POSSAMAI Dylan

This thesis is divided into three parts. In the first part, we apply Principal-Agent theory to some market microstructure problems. First, we develop an incentive policy to improve the quality of market liquidity in the context of market-making activity in a bed and a dark pool managed by the same exchange. We then adapt this incentive design to the regulation of market-making activity when several market-makers compete on a platform. We also propose a form of incentive based on the choice of asymmetric tick sizes for buying and selling an asset. We then address the issue of designing a derivatives market, using a quantization method to select the options listed on the platform, and Principal-Agent theory to create incentives for an options market-maker. Finally, we develop an incentive mechanism robust to the model specification to increase investment in green bonds.The second part of this thesis is devoted to high-dimensional options market-making. The second part of this paper is devoted to the market-making of high-dimensional options. Assuming constant Greeks, we first propose a model to deal with long-maturity options. Then we propose an approximation of the value function to handle non-constant Greeks and short maturity options. Finally, we develop a model for the high frequency dynamics of the implied volatility surface. Using multidimensional Hawkes processes, we show how this model can reproduce many stylized facts such as the skew, the smile and the term structure of the surface.The last part of this thesis is devoted to optimal trading problems in high dimension. First, we develop a model for optimal trading of stocks listed on several platforms. For a large number of platforms, we use a deep reinforcement learning method to compute the optimal trader controls. Then, we propose a methodology to solve trading problems in an approximately optimal way without using stochastic control theory. We present a model in which an agent exhibits approximately optimal behavior if it uses the gradient of the macroscopic trajectory as a short-term signal. Finally, we present two new developments on the optimal execution literature. First, we show that we can obtain an analytical solution to the Almgren-Chriss execution problem with geometric Brownian motion and quadratic penalty. Second, we propose an application of the latent order book model to the optimal execution problem of a portfolio of assets, in the context of liquidity stress tests.

The aim of this short note is to fill in a gap in our earlier paper [7] on 2BSDEs with reflections, and to explain how to correct the subsequent results in the second paper [6]. We also provide more insight on the properties of 2RBSDEs, in the light of the recent contributions [5, 13] in the so-called G−framework.

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.

No summary available.

This thesis is formulated in three parts with eight chapters and presents a research theme dealing with controlled processes/particles/interacting agents.In the first part of the thesis, we focus our attention on the study of interacting controlled processes representing a cooperative equilibrium, also known as Pareto equilibrium. A cooperative equilibrium can be seen as a situation where there is no way to improve the preference criterion of one agent without lowering the preference criterion of at least one other agent. It is now well known that this type of optimization problem is related, when the number of agents goes to infinity, to McKean-Vlasov optimal control. In the first three chapters of this thesis, we provide a precise mathematical answer to the link between these two optimization problems in different frameworks improving the existing literature, in particular by taking into account the control law while allowing a common noise situation.After studying the behavior of cooperative equilibria, we conclude the first part where we spend time in the analysis of the limit problem i.e. McKean-Vlasov optimal control, through the establishment of the dynamic programming principle (DPP) for this stochastic control problem.The second part of this thesis is devoted to the study of interacting controlled processes now representing a Nash equilibrium, also known as competitive equilibrium. A Nash equilibrium situation in a game is one in which no one has anything to gain by unilaterally leaving his own position. Since the pioneering work of Larsy - Lions and Huang - Malhamé - Caines, the behavior of Nash equilibria when the number of agents reaches infinity has been intensively studied and the associated limit game is known as Mean Field Games (MFG). In this second part, we first analyze the convergence of competitive equilibria to MFGs in a framework with the control law and with volatility control, then, the question of the existence of the MFG equilibrium in this context is studied.Finally, the last part, which consists of only one chapter, is devoted to some numerical methods for solving the limit problem i.e. McKean - Vlasov optimal control. Inspired by the proof of convergence of the cooperative equilibrium, we give a numerical algorithm to solve the McKean-Vlasov optimal control problem and prove its convergence. Then, we implement our algorithm using neural networks and test its efficiency on some application examples, namely mean-variance portfolio selection, the interbank systemic risk model and optimal liquidation with market impact.

This paper studies a class of non-Markovian singular stochastic control problems, for which we provide a novel probabilistic representation. The solution of such control problem is proved to identify with the solution of a Z-constrained BSDE, with dynamics associated to a non singular underlying forward process. Due to the non-Markovian environment, our main argumentation relies on the use of comparison arguments for path dependent PDEs. Our representation allows in particular to quantify the regularity of the solution to the singular stochastic control problem in terms of the space and time initial data. Our framework also extends to the consideration of degenerate diffusions, leading to the representation of the solution as the infimum of solutions to Z-constrained BSDEs. As an application, we study the utility maximization problem with transaction costs for non-Markovian dynamics.

We study the utility indifference price of a European option in the context of small transaction costs. Considering the general setup allowing consumption and a general utility function at final time T, we obtain an asymptotic expansion of the utility indifference price as a function of the asymptotic expansions of the utility maximization problems with and without the European contingent claim. We use the tools developed in [54] and [48] based on homogenization and viscosity solutions to characterize these expansions. Finally we study more precisely the example of exponential utilities, in particular recovering under weaker assumptions the results of [6].

No summary available.

In this paper, we investigate a moral hazard problem in finite time with lump–sum and continuous payments, involving infinitely many Agents with mean field type interactions, hired by one Principal. By reinterpreting the mean-field game faced by each Agent in terms of a mean field forward backward stochastic differential equation (FBSDE for short), we are able to rewrite the Principal's problem as a control problem of McKean–Vlasov SDEs. We review two general approaches to tackle it: the first one introduced recently in [2, 66, 67, 68, 69] using dynamic programming and Hamilton–Jacobi– Bellman (HJB for short) equations, the second based on the stochastic Pontryagin maximum principle, which follows [16]. We solve completely and explicitly the problem in special cases, going beyond the usual linear–quadratic framework. We finally show in our examples that the optimal contract in the N −players' model converges to the mean–field optimal contract when the number of agents goes to +∞, this illustrating in our specific setting the general results of [12].

In a framework close to the one developed by Holmström and Milgrom [44], we study the optimal contracting scheme between a Principal and several Agents. Each hired Agent is in charge of one project, and can make efforts towards managing his own project, as well as impact (positively or negatively) the projects of the other Agents. Considering economic agents in competition with relative performance concerns, we derive the optimal contracts in both first best and moral hazard settings. The enhanced resolution methodology relies heavily on the connection between Nash equilibria and multidimensional quadratic BSDEs. The optimal contracts are linear and each agent is paid a fixed proportion of the terminal value of all the projects of the firm. Besides, each Agent receives his reservation utility, and those with high competitive appetence are assigned less volatile projects, and shall even receive help from the other Agents. From the principal point of view, it is in the firm interest in our model to strongly diversify the competitive appetence of the Agents.

We provide a unified approach to a priori estimates for supersolutions of BSDEs in general filtrations, which may not be quasi left-continuous. As an example of application, we prove that reflected BSDEs are well-posed in a general framework.

The problem of robust hedging requires to solve the problem of superhedging under a nondominated family of singular measures. Recent progress was achieved by van Handel, Neufeld, and Nutz. We show that the dual formulation of this problem is valid in a context suitable for martingale optimal transportation or, more generally, for optimal transportation under controlled stochastic dynamics.

This thesis presents two main independent research topics, the last one being declined as two distinct problems. In the first part of the thesis, we focus on the notion of second order stochastic backward differential equations (in the following 2EDSR), first introduced by Cheredito, Soner, Touzi and Victoir and recently reformulated by Soner, Touzi and Zhang. We first prove an extension of their existence and uniqueness results when the considered generator is only continuous and linearly growing. Then, we continue our study by a new extension to the case of a quadratic generator. These theoretical results then allow us to solve a utility maximization problem for an investor in an incomplete market, both because constraints are imposed on his investment strategies, and because the volatility of the market is assumed to be unknown. We prove in our framework the existence of optimal strategies, characterize the value function of the problem thanks to a second-order SRGE and solve explicitly some examples that allow us to highlight the modifications induced by the addition of volatility uncertainty compared to the usual framework. We end this first part by introducing the notion of second order EDSR with reflection on an obstacle. We prove the existence and uniqueness of the solutions of such equations, and provide a possible application to the problem of shorting American options in a market with uncertain volatility. The first chapter of the second part of this thesis deals with an option pricing problem in a model where market liquidity is taken into account. We provide asymptotic developments of these prices in the neighborhood of infinite liquidity and highlight a phase transition phenomenon depending on the regularity of the payoff of the considered options. Some numerical results are also proposed. Finally, we conclude this thesis by studying a Principal/Agent problem in a moral hazard framework. A bank (which plays the role of the agent) has a number of loans for which it is willing to exchange the interest for capital flows. The bank can influence the default probabilities of these loans by performing or not performing costly monitoring activity. These choices of the bank are known only to the bank. Investors (who play the role of principal) want to set up contracts that maximize their utility while implicitly incentivizing the bank to perform constant monitoring activity. We solve this optimal control problem explicitly, describe the associated optimal contract and its economic implications, and provide some numerical simulations.

This thesis presents two main independent research topics, the last one being declined as two distinct problems. In the first part of the thesis, we focus on the notion of second order stochastic backward differential equations (in the following 2EDSR), first introduced by Cheridito, Soner, Touzi and Victoir [25] and recently reformulated by Soner, Touzi and Zhang [107]. We first prove an extension of their existence and uniqueness results when the considered generator is only continuous and linearly growing. Then, we continue our study by a new extension to the case of a quadratic generator. These theoretical results then allow us to solve a utility maximization problem for an investor in an incomplete market, both because constraints are imposed on his investment strategies, and also because the market volatility is assumed to be unknown. We prove in our framework the existence of optimal strategies, characterize the value function of the problem thanks to a second-order RLS and solve explicitly some examples that allow us to highlight the modifications induced by the addition of volatility uncertainty compared to the usual framework. We end this first part by introducing the notion of second order EDSR with reflection on an obstacle. We prove the existence and uniqueness of the solutions of such equations, and provide a possible application to the problem of shorting American options in a market with uncertain volatility. The first chapter of the second part of this thesis deals with an option pricing problem in a model where market liquidity is taken into account. We provide asymptotic developments of these prices in the vicinity of infinite liquidity and highlight a phase transition phenomenon depending on the regularity of the payoff of the considered options. Some numerical results are also proposed. Finally, we conclude this thesis by studying a Principal/Agent problem in a moral hazard framework. A bank (playing the role of the agent) owning a number of loans, wishes to exchange their interest for capital flows. The bank can influence the default probabilities of these loans by performing or not performing costly monitoring activity. These choices of the bank are known only to the bank. Investors (who play the role of principal) wish to set up contracts that maximize their utility while implicitly incentivizing the bank to perform constant monitoring activity. We solve this optimal control problem explicitly, describe the associated optimal contract and its economic implications, and provide some numerical simulations.

This thesis presents two main independent research topics, the last one being declined as two distinct problems. In the first part of the thesis, we focus on the notion of second order stochastic backward differential equations (in the following 2EDSR), first introduced by Cheredito, Soner, Touzi and Victoir and recently reformulated by Soner, Touzi and Zhang. We first prove an extension of their existence and uniqueness results when the considered generator is only continuous and linearly growing. Then, we continue our study by a new extension to the case of a quadratic generator. These theoretical results then allow us to solve a utility maximization problem for an investor in an incomplete market, both because constraints are imposed on his investment strategies, and because the volatility of the market is assumed to be unknown. We prove in our framework the existence of optimal strategies, characterize the value function of the problem thanks to a second-order SRGE and solve explicitly some examples that allow us to highlight the modifications induced by the addition of volatility uncertainty compared to the usual framework. We end this first part by introducing the notion of second order EDSR with reflection on an obstacle. We prove the existence and uniqueness of the solutions of such equations, and provide a possible application to the problem of shorting American options in a market with uncertain volatility. The first chapter of the second part of this thesis deals with an option pricing problem in a model where market liquidity is taken into account. We provide asymptotic developments of these prices in the vicinity of infinite liquidity and highlight a phase transition phenomenon depending on the regularity of the payoff of the considered options. Some numerical results are also proposed. Finally, we conclude this thesis by studying a Principal/Agent problem in a moral hazard framework. A bank (which plays the role of the agent) has a number of loans for which it is willing to exchange the interest for capital flows. The bank can influence the default probabilities of these loans by performing or not performing costly monitoring activity. These choices of the bank are known only to the bank. Investors (who play the role of principal) wish to set up contracts that maximize their utility while implicitly incentivizing the bank to perform constant monitoring activity. We solve this optimal control problem explicitly, describe the associated optimal contract and its economic implications, and provide some numerical simulations.