Sur un problem de contrˆole optimal stochastique pour certain aspect des ´equations differentielles stochastiques de type mean-field et applications

Cette thèse de doctorat s’inscrit dans le cadre de l’analyse stochastique dont le thème central est: les conditions necessaires et suffisantes sous forme du maximum stochastique de type champ moyen d’optimalite et de presque optimalite et ces applications. L’objectif de ce travail est d’etudier des problemes d’optimisation stochastique. Il s’agira ensuite de faire le point sur les conditions necessaires et suffisantes d’optimalite et de presque optimalite pour un system gouverne par des equations differentielles stochastiques de type champ moyen. Cette these s’articule autour de qua¬tre chapitres: Le chapitre 1 est essentiellement un rappel. La candidate présente quelques concepts et résultats qui lui permettent d’aborder son travail; tels que les processus stochastiques, l’esperance condition-nelle, les martingales, les formules d’Ito, les classes de contrôle stochastique,... etc. Dans le deuxieme chapitre, on a etablie et on a prouve les conditions necessaires et suffisantes de presque optimalite d’order 3b5{ 3bb} verifiees par un contrôle optimal stochastique, pour un system différentiel gouverne par des equations differentielles stochastiques EDSs. Le domaine de contrôle stochastique est suppose convexe. La methode utilisee est basee sur le lemme d’Ekeland. Les résultats obtenus dans le chapitre 2, sont tous nouveaux et font l’objet d’un premier article intitule : Boukaf Samira & Mokhtar Hafayed, & Ghebouli Messaoud: A study on optimal control problem with ex-error bound for stochastic systems with application to linear quadratic problem, International Journal of Dynamics and Control, Springer DOI: 10.1007/s40435-015-0178-x (2015). Dans le troisieme chapitre, on a demontré le principe du maximum stochastique de presque optimalite, oh le system est gouverne par des equations differentielles stochastiques progressive rétrogrades avec saut (FBSDEs). Ces resultats ont ete appliques pour résoudre un probleme d’optimisation en finance. Ces resultats generalisent le principe du maximum de Zhou (SIAM. Control. Optim. (36)-3, 929-947 (1998)). Les resultats obtenus dans le chapitre 3 sont tous nouveaux et font l’objet d’un deuxieme article intitule: Mokhtar Hafayed, & Abdelmadjid Abba & Samira Boukaf: On Zhou’s maximumprinciple for near- optimal control of mean-field forward-backward stochastic systems with jumps and its applications International Journal of Modelling, Identification and Control. 25 (1), 1-16, (2016). De plus, et dans le chapitre 4, on a prouve un principe du maximum stochastique de type de Pontryagin pour des systems gouvernes par FBSDEs avec saut. Ces resultats ont ete etabli avec M. Hafayed, et M. Tabet, sous le titre : Mokhtar Hafayed, & Moufida Tabet & Samira Boukaf: Mean-field maximum principle for optimal control of forward-backward stochastic systems with jumps and its application to mean-variance portfolio problem, Communication in Mathematics and Statistics, Springer, Doi: 10.1007/s40304- 015-0054-1, Volume 3, Issue 2, pp 163-186 (2015). Dans le chapitre 5, on a aborde un problème de contrôle singulier, où le problème est d’établir des conditions necessaires et suffisantes d’optimalite pour un control singulier ou le system est gouverne par des equations differentielles stochastiques progressive retrograde de type McKean-Vlasov. Dans ces cas, le domaine de contrôle admissible est suppose convexe. Les résultats obtenus dans le chapitre 5 sont tous nouveaux et font l’objet d’un article intitule : Mokhtar Hafayed, & Samira Boukaf & Yan Shi, & Shahlar Meherrem.: A McKean-Vlasov optimal mixed regular-singular control problem, for nonlinear stochastic systems with Poisson jump pro-cesses, Neurocomputing. Doi 10.1016/j.neucom.2015.11.082, Volume 182,19, pages 133-144 (2016

Comparison theorems for multi-dimensional BSDEs with jumps and applications to constrained stochastic linear-quadratic control

In this paper, we, for the first time, establish two comparison theorems for multi-dimensional backward stochastic differential equations with jumps. Our approach is novel and completely different from the existing results for one-dimensional case. Using these and other delicate tools, we then construct solutions to coupled two-dimensional stochastic Riccati equation with jumps in both standard and singular cases. In the end, these results are applied to solve a cone-constrained stochastic linear-quadratic and a mean-variance portfolio selection problem with jumps. Different from no jump problems, the optimal (relative) state processes may change their signs, which is of course due to the presence of jumps

ANALYTICAL STUDY AND GENERALISATION OF SELECTED STOCK OPTION VALUATION MODELS

In this work, the classical Black-Scholes model for stock option valuation on the basis of some stochastic dynamics was considered. As a result, a stock option val- uation model with a non-�xed constant drift coe�cient was derived. The classical Black-Scholes model was generalised via the application of the Constant Elasticity of Variance Model (CEVM) with regard to two cases: case one was without a dividend yield parameter while case two was with a dividend yield parameter. In both cases, the volatility of the stock price was shown to be a non-constant power function of the underlying stock price and the elasticity parameter unlike the constant volatility assumption of the classical Black-Scholes model. The It^o's theorem was applied to the associated Stochastic Di�erential Equations (SDEs) for conversion to Partial Dif- ferential Equations (PDEs), while two approximate-analytical methods: the Modi�ed Di�erential Transformation Method (MDTM) and the He's Polynomials Technique (HPT) were applied to the Black-Scholes model for stock option valuation; in both cases the integer and time-fractional orders were considered, and the results obtained proved the latter as an extension of the former. In addition, a nonlinear option pric- ing model was obtained when the constant volatility assumption of the classical linear Black-Scholes option pricing model was relaxed through the inclusion of transaction cost (Bakstein and Howison model). Thereafter, this nonlinear option pricing model was extended to a time-fractional ordered form, and its approximate-analytical solu- tions were obtained via the proposed solution technique. For e�ciency and reliability of the method, two cases with �ve examples were considered: Case 1 with two ex- amples for time-integer order, and Case 2 with three examples for time-fractional order, and the results obtained show that the time-fractional order form generalises the time-integer order form. Thus, the Black-Scholes and the Bakstein and Howison models for stock option valuation were generalised and extended to time-fractional order, and analytical solutions of these generalised models were provided

Modeling from a trader's perspective

I was trading professionally in the years 2006-2014 in the equity derivatives market. This thesis deals with two of the ideas inspired by my experience as a professional trader. The first topic deals with the pricing of a derivatives product in the market with a specific risk concentration. We call the product that causes the concentration a market driver. When the market driver exists, not only the market driver itself, but any derivatives product will not be priced fairly. We introduced a new model based on the Heston model that accounts for the concentration. The model leads to a pair of partial differential equations (PDEs): one semilinear parabolic PDE to price the market driver and one linear parabolic PDE to price all the other products. In solving the semilinear PDE, we use the policy improvement algorithm (PIA) to approximate the solution with those of linear PDEs. We show that the approximated solutions satisfy quadratic local convergence (QLC) which explains the efficiency of the algorithm. This efficiency of the algorithm is proved in a more general setup. The other idea sparked by my experience that is explored in the last chapter of the thesis concerns modeling technical analysis. Technical analysis is a family of methods that traders use to make decisions to purchase/sell assets. There is no mathematical proof that shows that they are correct as far as I am aware. We focus on one of the methods, the method of support and resistance levels, and used the optimal stopping argument to show the validity of the method. As far as I know, this is one of the first results to mathematically prove the effectiveness of a method in technical analysis

Applied Intertemporal Optimization

This textbook provides all tools required to easily solve intertemporal optimization problems in economics, finance, business administration and related disciplines. The focus of this textbook is on 'learning through examples' and gives a very quick access to all methods required by an undergraduate student, a PhD student and an experienced researcher who wants to explore new fields or confirm existing knowledge. Given that discrete and continuous time problems are given equal attention, insights gained in one area can be used for learning solutions methods more quickly in other contexts. This step-by-step approach is especially useful for the transition from deterministic to stochastic worlds. When it comes to stochastic methods in continuous time, the applied focus of this book is the most useful. Formulating and solving problems under continuous time uncertainty has never been explained in such a non-technical and highly accessible way.Intertemporal optimization, maximization, discrete time, continuous time, certainty, uncertainty, inserting, Lagrange, Hamiltonian, Dynamic Programming, Bellman equation, Ito's Lemma, Brownian motion, Poisson process, natural volatility