Cemracs 2017: numerical probabilistic approach to MFG

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 [29], and by Huang, Caines, and Malhamé, see [26]. 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

Complementary Lateral‐Spin–Orbit Building Blocks for Programmable Logic and In‐Memory Computing

Current-driven switching of nonvolatile spintronic materials and devices based on spin-orbit torques offer fast data processing speed, low power consumption, and unlimited endurance for future information processing applications. Analogous to conventional CMOS technology, it is important to develop a pair of complementary spin-orbit devices with differentiated magnetization switching senses as elementary building blocks for realizing sophisticated logic functionalities. Various attempts using external magnetic field or complicated stack/circuit designs have been proposed, however, plainer and more feasible approaches are still strongly desired. Here we show that a pair of two locally laser annealed perpendicular Pt/Co/Pt devices with opposite laser track configurations and thereby inverse field-free lateral spin-orbit torques (LSOTs) induced switching senses can be adopted as such complementary spin-orbit building blocks. By electrically programming the initial magnetization states (spin down/up) of each sample, four Boolean logic gates of AND, OR, NAND and NOR, as well as a spin-orbit half adder containing an XOR gate, were obtained. Moreover, various initialization-free, working current intensity-programmable stateful logic operations, including material implication (IMP) gate, were also demonstrated by regarding the magnetization state as a logic input. Our complementary LSOT building blocks provide a potentially applicable way towards future efficient spin logics and in-memory computing architectures.

Electric field control of deterministic current-induced magnetization switching in a hybrid ferromagnetic/ferroelectric structure

All-electrical and programmable manipulations of ferromagnetic bits are highly pursued for the aim of high integration and low energy consumption in modern information technology1, 2, 3. Methods based on the spin–orbit torque switching4, 5, 6 in heavy metal/ferromagnet structures have been proposed with magnetic field7, 8, 9, 10, 11, 12, 13, 14, 15, and are heading toward deterministic switching without external magnetic field16, 17. Here we demonstrate that an in-plane effective magnetic field can be induced by an electric field without breaking the symmetry of the structure of the thin film, and realize the deterministic magnetization switching in a hybrid ferromagnetic/ferroelectric structure with Pt/Co/Ni/Co/Pt layers on PMN-PT substrate. The effective magnetic field can be reversed by changing the direction of the applied electric field on the PMN-PT substrate, which fully replaces the controllability function of the external magnetic field. The electric field is found to generate an additional spin–orbit torque on the CoNiCo magnets, which is confirmed by macrospin calculations and micromagnetic simulations

Étude de méthodes numériques pour certaines équations différentielles stochastiques en finance et modélisation de la distribution du capital dans le marché financier

Nous apportons dans cette thèse quelques contributions à la modélisation du marché finan-cier dans le cadre de la théorie stochastique du portefeuille et à l’étude des méthodes numé-riques pour quelques équations différentielles stochastiques en modélisation financière et en théorie des jeux. Nous modélisons le marché par des poids relatifs des actifs et nous étudions un schéma probabiliste pour les lois marginales des solutions des EDS de McKean-Vlasov. Nous proposons également une représentation probabiliste et des formules d'intégration par parties à des modèles à volatilité stochastique pour obtenir des estimateurs Monte-Carlo sans biais du prix et des sensibilités. Enfin nous présentons deux algorithmes pour la résolution numérique des EDSPRs issues des jeux à champ moyenIn this thesis, we make some contributions to the modeling of the financial market in the context of stochastic portfolio theory, as well as to the study of numerical methods for some stochastic differential equations in financial modeling and game theory. The market is mod-eled by relative weights of assets and we study a probabilistic scheme for marginal laws of solutions of McKean-Vlasov SDEs. We also develop a probabilistic representation and inte-gration by parts formulae for some stochastic volatility models to obtain unbiased Monte-Carlo estimators of price and sensitivities. Finally we present two algorithms for numerical resolution of FBSDEs arising from mean-field game

Étude de méthodes numériques pour certaines équations différentielles stochastiques en finance et modélisation de la distribution du capital dans le marché financier

In this thesis, we make some contributions to the modeling of the financial market in the context of stochastic portfolio theory, as well as to the study of numerical methods for some stochastic differential equations in financial modeling and game theory. The market is mod-eled by relative weights of assets and we study a probabilistic scheme for marginal laws of solutions of McKean-Vlasov SDEs. We also develop a probabilistic representation and inte-gration by parts formulae for some stochastic volatility models to obtain unbiased Monte-Carlo estimators of price and sensitivities. Finally we present two algorithms for numerical resolution of FBSDEs arising from mean-field gamesNous apportons dans cette thèse quelques contributions à la modélisation du marché finan-cier dans le cadre de la théorie stochastique du portefeuille et à l’étude des méthodes numé-riques pour quelques équations différentielles stochastiques en modélisation financière et en théorie des jeux. Nous modélisons le marché par des poids relatifs des actifs et nous étudions un schéma probabiliste pour les lois marginales des solutions des EDS de McKean-Vlasov. Nous proposons également une représentation probabiliste et des formules d'intégration par parties à des modèles à volatilité stochastique pour obtenir des estimateurs Monte-Carlo sans biais du prix et des sensibilités. Enfin nous présentons deux algorithmes pour la résolution numérique des EDSPRs issues des jeux à champ moye

Probabilistic representation of integration by parts formulae for some stochastic volatility models with unbounded drift

In this paper, we establish a probabilistic representation as well as some integration by parts formulae for the marginal law at a given time maturity of some stochastic volatility model with unbounded drift. Relying on a perturbation technique for Markov semigroups, our formulae are based on a simple Markov chain evolving on a random time grid for which we develop a tailor-made Malliavin calculus. Among other applications, an unbiased Monte Carlo path simulation method stems from our formulas so that it can be used in order to numerically compute with optimal complexity option prices as well as their sensitivities with respect to the initial values or Greeks in finance, namely the Delta and Vega, for a large class of non-smooth European payoff. Numerical results are proposed to illustrate the efficiency of the method

O-GlcNAcylation Is Essential for Autophagy in Cardiomyocytes

Since both O-GlcNAcylation and autophagy sense intracellular nutrient level, the alteration of those two pathways plays substantial roles in the progression of heart failure. Hence, determining the relationship between O-GlcNAcylation and autophagy is imperative to understand, prevent, and treat heart failure. However, the mechanism on how O-GlcNAcylation regulates autophagy in the heart is poorly investigated. In this study, we demonstrated that O-GlcNAcylation is required for autophagy in cardiomyocytes by utilizing an O-linked β-N-acetylglucosamine transferase (OGT) cardiomyocyte-specific knockout mouse model for the first time. We also identified that OGT might regulate the initiation of autophagy in cardiomyocytes through promoting the activity of ULK1 by O-GlcNAcylation. In conclusion, our findings provide new insights into the molecular mechanisms underlying heart dysfunction and benefit the development of treatments for heart failure