CHASSAGNEUX Jean Francois

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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.

This thesis is about Stochastic Retrograde Differential Equations (SRDEs) reflected with two obstacles and their applications to zero-sum switching games, systems of partial differential equations, mean-field problems. There are two parts to this thesis. The first part deals with stochastic optimal switching and is composed of two works. In the first work, we show the existence of the solution of a system of reflexive EDSRs with interconnected bilateral obstacles in the general probabilistic framework. This problem is related to a zero-sum switching game. Then we address the question of the uniqueness of the solution. And finally we apply the obtained results to show that the associated PDE system has a unique solution in the viscosity sense, without the usual monotonicity condition. In the second work, we also consider a system of reflected PDEs with interconnected bilateral obstacles in the Markovian framework. The difference with the first work lies in the fact that the switching does not take place in the same way. This time when the switching is done, the system is put in the next state no matter which of the players decides to switch. This difference is fundamental and complicates the problem of the existence of the solution of the system. Nevertheless, in the Markovian framework we show this existence and give a uniqueness result using mainly Perron's method. Then, the link with a specific switching set is established in two settings. In the second part we study the one-dimensional reflected two-obstacle mean-field EDSR. Using the fixed point method, we show the existence and uniqueness of the solution in two frames, depending on the integrability of the data.

We introduce and study a new class of optimal switching problems, namely switching problem with controlled randomisation, where some extra-randomness impacts the choice of switching modes and associated costs. We show that the optimal value of the switching problem is related to a new class of multidimensional obliquely reflected BSDEs. These BSDEs allow as well to construct an optimal strategy and thus to solve completely the initial problem. The other main contribution of our work is to prove new existence and uniqueness results for these obliquely reflected BSDEs. This is achieved by a careful study of the domain of reflection and the construction of an appropriate oblique reflection operator in order to invoke results from [7].

This thesis deals with the theory of mean-field games. Most of it is devoted to mean-field games in which players can interact through their state and control law. We will use the terminology mean-field control game to designate such games. First, we make a structural assumption, which essentially consists in saying that the optimal dynamics depends on the control law in a lipschitzian way with a constant less than one. In this case, we prove several existence results for solutions to the mean control field game system, and a uniqueness result in short time. In a second step, we set up a numerical scheme and perform simulations for population motion models. In a third step, we show the existence and uniqueness when the control interaction satisfies a monotonicity condition. The last chapter concerns a numerical solution algorithm for mean-field games of variational type and without interaction via the control law. We use a preconditioning strategy by a multigrid method to obtain a fast convergence.

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This thesis is devoted to the theoretical and numerical study of the weak error of time and particle discretization of nonlinear Stochastic Differential Equations in the McKean sense. In the first part, we analyze the weak convergence speed of the time discretization of standard SDEs. More specifically, we study the convergence in total variation of the Euler-Maruyama scheme applied to d-dimensional DEs with a measurable drift coefficient and additive noise. We obtain, assuming that the drift coefficient is bounded, a weak order of convergence 1/2. By adding more regularity on the drift, namely a spatial divergence in the sense of L[rho]-space-uniform distributions in time for some [rho] greater than or equal to d, we reach a convergence order equal to 1 (within a logarithmic factor) at terminal time. In dimension 1, this result is preserved when the spatial derivative of the drift is a measure in space with a total mass bounded uniformly in time. In the second part of the thesis, we analyze the weak discretization error in both time and particles of two classes of nonlinear DHSs in the McKean sense. The first class consists of multi-dimensional SDEs with regular drift and diffusion coefficients in which the law dependence occurs through moments. The second class consists of one-dimensional SDEs with a constant diffusion coefficient and a singular drift coefficient where the law dependence occurs through the distribution function. We approximate the EDS by the Euler-Maruyama schemes of the associated particle systems and we obtain for both classes a weak order of convergence equal to 1 in time and in particles. In the second class, we also prove a result of chaos propagation of optimal order 1/2 in particles and a strong order of convergence equal to 1 in time and 1/2 in particles. All our theoretical results are illustrated by numerical simulations.

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.

This thesis contains two parts. In the first part, we prove two limit theorems of optimal quantization. The first limit theorem is the characterization of the convergence under the Wasserstein distance of a sequence of probability measures by the simple convergence of the quantization error functions. These results are established in Rd and also in a separable Hilbert space. The second limit theorem shows the speed of convergence of the optimal grids and the quantization performance for a sequence of probability measures which converge under the Wasserstein distance, in particular the empirical measure. The second part of this thesis focuses on the approximation and simulation of the McKean-Vlasov equation. We start this part by proving, by Feyel's method (see Bouleau (1988) [Section 7]), the existence and uniqueness of a strong solution of the McKean-Vlasov equation dXt = b(t, Xt, μt)dt + σ(t, Xt, μt)dBt under the condition that the coefficient functions b and σ are lipschitzian. Then, the convergence speed of the theoretical Euler scheme of the McKean-Vlasov equation is established and also the convex order functional results for the McKean-Vlasov equations with b(t,x,μ) = αx+β, α,β ∈ R. In the last chapter, the error of the particle method, several quantization-based schemes and a hybrid particle-quantization scheme are analyzed. At the end, two example simulations are illustrated: the Burgers equation (Bossy and Talay (1997)) in dimension 1 and the FitzHugh-Nagumo neural network (Baladron et al. (2012)) in dimension 3.

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.

Mean-field game systems (MFG) describe equilibrium configurations in differential games with an infinite number of infinitesimal agents. This thesis is structured around three different contributions to the theory of mean-field games. The main goal is to explore applications and extensions of this theory, and to propose new approaches and ideas to deal with the underlying mathematical issues. The first chapter first introduces the key concepts and ideas that we use throughout the thesis. We introduce the MFG problem and briefly explain the asymptotic connection with N-player differential games when N → ∞. We then present our main results and contributions. Chapter 2 explores an MFG model with a non-anticipatory interaction mode (myopic players). Unlike classical MFG models, we consider less rational agents who do not anticipate the evolution of the environment, but only observe the current state of the system, undergo changes, and take actions accordingly. We analyze the coupled PDE system resulting from this model, and establish the rigorous link with the corresponding N-Players game. We show that the population of agents can self-organize through a self-correcting process and converge exponentially fast to a well-known MFG equilibrium configuration. Chapters 3 and 4 concern the application of the MFG theory to the modeling of production and marketing processes of products with exhaustible resources (e.g. fossil fuels). In Chapter 3, we propose a variational approach for the study of the corresponding MFG system and analyze the deterministic limit (without demand fluctuations) in a regime where resources are renewable or abundant. In Chapter 4 we treat the MFG approximation by analyzing the asymptotic link between the N-player Cournot model and the MFG Cournot model when N is large. Finally, Chapter 5 considers an MFG model for the optimal execution of a portfolio of assets in a financial market. We explain our MFG model and analyze the resulting PDE system, then we propose a numerical method to compute the optimal execution strategy for an agent given its initial inventory, and present several simulations. Furthermore, we analyze the influence of trading activity on the intraday variation of the covariance matrix of asset returns. Next, we verify our findings and calibrate our model using historical trading data for a pool of 176 US stocks.

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In this paper, we prove new convergence results improving the ones by Chassagneux, Elie and Kharroubi [Ann. Appl. Probab. 22 (2012) 971–1007] for the discrete-time approximation of multidimensional obliquely reflected BSDEs. These BSDEs, arising in the study of switching problems, were considered by Hu and Tang [Probab. Theory Related Fields 147 (2010) 89–121] and generalized by Hamadène and Zhang [Stochastic Process. Appl.

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This thesis is devoted to the theoretical and numerical study of two main research topics: mean-reflected stochastic differential equations (SDEs) with jumps and backward stochastic differential equations (SRDEs) of the McKean-Vlasov type.The first work of my thesis establishes the propagation of chaos for mean-reflected SDEs with jumps. We first studied the existence and uniqueness of a solution. We then developed a numerical scheme via the particle system. Finally we obtained a convergence speed for this scheme.The second work of my thesis consists in studying the McKean-Vlasov type EDSRs. We have proved the existence and uniqueness of solutions of such equations, and we have proposed a numerical approximation based on the Wiener chaos decomposition as well as its convergence speed.The third work of my thesis is interested in another type of simulation for McKean-Vlasov type EDSRs. We have proposed a numerical scheme based on the approximation of the Brownian motion by a random walk and we have obtained a convergence speed for this scheme.Moreover, some numerical examples in these three works allow to notice the efficiency of our schemes and the convergence speeds announced by the theoretical results.

This thesis consists of two parts that can be read independently. In the first part of the thesis, we study hedging and option pricing problems related to a risk measure. Our main approach is the use of an asymmetric risk function and an asymptotic framework in which we obtain optimal solutions through nonlinear partial differential equations (PDEs).In the first chapter, we focus on the valuation and hedging of European options. We consider the problem of optimizing the residual risk generated by a discrete-time hedge in the presence of an asymmetric risk criterion. Instead of analyzing the asymptotic behavior of the solution of the associated discrete problem, we study the asymmetric residual risk measure integrated in a Markovian framework. In this context, we show the existence of this asymptotic risk measure. We then describe an asymptotically optimal hedging strategy via the solution of a totally nonlinear PDE. The second chapter applies this hedging method to the problem of valuing the output of a power plant. Since the power plant generates maintenance costs whether it is on or off, we are interested in reducing the risk associated with the uncertain revenues of this power plant by hedging with futures contracts. In the second part of the thesis, we consider several control problems related to economics and finance.The third chapter is dedicated to the study of a class of McKean-Vlasov (MKV) type problem with common noise, called conditional polynomial MKV. We reduce this polynomial class by Markov folding to finite dimensional control problems.We compare three different probabilistic techniques for numerically solving the reduced problem: quantization, control randomization regression, and delayed regression. We provide many numerical examples, such as portfolio selection with uncertainty about an underlying trend.In the fourth chapter, we solve dynamic programming equations associated with financial valuations in the energy market. We consider that a calibrated model for the underlyings is not available and that a small sample obtained from historical data is accessible.Moreover, in this context, we assume that futures contracts are often governed by hidden factors modeled by Markov processes. We propose a non-intrusive method to solve these equations through empirical regression techniques using only the historical log price of observable futures contracts.

This paper is dedicated to the presentation and the analysis of a numerical scheme for forward-backward SDEs of the McKean-Vlasov type, or equivalently for solutions to PDEs on the Wasserstein space. Because of the mean field structure of the equation, earlier methods for classical forward-backward systems fail. The scheme is based on a variation of the method of continuation. The principle is to implement recursively local Picard iterations on small time intervals. We establish a bound for the rate of convergence under the assumption that the decoupling field of the forward-bakward SDE (or equivalently the solution of the PDE) satisfies mild regularity conditions. We also provide numerical illustrations.

This thesis is devoted to the theoretical and numerical study of two nonlinear problems in the McKean sense in finance. In the first part, we address the problem of calibrating a model with local and stochastic volatility to take into account the prices of European vanilla options observed on the market. This problem results in the study of a nonlinear stochastic differential equation (SDE) in the McKean sense because of the presence in the diffusion coefficient of a conditional expectation of the stochastic volatility factor with respect to the SDE solution. We obtain the existence of the process in the particular case where the stochastic volatility factor is a jump process with a finite number of states. We also obtain the weak convergence at order 1 of the time discretization of the nonlinear DHS in the McKean sense for general stochastic volatility factors. In the industry, the calibration is efficiently performed using a regularization of the conditional expectation by a Nadaraya-Watson type kernel estimator, as proposed by Guyon and Henry-Labordère in [JGPHL]. We also propose a half-time numerical scheme and study the associated particle system that we compare to the algorithm proposed by [JGPHL]. In the second part of the thesis, we focus on a problem of contract valuation with margin calls, a problem that appeared with the application of new regulations since the financial crisis of 2008. This problem can be modeled by an anticipatory stochastic differential equation (SDE) with dependence on the law of the solution in the generator. We show that this equation is well-posed and propose an approximation of its solution using standard linear SRDEs when the liquidation time of the option in case of default is small. Finally, we show that the computation of the solutions of these standard EDSRs can be improved using the multilevel Monte Carlo method introduced by Giles in [G].

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.

This project investigates numerical methods for solving fully coupled forward-backward stochastic differential equations (FBSDEs) of McKean-Vlasov type. Having numerical solvers for such mean field FBSDEs is of interest because of the potential application of these equations to optimization problems over a large population, say for instance mean field games (MFG) and optimal mean field control problems. Theory for this kind of problems has met with great success since the early works on mean field games by Lasry and Lions, see \cite{Lasry_Lions}, and by Huang, Caines, and Malham\'{e}, see \cite{Huang}. Generally speaking, the purpose is to understand the continuum limit of optimizers or of equilibria (say in Nash sense) as the number of underlying players tends to infinity. When approached from the probabilistic viewpoint, solutions to these control problems (or games) can be described by coupled mean field FBSDEs, meaning that the coefficients depend upon the own marginal laws of the solution. In this note, we detail two methods for solving such FBSDEs which we implement and apply to five benchmark problems. The first method uses a tree structure to represent the pathwise laws of the solution, whereas the second method uses a grid discretization to represent the time marginal laws of the solutions. Both are based on a Picard scheme. importantly, we combine each of them with a generic continuation method that permits to extend the time horizon (or equivalently the coupling strength between the two equations) for which the Picard iteration converges.

In this paper, we study existence and uniqueness to multidimensional Reflected Backward Stochastic Differential Equations in an open convex domain, allowing for oblique directions of reflection. In a Markovian framework, combining a priori estimates for penalised equations and compactness arguments, we obtain existence results under quite weak assumptions on the driver of the BSDEs and the direction of reflection, which is allowed to depend on both Y and Z. In a non Markovian framework, we obtain existence and uniqueness result for direction of reflection depending on time and Y. We make use in this case of stability estimates that require some smoothness conditions on the domain and the direction of reflection.

The goal of this CIFRE thesis is to build a portfolio of intraday algorithmic trading strategies. Instead of considering prices as a function of time and randomness generally modeled by a Brownian motion, our approach consists in identifying the main signals to which order givers are sensitive in their decision making and then proposing a price model in order to build dynamic portfolio allocation strategies. In a second, more academic part, we present pricing work on European and Asian options.

This thesis studies the optimal control of McKean-Vlasov type dynamics and its applications in financial mathematics. The thesis contains two parts. In the first part, we develop the dynamic programming method for solving McKean-Vlasov type stochastic control problems. By using the appropriate admissible controls, we can reformulate the value function in terms of the law (resp. the conditional law) of the process as the only state variable and obtain the flow property of the law (resp. the conditional law) of the process, which allow us to obtain the principle of dynamic programming in all generality. Then we obtain the corresponding Bellman equation, based on the notion of differentiability with respect to probability measures introduced by P.L. Lions [Lio12] and the Itô formula for the probability stream. Finally we show the viscosity property and the uniqueness of the value function of the Bellman equation. In the first chapter, we summarize some useful results from differential calculus and stochastic analysis on the Wasserstein space. In the second chapter, we consider stochastic optimal control of nonlinear mean-field systems in discrete time. The third chapter studies the stochastic optimal control problem of McKean-Vlasov type EDS without common noise in continuous time where the coefficients can depend on the joint state and control law, and finally in the last chapter of this part we are interested in the optimal control of McKean-Vlasov type stochastic dynamics in the presence of common noise in continuous time. In the second part, we propose a robust portfolio allocation model allowing for uncertainty in the expected return and the correlation matrix of multiple assets, in a continuous time mean-variance framework. This problem is formulated as a mean-field differential game. We then show a separation principle for the associated problem. Our explicit results provide a quantitative justification for underdiversification, as shown in empirical studies.

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In a Markovian framework, we consider the problem of finding the minimal initial value of a controlled process allowing to reach a stochastic target with a given level of expected loss. This question arises typically in approximate hedging problems. The solution to this problem has been characterised by Bouchard et al. (SIAM J Control Optim 48(5):3123–3150, 2009) and is known to solve an Hamilton–Jacobi–Bellman PDE with discontinuous operator. In this paper, we prove a comparison theorem for the corresponding PDE by showing first that it can be rewritten using a continuous operator, in some cases. As an application, we then study the quantile hedging price of Bermudan options in the non-linear case, pursuing the study initiated in Bouchard et al. (J Financial Math 7(1):215–235, 2016).

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This thesis deals with various control and optimization problems for which only approximate solutions exist to date. On the one hand, we are interested in techniques to reduce or eliminate approximations in order to obtain more precise or even exact solutions. On the other hand, we develop new approximation methods to deal more quickly with larger scale problems. We study numerical methods for simulating stochastic differential equations and for improving expectation calculations. We implement quantization-type techniques for the construction of control variables and the stochastic gradient method for solving stochastic control problems. We are also interested in clustering methods related to quantization, as well as in information compression by neural networks. The problems studied are not only motivated by financial issues, such as stochastic control for option hedging in incomplete markets, but also by the processing of large media databases commonly referred to as Big data in Chapter 5. Theoretically, we propose different majorizations of the convergence of numerical methods on the one hand for the search of an optimal hedging strategy in incomplete market in chapter 3, on the other hand for the extension of the Beskos-Roberts technique of differential equation simulation in chapter 4. We present an original use of the Karhunen-Loève decomposition for a variance reduction of the expectation estimator in chapter 2.

We present in this section a direct approach to obtain the hedging price of a contingent claim, in the almost sure sense of super-replication or in the sense of a risk criterion (quantile hedging, expected shortfall, utility indifference). This approach, based on the notion of stochastic target, was initiated by Soner and Touzi [55] for the super-replication criterion, and then extended by Bouchard, Elie and Touzi [10] for the hedging under risk control, see also [8, 13] and [14].

Within a Markovian complete financial market, we consider the problem of hedging a Bermudan option with a given probability. Using stochastic target and duality arguments, we derive a backward numerical scheme for the Fenchel transform of the pricing function. This algorithm is similar to the usual American backward induction, except that it requires two additional Fenchel transformations at each exercise date. We provide numerical illustrations.

This article deals with the numerical approximation of Markovian backward stochastic differential equations (BSDEs) with generators of quadratic growth with respect to $z$ and bounded terminal conditions. We first study a slight modification of the classical dynamic programming equation arising from the time-discretization of BSDEs. By using a linearization argument and BMO martingales tools, we obtain a comparison theorem, a priori estimates and stability results for the solution of this scheme. Then we provide a control on the time-discretization error of order $\frac{1}{2}-\varepsilon$ for all $\varepsilon>0$. In the last part, we give a fully implementable algorithm for quadratic BSDEs based on quantization and illustrate our convergence results with numerical examples.

In complete markets, the notion of viable price is very satisfying as it leads to the definition of a unique no-arbitrage price. Moreover, this price is the solution of a replication problem, which can be characterised quite simply. In particular, we have seen in Chap. 4 that this price is given as the unique solution of a linear PDE in a Markovian framework.

In this chapter, we study a class of Markovian models for complete markets. This type of model is the most commonly used in practice. In these models, the underlying price process is solution to a Stochastic Differential Equation.

Stochastic volatility models are used when the option price is very sensitive to volatility (smile) moves, and when they cannot be explained by the evolution of the underlying asset itself, see e.g. [34]. This is typically the case for exotic options.

We present here the main characteristics of local volatility models in which the volatility of the risky assets is a function of time and of the spot value of the underlying. It is a standard in the industry. They are flexible enough to fit the vanilla option prices of all maturities, while preserving the completeness of the market. This permits a clear identification of the hedging strategy, see Chap. 4.

This first chapter is dedicated to discrete time markets. We first relate the absence of arbitrage opportunities to the existence of equivalent martingale measures, i.e. of equivalent probability measures that turn discounted asset prices into martingales. These measures are the basis of the whole pricing theory. They define the price intervals for derivatives products that are acceptable for the market. When the market is complete, meaning that any source of risk can be hedged perfectly by trading liquid assets, these intervals are reduced to one single point. This unique price allows one to hedge the corresponding derivative perfectly. However, in general, these intervals are not reduced to a singleton, and only their upper-bound, the so-called super-hedging price, permits to offset all risks by using a suitable dynamic hedging strategy. We shall study in details both European and American options. At the end of the chapter, the impact of portfolio constraints will also be discussed.

In this chapter, we extend the results obtained in discrete time markets to a continuous time setting. We work with Ito semimartingale models in which the risky assets are modeled as a diffusion driven by a Brownian motion. Note however that most of the results presented below remain true in much more general setting, see e.g. [23] and [24]. The most technical results will be stated without proofs.

This book covers the theory of derivatives pricing and hedging as well as techniques used in mathematical finance. The authors use a top-down approach, starting with fundamentals before moving to applications, and present theoretical developments alongside various exercises, providing many examples of practical interest.A large spectrum of concepts and mathematical tools that are usually found in separate monographs are presented here. In addition to the no-arbitrage theory in full generality, this book also explores models and practical hedging and pricing issues. Fundamentals and Advanced Techniques in Derivatives Hedging further introduces advanced methods in probability and analysis, including Malliavin calculus and the theory of viscosity solutions, as well as the recent theory of stochastic targets and its use in risk management, making it the first textbook covering this topic. Graduate students in applied mathematics with an understanding of probability theory and stochastic calculus will find this book useful to gain a deeper understanding of fundamental concepts and methods in mathematical finance.

This book covers the theory of derivatives pricing and hedging as well as techniques used in mathematical finance. The authors use a top-down approach, starting with fundamentals before moving to applications, and present theoretical developments alongside various exercises, providing many examples of practical interest.A large spectrum of concepts and mathematical tools that are usually found in separate monographs are presented here. In addition to the no-arbitrage theory in full generality, this book also explores models and practical hedging and pricing issues. Fundamentals and Advanced Techniques in Derivatives Hedging further introduces advanced methods in probability and analysis, including Malliavin calculus and the theory of viscosity solutions, as well as the recent theory of stochastic targets and its use in risk management, making it the first textbook covering this topic.Graduate students in applied mathematics with an understanding of probability theory and stochastic calculus will find this book useful to gain a deeper understanding of fundamental concepts and methods in mathematical finance.

This chapter is dedicated to the resolution of portfolio management problems and to the study of partial hedging strategies, based on a risk criteria. We shall mainly appeal to convex duality and calculus of variations arguments that turn out to be very powerful in complete markets: they will allow us to find explicit solutions.

We analyze a class of nonlinear partial dierential equations (PDEs) defined on the Euclidean space of dimension d times the Wasserstein space of d-dimensional probability measures with a finite second-order moment. We show that such equations admit a classical solutions for sufficiently small time intervals. Under additional constraints, we prove that their solution can be extended to arbitrary large intervals. These nonlinear PDEs arise in the recent developments in the theory of large population stochastic control. More precisely they are the so-called master equations corresponding to asymptotic equilibria for a large population of controlled players with mean-field interaction and subject to minimization constraints. The results in the paper are deduced by exploiting this connection. In particular, we study the differentiability with respect to the initial condition of the flow generated by a forward-backward stochastic system of McKean-Vlasov type. As a byproduct, we prove that the decoupling field generated by the forward-backward system is a classical solution of the corresponding master equation. Finally, we give several applications to mean-field games and to the control of McKean-Vlasov diffusion processes.

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.

In this paper, we study the qualitative behaviour of approximation schemes for Backward Stochastic Differential Equations (BSDEs) by introducing a new notion of numerical stability. For the Euler scheme, we provide sufficient conditions in the one-dimensional and multidimensional case to guarantee the numerical stability. We then perform a classical Von Neumann stability analysis in the case of a linear driver $f$ and exhibit necessary conditions to get stability in this case. Finally, we illustrate our results with numerical applications.

This book provides a comprehensive overview of the mathematical fundamentals of financial asset pricing and derivative hedging. The abstract theory of asset pricing is presented in detail in a general framework of discrete-time models before being extended to continuous-time models. It then presents in depth the hedging techniques applied in Markovian models: in complete market, in incomplete market or in the presence of portfolio constraints. The study relies heavily on the characterization of prices as solutions of partial differential equations in the classical sense or in the sense of viscosity solutions. The originality of this book also lies in the presentation of recent stochastic target techniques that allow the study of non-classical models (such as those used in high frequency trading) and the calculation of prices defined according to a risk criterion. Essential practical notions such as calibration, the impact of a model specification error or the ability to set up a dynamic hedge are also discussed. Each chapter is completed with a series of exercises and examples corresponding to industry standards.

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This thesis is composed of two independent parts that focus on the application of probability to the field of finance. The first part studies the regularity of the solutions of certain types of backward-looking stochastic differential equations (SRDEs) and reflexive differential equations, as well as numerical approximation schemes of these solutions. In finance, the main application is the pricing and hedging of American and gambling options, but our work is not limited to this framework. The proposed systematic method is based on the study of equations that are reflected only on a discrete time grid. In finance, these equations are interpreted as Bermuda options. In a general framework of multidimensional convex domains that can, under certain conditions, evolve randomly, we obtain convergence and regularity results for these discretely reflected equations that we extend to continuously reflected SDEs. The second part deals with a theoretical problem in mathematical finance. We deal with the valuation of American options in the framework of market models with proportional transaction costs, both for discrete and continuous time. We obtain an over-replication theorem for these contingent assets in the very general framework of ladlag option processes.