ELIE Romuald

Algorithmic Fairness is an established area of machine learning, willing to reduce the influence of biases in the data. Yet, despite its wide range of applications, very few works consider the multi-class classification setting from the fairness perspective. We address this question by extending the definition of Demographic Parity to the multi-class problem while specifying the corresponding expression of the optimal fair classifier. This suggests a plug-in data-driven procedure, for which we establish theoretical guarantees. Specifically, we show that the enhanced estimator mimics the behavior of the optimal rule, both in terms of fairness and risk. Notably, fairness guarantee is distribution-free. We illustrate numerically the quality of our algorithm. The procedure reveals to be much more suitable than an alternative approach enforcing fairness constraints on the score associated to each class. This shows that our method is empirically very effective in fair decision making on both synthetic and real datasets.

Artificial intelligence has become increasingly popular in quantitative finance with the increase in computational capabilities as well as model complexity and has led to many financial applications. In this thesis, we explore three different applications to solve challenges in the financial derivatives domain ranging from model selection, to model calibration, to derivative valuation. In Part I, we focus on a volatility regime-switching model to value equity derivatives. The model parameters are estimated using the Expectation-Maximization (EM) algorithm and a local volatility component is added so that the model is calibrated to vanilla option prices using the particle method. In Part II, we then use deep neural networks to calibrate a stochastic volatility model, in which volatility is represented by the exponential of an Ornstein-Uhlenbeck process, to approximate the function that relates the model parameters to the corresponding implied volatilities offline. Once the costly approximation is done offline, the calibration reduces to a standard and fast optimization problem. In Part III, we finally use deep neural networks to value American options on large baskets of stocks to overcome the curse of dimension. Different methods are studied with a Longstaff-Schwartz approach, where we approximate the continuation values, and a stochastic control approach, where we solve the partial differential valuation equation by reformulating it into a stochastic control problem using the nonlinear Feynman-Kac formula.

No summary available.

The treatment of uncertainties is a fundamental problem in the financial context. The variables studied are often time-dependent, with thick distribution tails. In this thesis, we are interested in tools to take into account uncertainties in its main forms: statistical uncertainties, parametric uncertainties and model error, keeping in mind that we wish to apply them to this context. The first part is devoted to the establishment of concentration inequalities in the context of variables with thick tails. The objective of these inequalities is to quantify the confidence that can be given to an estimator based on a finite size of observations. In this thesis, we establish new concentration inequalities, which cover in particular the case of estimators with lognormal distribution.In the second part, we deal with the impact of the model error for the estimation of the covariance matrix on stock returns, under the assumption that there is an instantaneous covariance process between the returns whose present value depends on its past value. We can then explicitly construct the best estimate of the covariance matrix for a given time and investment horizon, and show that it provides the smallest realized variance with high probability in the minimum variance portfolio framework.In the third part, we propose an approach to estimate the Sharpe ratio and the portfolio allocation when they depend on parameters considered uncertain. Our approach involves the adaptation of a stochastic approximation technique for the computation of the polynomial decomposition of the quantity of interest.Finally, in the last part of this thesis, we focus on portfolio optimization with target distribution. This technique can be formalized without any model assumptions on the returns. We propose to find these portfolios by minimizing divergence measures based on kernel functions and optimal transport theory.

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.

We consider the control of the COVID-19 pandemic, modeled by a standard SIR com-partmental model. The control of the epidemic is induced by the aggregation of individuals' decisions to limit their social interactions: on one side, when the epidemic is ongoing, an individual is encouraged to diminish his/her contact rate in order to avoid getting infected, but, on the other side, 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. However, it is not clear that the individual's interest aligns with that of the society. We prove that the equilibrium exists and compute it numerically. The 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. We show that the divergence between the individual and societal strategies happens after the epidemic peak but while significant propagation is still underway.

The use of statistical magnitudes to inform decision-making in risk situations, which first appeared in the 18th century and was then disqualified in the 19th century, was reintroduced in the middle of the 20th century and has since then gradually gained acceptance in the financial industry, percolating through the insurance industry to the point where it has become widespread via Solvency 2. However, numerous dysfunctions linked to the mobilization of these tools for the management and regulation of risks have been highlighted by the economic, actuarial and management science literature. Consequently, this extension of the scope of application of such tools to the regulation of insurance raises questions. This thesis (i) empirically extends to the insurance sector the literature produced in other sectors in order to identify the limits of the use of statistics for risk management and prudential regulation, (ii) proposes a unified theoretical framework for the dysfunctions linked to these uses, and (iii) sheds light on the reasons for the adoption of these tools within the insurance sector.

No summary available.

We consider here an extended $SIR$ model, including several features of the recent COVID-19 outbreak: in particular the infected and recovered individuals can either be detected (+) or undetected (-) and we also integrate an intensive care unit (ICU) capacity. Our model enables a tractable quantitative analysis of the optimal policy for the control of the epidemic dynamics using both lockdown and detection intervention levers. With parametric specification based on literature on COVID-19, we investigate the sensitivities of various quantities on the optimal strategies, taking into account the subtle trade-off between the sanitary and the socio-economic cost of the pandemic, together with the limited capacity level of ICU. We identify the optimal lockdown policy as an intervention structured in 4 successive phases: First a quick and strong lockdown intervention to stop the exponential growth of the contagion. second a short transition phase to reduce the prevalence of the virus. third a long period with full ICU capacity and stable virus prevalence. finally a return to normal social interactions with disappearance of the virus. The optimal scenario hereby avoids the second wave of infection, provided the lockdown is released sufficiently slowly. We also provide optimal intervention measures with increasing ICU capacity, as well as optimization over the effort on detection of infectious and immune individuals. Whenever massive resources are introduced to detect infected individuals, the pressure on social distancing can be released, whereas the impact of detection of immune individuals reveals to be more moderate.

No summary available.

In this paper, we deepen the analysis of continuous time Fictitious Play learning algorithm to the consideration of various finite state Mean Field Game settings (finite horizon, $\gamma$-discounted), allowing in particular for the introduction of an additional common noise. We first present a theoretical convergence analysis of the continuous time Fictitious Play process and prove that the induced exploitability decreases at a rate $O(\frac{1}{t})$. Such analysis emphasizes the use of exploitability as a relevant metric for evaluating the convergence towards a Nash equilibrium in the context of Mean Field Games. These theoretical contributions are supported by numerical experiments provided in either model-based or model-free settings. We provide hereby for the first time converging learning dynamics for Mean Field Games in the presence of common noise.

We consider the control of the COVID-19 pandemic, modeled by a standard SIR com-partmental model. The control of the epidemic is induced by the aggregation of individuals' decisions to limit their social interactions: on one side, when the epidemic is ongoing, an individual is encouraged to diminish his/her contact rate in order to avoid getting infected, but, on the other side, 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. However, it is not clear that the individual's interest aligns with that of the society. We prove that the equilibrium exists and compute it numerically. The 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. We show that the divergence between the individual and societal strategies happens after the epidemic peak but while significant propagation is still underway.

We consider here an extended SIR model, including several features of the recent COVID-19 outbreak: in particular the infected and recovered individuals can either be detected (+) or undetected (-) and we also integrate an intensive care unit capacity. Our model enables a tractable quantitative analysis of the optimal policy for the control of the epidemic dynamics using both lockdown and detection intervention levers. With parametric specification based on literature on COVID-19, we investigate sensitivity of various quantities on optimal strategies, taking into account the subtle tradeoff between the sanitary and the economic cost of the pandemic, together with the limited capacity level of ICU. We identify the optimal lockdown policy as an intervention structured in 4 successive phases: First a quick and strong lockdown intervention to stop the exponential growth of the contagion. second a short transition phase to reduce the prevalence of the virus. third a long period with full ICU capacity and stable virus prevalence. finally a return to normal social interactions with disappearance of the virus. We also provide optimal intervention measures with increasing ICU capacity, as well as optimization over the effort on detection of infectious and immune individuals.

The thesis deals with numerical schemes for Markovian decision problems (MDPs), partial differential equations (PDEs), backward stochastic differential equations (SRs), as well as reflected backward stochastic differential equations (SRDEs). The thesis is divided into three parts.The first part deals with numerical methods for solving MDPs, based on quantization and local or global regression. A market-making problem is proposed: it is solved theoretically by rewriting it as an MDP. and numerically by using the new algorithm. In a second step, a Markovian embedding method is proposed to reduce McKean-Vlasov type probabilities with partial information to MDPs. This method is implemented on three different McKean-Vlasov type problems with partial information, which are then numerically solved using numerical methods based on regression and quantization.In the second part, new algorithms are proposed to solve MDPs in high dimension. The latter are based on neural networks, which have proven in practice to be the best for learning high dimensional functions. The consistency of the proposed algorithms is proved, and they are tested on many stochastic control problems, which allows to illustrate their performances.In the third part, we focus on methods based on neural networks to solve PDEs, EDSRs and reflected EDSRs. The convergence of the proposed algorithms is proved and they are compared to other recent algorithms of the literature on some examples, which allows to illustrate their very good performances.

In this thesis, we make some contributions to the theoretical and numerical study of some stochastic control problems, as well as their applications to financial mathematics and financial risk management. These applications concern problems of valuation and weak hedging of financial products, as well as regulatory issues. We propose numerical methods to efficiently compute these quantities for which no explicit formula exists. Finally, we study backward stochastic differential equations related to new switching problems with cost uncertainty.

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 the summer of 2013, the term "Big Data" made its appearance and aroused strong interest among companies. This thesis studies the contribution of these methods to actuarial sciences. It addresses both theoretical and practical issues on high-potential topics such as textit{Optical Character Recognition} (OCR), text analysis, data anonymization or model interpretability. Starting with the application of machine learning methods in the calculation of economic capital, we then try to better illustrate the frontality that can exist between machine learning and statistics. Putting forward some advantages and different techniques, we then study the application of deep neural networks in the optical analysis of documents and text, once extracted. The use of complex methods and the implementation of the General Data Protection Regulation (GDPR) in 2018 led us to study the potential impacts on pricing models. By applying anonymization methods on pure premium calculation models in non-life insurance, we explored different generalization approaches based on unsupervised learning. Finally, as the regulation also imposes criteria in terms of model explanation, we conclude with a general study of the methods that allow today to better understand complex methods such as neural networks.

We begin this thesis report by evaluating the expected exposure, which represents one of the major components of XVA adjustments. Under the assumption of independence between exposure and financing and credit costs, we derive in Chapter 3 a new representation of expected exposure as the solution of an ordinary differential equation with respect to the time of default observation. For the one-dimensional case, we rely on arguments similar to those for Dupire's local volatility. And for the multidimensional case, we refer to the Co-aire formula. This representation allows us to explain the impact of volatility on the expected exposure: this time value involves the volatility of the underlyings as well as the first-order sensitivity of the price, evaluated on a finite set of points. Despite numerical limitations, this method is an accurate and fast approach for valuing unit XVA in dimension 1 and 2.The following chapters are dedicated to the risk aspects of correlations between exposure envelopes and XVA costs. We present a model of the general correlation risk through a multivariate stochastic diffusion, including both the underlying assets of the derivatives and the default intensities. In this framework, we present a new approach to valuation by asymptotic developments, such that the price of an XVA adjustment corresponds to the price of the zero-correlation adjustment, plus an explicit sum of corrective terms. Chapter 4 is devoted to the technical derivation and study of the numerical error in the context of the valuation of default contingent derivatives. The quality of the numerical approximations depends solely on the regularity of the credit intensity diffusion process, and is independent of the regularity of the payoff function. The valuation formulas for CVA and FVA are presented in Chapter 5. A generalization of the asymptotic developments for the bilateral default framework is addressed in Chapter 6.We conclude this dissertation by addressing a case of the specific correlation risk related to rating migration contracts. Beyond the valuation formulas, our contribution consists in presenting a robust approach for the construction and calibration of a rating transition model consistent with market implied default probabilities.

The objective of this manuscript is to present several results on uniqueness restoration and equilibrium selection in mean field games. The theory of mean-field games was initiated in the 2000s by two groups of researchers, Lasry and Lions in France, and Huang, Caines and Malhamé in Canada. The objective of this theory is to describe Nash equilibria in stochastic differential games including a large number of players interacting with each other through their common empirical measure and presenting sufficient symmetry. While the existence of equilibria in mean-field games is now well understood, the uniqueness remains known in a very limited number of cases. In this respect, the best known condition is the so-called monotonicity condition, due to Lasry and Lions. In this thesis, we show that, for a certain class of mean-field games, uniqueness can be restored using a random forcing of the dynamics, common to all players. Such a forcing is called "common noise". We also show that, in some cases, it is possible to select equilibria in the absence of common noise by making the common noise tend to zero. Finally, we show how these results apply to principal-agent problems, with a large number of interacting agents.

In this paper, we study a new type of BSDE, where the distribution of the Y-component of the solution is required to satisfy an additional constraint, written in terms of the expectation of a loss function. This constraint is imposed at any deterministic time t and is typically weaker than the classical pointwise one associated to reflected BSDEs. Focusing on solutions (Y, Z, K) with deterministic K, we obtain the well-posedness of such equation, in the presence of a natural Skorokhod type condition. Such condition indeed ensures the minimality of the enhanced solution, under an additional structural condition on the driver. Our results extend to the more general framework where the constraint is written in terms of a static risk measure on Y. In particular, we provide an application to the super hedging of claims under running risk management constraint.

Artificial Intelligence is present in every aspect of our lives in the era of Big Data. It has led to revolutionary changes in various sectors, including e-commerce and finance. In this thesis, we present four applications of AI that improve existing goods and services, enable automation, and dramatically increase the efficiency of many tasks in both fields. First, we improve the product search service offered by most e-commerce sites by using a novel term weighting system to better evaluate the importance of terms in a search query. Next, we build a predictive model on daily sales using a time series forecasting approach and leverage the predicted results to rank product search results to maximize a company's revenue. Next, we propose the difficulty of online product classification and analyze winning solutions, consisting of state-of-the-art classification algorithms, on our real-world dataset. Finally, we combine the skills previously learned from time series-based sales prediction and classification to predict one of the most challenging but attractive time series: inventory. We perform an in-depth study on each stock in the S&P 500 Index using four state-of-the-art classification algorithms and report very promising results.

This thesis is composed of three chapters that deal with impulse control problems. In the first chapter, we introduce a general framework for impulse control with uncertainty. Knowing an a priori law on unknown parameters, we explain how it should evolve and integrate it to the optimal control problem. We characterize the solution through a quasivariational parabolic equation that can be solved numerically and give examples of applications to finance. In the second chapter, we introduce an impulse control problem with uncertainty in an actuarial setting. An (re)insurer faces natural catastrophes and can issue CAT bonds to reduce the risk taken. We again characterize the optimal control problem through a numerically solvable quasi-variational parabolic equation and give some application examples. In the last chapter, we propose a model of the price through a completely endogenous order book. We solve impulse optimal control problems (order placement) of rational economic agents that we gather on a same market.

This thesis is divided into three main parts, dealing with quasi-linear parabolic PDEs on a junction, stochastic diffusions on a junction, and optimal control also on a junction, with control at the junction point. We start in the first chapter by introducing a new class of non-degenerate quasi-linear PDEs, satisfying a non-linear and non-dynamic Neumann (or Kirchoff) condition at the junction point. We prove the existence of a classical solution, as well as its uniqueness. One of the motivations for studying this type of PDE is to make the link with the theory of optimal control on junctions, and to characterize the value function of this type of problem using the Hamilton Jacobi Bellman equations. Thus, in the next chapter, we formulate a proof giving the existence of a diffusion on a junction. This process admits a local time, whose existence and quadratic variation depend essentially on the assumption of ellipticity of the second order terms at the junction. We will formulate an Itô formula for this process. Thus, thanks to the results of these two Chapters, we will formulate in the last Chapter a stochastic control problem on junctions, with control at the junction point. The control space is that of probability measures solving a martingale problem. We prove the compactness of the space of admissible controls, as well as the principle of dynamic programming.

In this paper, we investigate the impact of the claim reporting strategy of drivers, within a bonus malus system. We exhibit the induced modification of the corresponding class level transition matrix and derive the optimal reporting strategy for rational drivers. The hunger for bonuses induces optimal thresholds under which, drivers do not claim their losses. A numerical algorithm is provided for computing such thresholds and realistic numerical applications are discussed.

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.

In this paper, we investigate the impact of the claim reporting strategy of drivers, within a bonus malus system. We exhibit the induced modification of the corresponding class level transition matrix and derive the optimal reporting strategy for rational drivers. The hunger for bonuses induces optimal thresholds under which, drivers do not claim their losses. A numerical algorithm is provided for computing such thresholds and realistic numerical applications are discussed.

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 this paper, we investigate the impact of the claim reporting strategy of drivers, within a bonus malus system. We exhibit the induced modification of the corresponding class level transition matrix and derive the optimal reporting strategy for rational drivers. The hunger for bonuses induces optimal thresholds under which, drivers do not claim their losses. A numerical algorithm is provided for computing such thresholds and realistic numerical applications are discussed.

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.

This paper is dedicated to the replication of a convex contingent claim h(S 1) in a financial market with frictions, due to deterministic order books or regulatory constraints. The corresponding transaction costs can be rewritten as a nonlinear function G of the volume of traded assets, with G′(0)>0. For a stock with Black–Scholes midprice dynamics, we exhibit an asymptotically convergent replicating portfolio, defined on a regular time grid with n trading dates. Up to a well-chosen regularization h n of the payoff function, we first introduce the frictionless replicating portfolio of hn(Sn1), where S n is a fictitious stock with enlarged local volatility dynamics. In the market with frictions, a suitable modification of this portfolio strategy provides a terminal wealth that converges in L2 to the claim h(S 1) as n goes to infinity. In terms of order book shapes, the exhibited replicating strategy only depends on the size 2G′(0) of the bid–ask spread. The main innovation of the paper is the introduction of a “Leland-type” strategy for nonvanishing (nonlinear) transaction costs on the volume of traded shares, instead of the commonly considered traded amount of money. This induces lots of technicalities, which we overcome by using an innovative approach based on the Malliavin calculus representation of the Greeks.

We introduce a new class of Backward Stochastic Differential Equations in which the $T$-terminal value $Y_{T}$ of the solution $(Y,Z)$ is not fixed as a random variable, but only satisfies a weak constraint of the form $E[\Psi(Y_{T})]\ge m$, for some (possibly random) non-decreasing map $\Psi$ and some threshold $m$. We name them \textit{BSDEs with weak terminal condition} and obtain a representation of the minimal time $t$-values $Y_{t}$ such that $(Y,Z)$ is a supersolution of the BSDE with weak terminal condition. It provides a non-Markovian BSDE formulation of the PDE characterization obtained for Markovian stochastic target problems under controlled loss in Bouchard, Elie and Touzi \cite{BoElTo09}. We then study the main properties of this minimal value. In particular, we analyze its continuity and convexity with respect to the $m$-parameter appearing in the weak terminal condition, and show how it can be related to a dual optimal control problem in Meyer form. These last properties generalize to a non Markovian framework previous results on quantile hedging and hedging under loss constraints obtained in F\"{o}llmer and Leukert \cite{FoLe99,FoLe00}, and in Bouchard, Elie and Touzi \cite{BoElTo09}.

This paper deals with the superreplication of non-path-dependent European claims under additional convex constraints on the number of shares held in the portfolio. The corresponding superreplication price of a given claim has been widely studied in the literature, and its terminal value, which dominates the claim of interest, is the so-called facelift transform of the claim. We investigate under which conditions the superreplication price and strategy of a large class of claims coincide with the exact replication price and strategy of the facelift transform of this claim. In one dimension, we observe that this property is satisfied for any local volatility model. In any dimension, we exhibit an analytical necessary and sufficient condition for this property, which combines the dynamics of the stock together with the characteristics of the closed convex set of constraints. To obtain this condition, we introduce the notion of first order viability property for linear parabolic PDEs. We investigate in detail several practical cases of interest: multidimensional Black–Scholes model, non-tradable assets, and short-selling restrictions.

We consider the minimal super-solution of a backward stochastic differential equation with constraint on the gains-process. The terminal condition is given by a function of the terminal value of a forward stochastic differential equation. Under boundedness assumptions on the coefficients, we show that the first component of the solution is Lipschitz in space and 1/2-Hölder in time with respect to the initial data of the forward process. Its path is continuous before the time horizon at which its left-limit is given by a face-lifted version of its natural boundary condition. This first component is actually equal to its own face-lift. We only use probabilistic arguments. In particular, our results can be extended to certain non-Markovian settings.

No summary available.

This thesis deals with two areas of stochastic analysis and financial mathematics: the Malliavin calculus for Markov chains (Part I) and counterparty risk (Part II). Part I aims at studying the Malliavin calculus for Markov chains in continuous time. Two points are presented: proving the existence of the density for the solutions of a stochastic differential equation and computing the sensitivities of derivatives. Part II deals with current topics in the field of market risk, namely XVA (price adjustments) and multi-curve modeling.

This paper provides two different strong BSDE representations for optimal switching problems in the case where the dynamics of the underlying diffusion process depends on the current value of the switching mode. These new representations are valid in a non-Markovian framework and make use of either one-dimensional constrained BS-DEs with jumps or multidimensional BSDEs with oblique reflections, thus extending the framework considered by Hu and Tang [12]. In particular, the numerical resolu-tion of the corresponding switching problem can therefore be treated via the entirely probabilistic schemes presented in [4] or [8].

Let B be a Brownian motion and T1 its first hitting time of the level 1. For U a uniform random variable independent of B, we study in depth the distribution of B UT1/√T1, that is the rescaled Brownian motion sampled at uniform time. In particular, we show that this variable is centered.

We introduce a new class of Backward Stochastic Differential Equations in which the $T$-terminal value $Y_{T}$ of the solution $(Y,Z)$ is not fixed as a random variable, but only satisfies a weak constraint of the form $E[\Psi(Y_{T})]\ge m$, for some (possibly random) non-decreasing map $\Psi$ and some threshold $m$. We name them \textit{BSDEs with weak terminal condition} and obtain a representation of the minimal time $t$-values $Y_{t}$ such that $(Y,Z)$ is a supersolution of the BSDE with weak terminal condition. It provides a non-Markovian BSDE formulation of the PDE characterization obtained for Markovian stochastic target problems under controlled loss in Bouchard, Elie and Touzi \cite{BoElTo09}. We then study the main properties of this minimal value. In particular, we analyze its continuity and convexity with respect to the $m$-parameter appearing in the weak terminal condition, and show how it can be related to a dual optimal control problem in Meyer form. These last properties generalize to a non Markovian framework previous results on quantile hedging and hedging under loss constraints obtained in F\"{o}llmer and Leukert \cite{FoLe99,FoLe00}, and in Bouchard, Elie and Touzi \cite{BoElTo09}.

This paper is dedicated to the analysis of backward stochastic differential equations (BSDEs) with jumps, subject to an additional global constraint involving all the com-ponents of the solution. We study the existence and uniqueness of a minimal solution for these so-called constrained BSDEs with jumps via a penalization procedure. This new type of BSDE offers a nice and practical unifying framework to the notions of constrained BSDEs presented in [22] and BSDEs with constrained jumps introduced in [17]. More remarkably, the solution of a multidimensional Brownian reflected BSDE studied in [16] and [14] can also be represented via a well chosen one-dimensional con-strained BSDE with jumps. This last result is very promising from a numerical point of view for the resolution of high dimensional optimal switching problems and more generally for systems of coupled variational inequalities.

The development of organized electronic markets puts constant pressure on academic research in finance. The price impact of a stock market transaction involving a large quantity of shares over a short period of time is a central topic. Controlling and monitoring the price impact is of great interest to practitioners, and its modeling has thus become a central focus of quantitative finance research. Historically, stochastic calculus has gradually been imposed in finance, under the implicit assumption that asset prices satisfy diffusive dynamics. But these hypotheses do not hold at the level of "price formation", i.e. when one considers the fine scales of market participants. New mathematical techniques derived from the statistics of point processes are therefore progressively imposed. The observables (processed price, middle price) appear as events taking place on a discrete network, the order book, and this at very short time scales (a few tens of milliseconds). The approach of prices seen as Brownian diffusions satisfying equilibrium conditions becomes rather a macroscopic description of complex phenomena arising from price formation. In the first chapter, we review the properties of electronic markets. We recall the limitations of diffusive models and introduce Hawkes processes. In particular, we review the research on the maket impact and present the progress of this thesis. In a second part, we introduce a new continuous time and discrete space impact model using Hawkes processes. We show that this model takes into account the microstructure of markets and is able to reproduce recent empirical results such as the concavity of the temporary impact. In the third chapter, we study the impact of a large volume of action on the price formation process at the daily scale and at a larger scale (several days after the execution). Furthermore, we use our model to highlight new stylized facts discovered in our database. In a fourth part, we focus on a non-parametric estimation method for a one-dimensional Hawkes process. This method relies on the link between the self-covariance function and the kernel of the Hawkes process. In particular, we study the performance of this estimator in the direction of the squared error on Sobolev spaces and on a certain class containing "very" smooth functions.

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Let B be a Brownian motion and T its first hitting time of the level 1. For U a uniform random variable independent of B, we study in depth the distribution of T^{-1/2}B_{UT}, that is the rescaled Brownian motion sampled at uniform time. In particular, we show that this variable is centered.

This paper provides a simple approach for the consideration of quadratic BSDEs with bounded terminal conditions. Using solely probabilistic arguments, we retrieve the existence and uniqueness result derived via PDE-based methods by Kobylanski (2000) [14]. This approach is related to the study of quadratic BSDEs presented by Tevzadze (2008) [19]. Our argumentation, as in Tevzadze (2008) [19], highly relies on the theory of BMO martingales which was used for the first time for BSDEs by Hu et al. (2005) [12]. However, we avoid in our method any fixed point argument and use Malliavin calculus to overcome the difficulty. Our new scheme of proof allows also to extend the class of quadratic BSDEs, for which there exists a unique solution: we incorporate delayed quadratic BSDEs, whose driver depends on the recent past of the Y component of the solution. When the delay vanishes, we verify that the solution of a delayed quadratic BSDE converges to the solution of the corresponding classical non-delayed quadratic BSDE.

Considering a positive portfolio diffusion $X$ with negative drift, we investigate optimal stopping problems of the form $$ \inf_\theta \Esp{f\left(\frac{X_\theta}{\Sup_{s\in[0,\tau]}{X_s}}\right)}\.,$$ where $f$ is a non-increasing function, $\tau$ is the next random time where the portfolio $X$ crosses zero and $\theta$ is any stopping time smaller than $\tau$. Hereby, our motivation is the obtention of an optimal selling strategy minimizing the relative distance between the liquidation value of the portfolio and its highest possible value before it reaches zero. This paper unifies optimal selling rules observed for quadratic absolute distance criteria with bang-bang type ones. More precisely, we provide a verification result for the general stopping problem of interest and derive the exact solution for two classical criteria $f$ of the literature. For the power utility criterion $f:y \mapsto - {y^\la}$ with $\la>0$, instantaneous selling is always optimal, which is consistent with the observations of \cite{DaiJinZhoZho10} or \cite{ShiXuZho08} for the Black-Scholes model in finite horizon. On the contrary, for a relative quadratic error criterion, $f:y \mapsto {(1-y)^2}$, selling is optimal as soon as the process $X$ crosses a specified function $\varphi$ of its running maximum $X^*$. These results reinforce the idea that optimal stopping problems of similar type lead easily to selling rules of very different nature. Nevertheless, our numerical experiments suggest that the practical optimal selling rule for the relative quadratic error criterion is in fact very close to immediate selling.

The financial markets have expanded considerably over the last three decades and have seen the emergence of a variety of derivative products. The most widely used of these derivatives are American options.

In the first part, we present a non-parametric method for estimating option price sensitivities using random perturbation of the parameter of interest, Monte Carlo simulations and kernel regression. For an irregular function, the estimator converges faster than finite difference estimators, which is numerically verified. The 2nd part proposes an algorithm for solving decoupled EDSPR systems with jumps. The discretization error in time is parametric. And the statistical error is controlled. We present numerical examples on coupled systems of semi-linear PDE. The 3rd part studies the behavior of a fund manager, maximizing the intertemporal utility of his consumption, under a drawdown constraint. We obtain in explicit form the optimal strategy in infinite horizon, and we characterize the value function in finite horizon as the unique viscosity solution of the corresponding HJB equation.

This thesis presents three independent research topics belonging to the field of numerical methods and stochastic control with applications in financial mathematics. In the first part, we present a non-parametric method for estimating option price sensitivities. Using a random perturbation of the parameter of interest, we represent these sensitivities as conditional expectations, which we estimate using Monte Carlo simulations and kernel regression. Using integration by parts arguments, we propose several kernel estimators of these sensitivities, which do not require knowledge of the density of the underlying, and we obtain their asymptotic properties. When the payoff function is irregular, they converge faster than the finite difference estimators, which is verified numerically. The second part focuses on the numerical solution of decoupled systems of backward progressive stochastic differential equations. For Lipschitz coefficients, we propose a discretization scheme that converges faster than $n^{-1/2+e}$, for any $e>0$, when the time step $1/n$ tends to $0$, and under stronger regularity assumptions, the scheme reaches the parametric convergence speed. The statistical error of the algorithm due to the non-parametric approximation of conditional expectations is also controlled and we present examples of numerical solution of coupled systems of semi-linear PDEs. Finally, the last part of this thesis studies the behavior of a fund manager, maximizing the intertemporal utility of his consumption, under the constraint that the value of his portfolio does not fall below a fixed fraction of its current maximum. We consider a general class of utility functions, and a financial market composed of a risky asset with black-Scholes dynamics. When the manager sets an infinite time horizon, we obtain in explicit form his optimal investment and consumption strategy, as well as the value function of the problem. In a finite horizon, we characterize the value function as the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation.