touzi (2018) in a single-agent model.In the second part, we present two applications of principal-agent problems to the energy domain. The first one, developed in Chapter 5, uses the formalism and theoretical results introduced in the previous chapter to improve electricity demand response programs, already considered by Aïd, Possamaï and Touzi (2019). Indeed, by taking into account the infinite number of consumers that a producer has to supply with electricity, it is possible to use this additional information to build the optimal incentives, in particular to better manage the residual risk implied by weather hazards. In a second step, chapter 6 proposes, through a principal-agent model with adverse selection, an insurance that could prevent some forms of precariousness, in particular fuel precariousness.Finally, we end this thesis by studying in the last part a second field of application, that of epidemiology, and more precisely the control of the diffusion of a contagious disease within a population. In chapter 7, we first consider the point of view of individuals, through a mean-field game: each individual can choose his rate of interaction with others, reconciling on the one hand his need for social interactions and on the other hand his fear of being contaminated in turn, and of contributing to the wider diffusion of the disease. We prove the existence of a Nash equilibrium between individuals, and exhibit it numerically. In the last chapter, we take the point of view of the government, wishing to incite the population, now represented as a whole, to decrease its interactions in order to contain the epidemic. We show that the implementation of sanctions in case of non-compliance with containment can be effective, but that, for a total control of the epidemic, it is necessary to develop a conscientious screening policy, accompanied by a scrupulous isolation of the individuals tested positive.

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.

We study a dynamic corporate finance contracting model in which the firm's profitability fluctuates and is impacted by the unobservable managerial effort. Thereby, we introduce in an agency framework the issue of strategic liquidation. We show that the principal's problem takes the form of a two-dimensional fully degenerate Markov control problem. We prove regularity properties of the value function and derive explicitly the optimal contract that implements full effort. Our regularity results appear in some recent studies, but with heuristic proofs that do not clarify the importance of the regularity of the value function at the boundaries.

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.

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We model the financing, cash holdings, and hedging policies of a firm facing financing frictions and subject to permanent and transitory cash flow shocks. The permanent and transitory shocks generate distinct, sometimes opposite, effects on corporate policies. We use the model to develop a rich set of empirical predictions. In our model, correlated permanent and transitory shocks imply less risk, lower cash savings, and a drop in the value of credit lines. The composition of cash-flow shocks affects the cash-flow sensitivity of cash, which can be positive or negative. Optimal hedging of permanent and transitory shocks may involve opposite positions.

This paper studies the impact of the confirmatory bias on financial markets. We propose a model in which some traders may ignore new evidence inconsistent with their favorite hypothesis regarding the state of the world. The confirmatory bias provides a unified rationale for several existing stylized facts, including excess volatility, excess volume, and momentum. It also delivers novel predictions for which we find empirical support using data on analysts’ earnings forecasts: traders update beliefs depending on the sign of past signals and previous beliefs, and, at the stock level, differences of opinion are larger when past signals have different signs.

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This paper examines the dividend and investment policies of a cash constrained firm, assuming a decreasing-returns-to-scale technology and adjustment costs. We extend the literature by allowing the firm to draw on a secured credit line both to hedge against cash-flow shortfalls and to invest/disinvest in a productive asset. We formulate this problem as a two-dimensional singular control problem and use both a viscosity solution approach and a verification technique to get qualitative properties of the value function. We further solve quasi-explicitly the control problem in two special cases.

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This thesis studies some stochastic control problems in a context of liquidity risk and impact on asset prices. The thesis is composed of four chapters.In the second chapter, we propose a modeling of a market making problem in a liquidity risk context in the presence of inventory constraints and regime shifts. This formulation can be considered as an extension of previous studies on this subject. The main result of this part is the characterization of the value function as a unique solution, in the sense of viscosity, of a system of Hamilton-Jacobi-Bellman equations . In the third chapter, we propose a numerical approximation scheme to solve a portfolio optimization problem in a context of liquidity risk and impact on asset prices. We show that the value function can be obtained as a limit of an iterative procedure where each iteration represents an optimal stopping problem and we use a numerical algorithm, based on optimal quantization, to compute the value function and the control policy. The convergence of the numerical scheme is obtained via monotonicity, stability and consistency criteria.In the fourth chapter, we focus on a coupled problem of singular control and impulse control in an illiquidity context. We propose a mathematical formulation to model the dividend distribution and the investment policy of a firm subject to liquidity constraints. We show that, under transaction costs and an impact on the price of illiquid assets, the firm's value function is the unique viscosity solution of a Hamilton-Jacobi-Bellman equation. An iterative numerical method is also proposed to compute the optimal buy, sell and dividend strategy.

The purpose of this thesis is to model and optimize the investment strategies of an agent subject to a Markovian environment, and to a liquidity risk that arises when he can no longer face a cash outflow due to a lack of liquid assets. During this study, we will assume that his objective is to avoid bankruptcy. To do so, he has investment opportunities, allowing him to increase his future earnings in exchange for an immediate expense, thus risking a premature ruin since the investment is assumed to be illiquid: the goal of the work is to determine the conditions under which it is more judicious to run such a liquidity risk than to give up a permanent income.

This thesis is composed of three essays. The first essay deals with the problem of the risk of extreme losses in the banking sector in the context of the agency problem between the shareholders and the top managers of banks. In order to induce banks not to take the risk of extreme losses, it is proposed to apply capital regulation in the form of a mandatory recapitalization policy, the parameters of which are chosen to induce shareholders to remunerate their managers in such a way as to steer them away from extreme loss strategies.The 2nd essay develops the design of bank supervision that aims to eliminate the moral hazard problem within a bank, while ensuring a minimum cost of supervision. Banks, whose financial condition begins to deteriorate, should be subject to random audits. Banks whose asset values have deteriorated significantly should be placed under conservatorship for financial recovery. External auditors can be involved in the supervisory process, but should not completely replace regulators. The third essay studies how the borrowing capacity of the non-financial firm affects its investment policy in the presence of debt issuance costs. It is shown that firms, with medium borrowing capacity, have an incentive to make a larger investment compared to firms with relatively low/high borrowing capacity. This is entirely due to the effect of fixed debt issuance costs, which emerges in the dynamic investment environment.

In this thesis, we treat two stochastic optimal control problems. Each problem corresponds to a part of this paper. The first problem is very specific, it is the valuation of American put contracts in the presence of discrete dividends (Part I). The second one is more general, since it is about proving the existence of a dynamic programming principle under probability constraints in a discrete time framework (Part II). Although the two problems are quite distinct, the dynamic programming principle is at the heart of both problems. The relation between the valuation of an American Put and a free boundary problem has been proved by McKean. The frontier of this problem has a clear economic meaning since it corresponds at any moment to the upper bound of the set of asset prices for which it is preferable to exercise one's right to sell immediately. The shape of this frontier in the presence of discrete dividends has not been solved to our knowledge. Under the assumption that the dividend is a deterministic function of the asset price at the time preceding its payment, we study how the frontier is modified. In the vicinity of the dividend dates, and in the model of Chapter 3, we know how to qualify the monotonicity of the frontier, and in some cases quantify its local behavior. In Chapter 3, we show that the smooth-fit property is satisfied at all dates except the dividend dates. In both Chapters 3 and 4, we give conditions to guarantee the continuity of this frontier outside the dividend dates. Part II is originally motivated by the optimal management of the production of a hydro-electric plant with a constraint in probability on the water level of the dam at certain dates. Using Balder's work on Young's relaxation of optimal control problems, we focus more specifically on solving them by dynamic programming. In Chapter 5, we extend the results of Evstigneev to the framework of Young's measurements. We then establish that it is possible to solve by dynamic programming some problems with constraints in conditional expectations. Thanks to the work of Bouchard, Elie, Soner and Touzi on stochastic target problems with controlled loss, we show in Chapter 6 that a problem with expectation constraints can be reduced to a problem with conditional expectation constraints. As a special case, we prove that the initial dam management problem can be solved by dynamic programming.

In this paper, we develop a dynamic model that captures the interaction between a firm’s cash reserves, the risk management policy and the profitability of a non-predictable irreversible investment opportunity. We consider a firm that has assets in place generating a stochastic cash-flow stream. The firm has a non-predictable growth opportunity to expand its operation size by paying a sunk cost. When the opportunity is available, the firm can finance it either by cash or by costly equity issuance. We provide an explicit characterization of the firm strategy in terms of investment, hedging, equity issuance and dividend distribution.

The aim of this thesis is to study American options in a multi-dimensional diffusion model. Mathematically, this study is related to an optimal stopping problem with a finite or non finite horizon. The first part of the paper is concerned with the description of the valuation model of American options as a solution of a parabolic variational inequation and with the existence or not of a stopping region more commonly called the exercise region in finance. The first chapter provides a necessary and sufficient condition on the infinitesimal generator of the diffusion for the stopping region to be non empty. The following chapters study the properties of exercise regions associated with some types of options commonly traded on the markets: convexity, regularity and asymptotic behavior for infinite or near zero horizons. The second part concerns the numerical analysis of American options in dimension two. After recalling the different formulations using partial differential equations (solution in sobolev spaces or viscosity solution), two approximation methods of the alternate directions type are proposed and two convergence theorems are established. A comparison result between these methods ends this part. The last part studies the critical price of the American put in the vicinity of the maturity when the stock pays dividends. A result concerning the strict monotonicity of the critical price is proved as well as a framework of this price in the vicinity of the maturity.