OUDJANE Nadia

We propose a fully backward representation of semilinear PDEs with application to stochastic control. Based on this, we develop a fully backward Monte-Carlo scheme allowing to generate the regression grid, backwardly in time, as the value function is computed. This offers two key advantages in terms of computational efficiency and memory. First, the grid is generated adaptively in the areas of interest and second, there is no need to store the entire grid. The performances of this technique are compared in simulations to the traditional Monte-Carlo forward-backward approach on a control problem of thermostatic loads.

Aggregative games have many industrial applications, and computing an equilibrium in those games is challenging when the number of players is large. In the framework of atomic aggregative games with coupling constraints, we show that variational Nash equilibria of a large aggregative game can be approximated by a Wardrop equilibrium of an auxiliary population game of smaller dimension. Each population of this auxiliary game corresponds to a group of atomic players of the initial large game. This approach enables an efficient computation of an approximated equilibrium, as the variational inequality characterizing the Wardrop equilibrium is of smaller dimension than the initial one. This is illustrated on an example in the smart grid context.

Usually Fokker-Planck type partial differential equations (PDEs) are well-posed if the initial condition is specified. In this paper, alternatively, we consider the inverse problem which consists in prescribing final data: in particular we give sufficient conditions for existence and uniqueness. In the second part of the paper we provide a probabilistic representation of those PDEs in the form a solution of a McKean type equation corresponding to the time-reversal dynamics of a diffusion process.

We consider the framework of high dimensional stochastic control problem, in which the controls are aggregated in the cost function. As first contribution we introduce a modified problem, whose optimal control is under some reasonable assumptions an ε-optimal solution of the original problem. As second contribution, we present a decentralized algorithm whose convergence to the solution of the modified problem is established. Finally, we study the application to a problem of coordination of energy consumption and production of domestic appliances.

This paper shows the existence of O(1/n^γ)-Nash equilibria in n-player noncooperative aggregative games where the players' cost functions depend only on their own action and the average of all the players' actions, and is lower semicontinuous in the former while γ-Hölder continuous in the latter. Neither the action sets nor the cost functions need to be convex. For an important class of aggregative games which includes congestion games with γ being 1, a proximal best-reply algorithm is used to construct an O(1/n)-Nash equilibria with at most O(n^3) iterations. These results are applied in a numerical example of demand-side management of the electricity system. The asymptotic performance of the algorithm is illustrated when n tends to infinity.

No summary available.

An important part of the Smart Grid literature on residential Demand Response deals with game-theoretic consumption models. Among those papers, the hourly billing model is of special interest as an intuitive and fair mechanism. We focus on this model and answer to several theoretical and practical questions. First, we prove the uniqueness of the consumption profile corresponding to the Nash equilibrium, and we analyze its efficiency by providing a bound on the Price of Anarchy. Next, we address the computational issue of the equilibrium profile by providing two algorithms: the cycling best response dynamics and a projected gradient descent method, and by giving an upper bound on their convergence rate to the equilibrium. Last, we simulate this demand response framework in a stochastic environment where the parameters depend on forecasts. We show numerically the relevance of an online demand response procedure, which reduces the impact of inaccurate forecasts.

We consider a resource allocation problem involving a large number of agents with individual constraints subject to privacy, and a central operator whose objective is to optimizing a global, possibly non-convex, cost while satisfying the agents' constraints. We focus on the practical case of the management of energy consumption flexibilities by the operator of a microgrid. This paper provides a privacy-preserving algorithm that does compute the optimal allocation of resources, avoiding each agent to reveal her private information (constraints and individual solution profile) neither to the central operator nor to a third party. Our method relies on an aggregation procedure: we maintain a global allocation of resources, and gradually disaggregate this allocation to enforce the satisfaction of private constraints, by a protocol involving the generation of polyhedral cuts and secure multiparty computations (SMC). To obtain these cuts, we use an alternate projections method à la Von Neumann, which is implemented locally by each agent, preserving her privacy needs. Our theoretical and numerical results show that the method scales well as the number of agents gets large, and thus can be used to solve the allocation problem in high dimension, while addressing privacy issues.

This paper presents a partial state of the art about the topic of representation of generalized Fokker-Planck Partial Differential Equations (PDEs) by solutions of McKean Feynman-Kac Equations (MFKEs) that generalize the notion of McKean Stochastic Differential Equations (MSDEs). While MSDEs can be related to non-linear Fokker-Planck PDEs, MFKEs can be related to non-conservative non-linear PDEs. Motivations come from modeling issues but also from numerical approximation issues in computing the solution of a PDE, arising for instance in the context of stochastic control. MFKEs also appear naturally in representing final value problems related to backward Fokker-Planck equations.

We consider a resource allocation problem involving a large number of agents with individual constraints subject to privacy, and a central operator whose objective is to optimizing a global, possibly non-convex, cost while satisfying the agents' constraints. We focus on the practical case of the management of energy consumption flexibilities by the operator of a microgrid. This paper provides a privacy-preserving algorithm that does compute the optimal allocation of resources, avoiding each agent to reveal her private information (constraints and individual solution profile) neither to the central operator nor to a third party. Our method relies on an aggregation procedure: we maintain a global allocation of resources, and gradually disaggregate this allocation to enforce the satisfaction of private constraints, by a protocol involving the generation of polyhedral cuts and secure multiparty computations (SMC). To obtain these cuts, we use an alternate projections method à la Von Neumann, which is implemented locally by each agent, preserving her privacy needs. Our theoretical and numerical results show that the method scales well as the number of agents gets large, and thus can be used to solve the allocation problem in high dimension, while addressing privacy issues.

We consider a resource allocation problem involving a large number of agents with individual constraints subject to privacy, and a central operator whose objective is to optimize a global, possibly nonconvex, cost while satisfying the agents' constraints, for instance an energy operator in charge of the management of energy consumption flexibilities of many individual consumers. We provide a privacy-preserving algorithm that does compute the optimal allocation of resources, avoiding each agent to reveal her private information (constraints and individual solution profile) neither to the central operator nor to a third party. Our method relies on an aggregation procedure: we compute iteratively a global allocation of resources, and gradually ensure existence of a disaggregation, that is individual profiles satisfying agents' private constraints, by a protocol involving the generation of polyhedral cuts and secure multiparty computations (SMC). To obtain these cuts, we use an alternate projection method, which is implemented locally by each agent, preserving her privacy needs. We adress especially the case in which the local and global constraints define a transportation polytope. Then, we provide theoretical convergence estimates together with numerical results, showing that the algorithm can be effectively used to solve the allocation problem in high dimension, while addressing privacy issues.

The evolution of energy storage allows the development of innovative methods of energy management at a local scale. Microgrids are an emerging form of small-scale power systems with local generation, energy storage and in particular an Energy Management System (EMS). Numerous studies and scientific researches have been conducted to propose various strategies for the implementation of these EMS. Nevertheless, there is no clear and formal articulation of these methods that would allow their comparison. One of the main difficulties for EMS is the management of the dynamics of the different energy systems. The current variations go at the speed of the electron, the solar photovoltaic energy production varies according to the clouds and different storage technologies can react more or less quickly to these unpredictable phenomena. In this manuscript, we study a mathematical formalism and algorithms based on the theory of multi-step stochastic optimization and Dynamic Programming. This formalism allows to model and solve inter-temporal decision problems in the presence of uncertainties, using temporal decomposition methods that we apply to energy management problems. In the first part of this thesis, "Contributions to time decomposition in multi-step stochastic optimization", we present the general formalism we use to time decompose stochastic optimization problems with a large number of time steps. We then classify different optimal control methods within this formalism. In the second part, "Stochastic optimization of energy storage for microgrid management", we compare different methods, introduced in the first part, on real cases. In a first step, we control a battery as well as ventilations in a subway station recovering energy from train braking, by comparing four different algorithms. In a second step, we show how these algorithms could be implemented on a real system using a hierarchical control architecture of DC microgrids. The studied microgrid connects this time photovoltaic energy to a battery, a super-capacitor and an electrical load. Finally, we apply the temporal block decomposition formalism presented in the first part to address a battery charge management problem but also its long term aging. This last chapter introduces two algorithms based on temporal block decomposition that could be used for hierarchical control of micro networks or stochastic optimization problems with a large number of time steps. In the third and last part, "Software and Experiments", we present DynOpt.jl a package developed in the Julia language which has allowed the development of all the applications of this thesis and many others. We finally study the use of this package in a real case of energy system control: the intelligent management of the temperature in a house with the Sense City equipment.

We analyze the well-posedness of a so called McKean Feynman-Kac Equation (MFKE), which is a McKean type equation with a Feynman-Kac perturbation. We provide in particular weak and strong existence conditions as well as pathwise uniqueness conditions without strong regularity assumptions on the coefficients. One major tool to establish this result is a representation theorem relating the solutions of MFKE to the solutions of a nonconservative semilinear parabolic Partial Differential Equation (PDE).

Computing an equilibrium in congestion games can be challenging when the number of players is large. Yet, it is a problem to be addressed in practice, for instance to forecast the state of the system and be able to control it. In this work, we analyze the case of generalized atomic congestion games, with coupling constraints, and with players that are heterogeneous through their action sets and their utility functions. We obtain an approximation of the variational Nash equilibria-a notion generalizing Nash equilibria in the presence of coupling constraints-of a large atomic congestion game by an equilibrium of an auxiliary population game, where each population corresponds to a group of atomic players of the initial game. Because the variational inequalities characterizing the equilibrium of the auxiliary game have smaller dimension than the original problem, this approach enables the fast computation of an estimation of equilibria in a large congestion game with thousands of heterogeneous players.

Electricity grids have to absorb an increasing production of renewable energy in a decentralized way. Their optimal management leads to specific problems. We study in this thesis the mathematical formulation of such problems as multi-step stochastic optimization problems. We analyze more specifically the time and space decomposition of such problems. In the first part of this manuscript, Time Decomposition for the Optimization of Domestic Microgrid Management, we apply stochastic optimization methods to small microgrid management. We compare different optimization algorithms on two examples: the first one considers a domestic microgrid equipped with a battery and a micro-cogeneration plant. The second one considers another domestic microgrid, this time equipped with a battery and solar panels. In the second part, Temporal and spatial decomposition of large optimization problems, we extend the previous studies to larger microgrids, with different units and storages connected together. The frontal solution of such large problems by Dynamic Programming proves impractical. We propose two original algorithms to overcome this problem by mixing a temporal decomposition with a spatial decomposition --- by prices or by resources. In the last part, Contributions to the Stochastic Dual Dynamic Programming algorithm, we focus on the emph{Stochastic DualDynamic Programming} (SDDP) algorithm which is currently a reference method for solving multi-time step stochastic optimization problems. We study a new stopping criterion for this algorithm based on a dual version of SDDP, which allows to obtain a deterministic upper bound for the primal problem.

No summary available.

We propose a nonlinear forward Feynman-Kac type equation, which represents the solution of a non-conservative semilinear parabolic Partial Differential Equations (PDE). We show in particular existence and uniqueness. The solution of that type of equation can be approached via a weighted particle system.

The paper is devoted to the construction of a probabilistic particle algorithm. This is related to nonlin-ear forward Feynman-Kac type equation, which represents the solution of a nonconservative semilinear parabolic Partial Differential Equations (PDE). Illustrations of the efficiency of the algorithm are provided by numerical experiments.

We provide a representation result of parabolic semi-linear PD-Es, with polynomial nonlinearity, by branching diffusion processes. We extend the classical representation for KPP equations, introduced by Skorokhod [23], Watanabe [27] and McKean [18], by allowing for polynomial nonlinearity in the pair (u, Du), where u is the solution of the PDE with space gradient Du. Similar to the previous literature, our result requires a non-explosion condition which restrict to " small maturity " or " small nonlinearity " of the PDE. Our main ingredient is the automatic differentiation technique as in [15], based on the Malliavin integration by parts, which allows to account for the nonlin-earities in the gradient. As a consequence, the particles of our branching diffusion are marked by the nature of the nonlinearity. This new representation has very important numerical implications as it is suitable for Monte Carlo simulation. Indeed, this provides the first numerical method for high dimensional nonlinear PDEs with error estimate induced by the dimension-free Central limit theorem. The complexity is also easily seen to be of the order of the squared dimension. The final section of this paper illustrates the efficiency of the algorithm by some high dimensional numerical experiments.

We compare two Demand Side Management (DSM) mechanisms, introduced respectively by Mohsenian-Rad et al (2010) and Baharlouei et al (2012), in terms of efficiency and fairness. Each mechanism defines a game where the consumers optimize their flexible consumption to reduce their electricity bills. Mohsenian-Rad et al propose a daily mechanism for which they prove the social optimality. Baharlouei et al propose a hourly billing mechanism for which we give theoretical results: we prove the uniqueness of an equilibrium in the associated game and give an upper bound on its price of anarchy. We evaluate numerically the two mechanisms, using real consumption data from Pecan Street Inc. The simulations show that the equilibrium reached with the hourly mechanism is socially optimal up to 0.1%, and that it achieves an important fairness property according to a quantitative indicator we define. We observe that the two DSM mechanisms avoid the synchronization effect induced by non- game theoretic mechanisms, e.g. Peak/OffPeak hours contracts.

This thesis focuses on valuation and financial strategies for hedging risks in energy markets. These markets present particularities that distinguish them from standard financial markets, notably illiquidity and incompleteness. Illiquidity is reflected in high transaction costs and constraints on volumes traded. Incompleteness is the inability to perfectly replicate derivatives. We focus on different aspects of market incompleteness. The first part deals with valuation in Lévy models. We obtain an approximate formula for the indifference price and we measure the minimum premium to be brought over the Black-Scholes model. The second part concerns the valuation of spread options in the presence of stochastic correlation. Spread options deal with the price difference between two underlying assets -- for example gas and electricity -- and are widely used in the energy markets. We propose an efficient numerical procedure to calculate the price of these options. Finally, the third part deals with the valuation of a product with an exogenous risk for which forecasts exist. We propose an optimal dynamic strategy in the presence of volume risk, and apply it to the valuation of wind farms. In addition, a section is devoted to asymptotic optimal strategies in the presence of transaction costs.

In Demand Response programs, price incentives might not be sufficient to modify residential consumers load profile. Here, we consider that each consumer has a preferred profile and a discomfort cost when deviating from it. Consumers can value this discomfort at a varying level that we take as a parameter. This work analyses Demand Response as a game theoretic environment. We study the equilibria of the game between consumers with preferences within two different dynamic pricing mechanisms, respectively the daily proportional mechanism introduced by Mohsenian-Rad et al, and an hourly proportional mechanism. We give new results about equilibria as functions of the preference level in the case of quadratic system costs and prove that, whatever the preference level, system costs are smaller with the hourly mechanism. We simulate the Demand Response environment using real consumption data from PecanStreet database. While the Price of Anarchy remains always close to one up to 0.1% with the hourly mechanism, it can be more than 10% bigger with the daily mechanism.

No summary available.

We discuss numerical aspects related to a new class of nonlinear Stochastic Differential Equations in the sense of McKean, which are supposed to represent non conservative nonlinear Partial Differential equations (PDEs). We propose an original interacting particle system for which we discuss the propagation of chaos. We consider a time-discretized approximation of this particle system to which we associate a random function which is proved to converge to a solution of a regularized version of a nonlinear PDE.

In this thesis, we propose a progressive (forward) approach for the probabilistic representation of nonlinear and nonconservative Partial Differential Equations (PDEs), allowing to develop a particle-based algorithm to numerically estimate their solutions. The Nonlinear Stochastic Differential Equations of McKean type (NLSDE) studied in the literature constitute a microscopic formulation of a phenomenon modeled macroscopically by a conservative PDE. A solution of such a NLSDE is the data of a couple $(Y,u)$ where $Y$ is a solution of a stochastic differential equation (SDE) whose coefficients depend on $u$ and $t$ such that $u(t,cdot)$ is the density of $Y_t$. The main contribution of this thesis is to consider nonconservative PDEs, i.e. conservative PDEs perturbed by a nonlinear term of the form $Lambda(u,nabla u)u$. This implies that a pair $(Y,u)$ will be a solution of the associated probabilistic representation if $Y$ is still a stochastic process and the relation between $Y$ and the function $u$ will then be more complex. Given the law of $Y$, the existence and uniqueness of $u$ are proved by a fixed point argument via an original Feynmann-Kac formulation.

We introduce a new class of nonlinear Stochastic Differential Equations in the sense of McKean, related to non conservative nonlinear Partial Differential equations (PDEs). We discuss existence and uniqueness pathwise and in law under various assumptions.

We introduce a new class of nonlinear Stochastic Differential Equations in the sense of McKean, related to non conservative nonlinear Partial Differential equations (PDEs). We discuss existence and uniqueness pathwise and in law under various assumptions. We propose an original interacting particle system for which we discuss the propagation of chaos. To this system, we associate a random function which is proved to converge to a solution of a regularized version of PDE.

We investigate the problem of pricing and hedging derivatives of Electricity Futures contract when the underlying asset is not available. We propose to use a cross hedging strategy based on the Futures contract covering the larger delivery period. A quick overview of market data shows a basis risk for this market incompleteness. For that purpose we formulate the pricing problem in a stochastic target form along the lines of Bouchard and al. (2008), with a moment loss function. Following the same techniques as in the latter, we avoid to demonstrate the uniqueness of the value function by comparison arguments and explore convex duality methods to provide a semi-explicit solution to the problem. We then propose numerical results to support the new hedging strategy and compare our method to the Black-Scholes naive approach.

We investigate the problem of pricing and hedging derivatives of Electricity Futures contract when the underlying asset is not available. We propose to use a cross hedging strategy based on the Futures contract covering the larger delivery period. A quick overview of market data shows a basis risk for this market incompleteness. For that purpose we formulate the pricing problem in a stochastic target form along the lines of Bouchard and al. (2008), with a moment loss function. Following the same techniques as in the latter, we avoid to demonstrate the uniqueness of the value function by comparison arguments and explore convex duality methods to provide a semi-explicit solution to the problem. We then propose numerical results to support the new hedging strategy and compare our method to the Black-Scholes naive approach.

For a large class of vanilla contingent claims, we establish an explicit F\"ollmer-Schweizer decomposition when the underlying is an exponential of an additive process. This allows to provide an efficient algorithm for solving the mean variance hedging problem. Applications to models derived from the electricity market are performed.

Given a process with independent increments $X$ (not necessarily a martingale) and a large class of square integrable r.v. $H=f(X_T)$, $f$ being the Fourier transform of a finite measure $\mu$, we provide explicit Kunita-Watanabe and Follmer-Schweizer decompositions. The representation is expressed by means of two significant maps: the expectation and derivative operators related to the characteristics of $X$. We also provide an explicit expression for the variance optimal error when hedging the claim $H$ with underlying process $X$. Those questions are motivated by finding the solution of the celebrated problem of global and local quadratic risk minimization in mathematical finance.

For a large class of vanilla contingent claims, we establish an explicit Föllmer-Schweizer decomposition when the underlying is an exponential of an additive process.
This allows to provide an efficient algorithm for solving the
mean variance hedging problem.
Applications to models derived from the electricity market are performed.

The problem of nonlinear filtering consists in computing in an approximate way the conditional law of a Markovian process called signal, indirectly related to an observational process of which a realization is known. The aim of this thesis is to propose new particle algorithms for the solution of the nonlinear filtering problem. The first idea developed here underlines the strong link between the stability properties of the optimal filter and the long time behavior of some approach filters. We study the sensitivity of the optimal filter to different types of local perturbations intervening in its evolution at each time step. The key role of the Hilbert projective metric is highlighted, which allows the time-uniform control of the global error induced in the perturbed filter provided that the signal verifies certain ergodicity conditions. In particular, these results show that under these same ergodicity conditions, the two particle filters initially proposed in the literature (weighted monte carlo filter and particle filter with interaction) converge uniformly in time, towards the optimal filter. However, a more general analysis of the classical particle methods shows their weakness especially in the case of low noise systems and leads us to propose a new type of particle filters using a finer local perturbation. Regularization is the key step of this perturbation. It is based on an extension of the theory of density estimation by kernels and allows to replace the discrete approximation provided by the classical particle filters in a smooth approximation. The two types of resulting particle filters called pre-regularized and post-regularized filters are analyzed. They are then applied in simulations to different tracking problems.