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Author: Brahim El Asri
Publisher:
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Languages : en
Pages : 107
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We study optimal switching and Lр-solution for doubly reflected backward stochastic differential equations. In the first part, we show existence and uniqueness of a solution for a system of m variational partial differential inequalities with inter-connected obstacles. This system is the deterministic version of the Verification Theorem of the Markovian optimal m-states switching problem. The switching cost functions are arbitrary. In the second part we study the problem of the deterministic version of the Verification Theorem for the optimal m-states switching in infinite horizon under Markovian framework with arbitrary switching cost functions. The problem is formulated as an extended impulse control problem and solved by means of probabilistic tools such as the Snell envelop of processes and reflected backward stochastic differential equations. A viscosity solutions approach is employed to carry out a fine analysis on the associated system of m variational inequalities with inter-connected obstacles. We show that the vector of value functions of the optimal problem is the unique viscosity solution to the system. Finally in the third part, we deal the problem of existence and uniqueness of a solution for à backward stochastic differential equation (BSDE for short) with two strictly separated continuous reflecting barriers in the case when the terminal value, the generator and the obstacle process are Lр-integrable with р Є (1, 2). The main idea is to use the concept of local solution to construct the global one. As applications, we obtain new results in zerosum Dynkin games and in double obstacle variational inequalities theories.
Author: Brahim El Asri
Publisher:
ISBN:
Category :
Languages : en
Pages : 107
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Book Description
We study optimal switching and Lр-solution for doubly reflected backward stochastic differential equations. In the first part, we show existence and uniqueness of a solution for a system of m variational partial differential inequalities with inter-connected obstacles. This system is the deterministic version of the Verification Theorem of the Markovian optimal m-states switching problem. The switching cost functions are arbitrary. In the second part we study the problem of the deterministic version of the Verification Theorem for the optimal m-states switching in infinite horizon under Markovian framework with arbitrary switching cost functions. The problem is formulated as an extended impulse control problem and solved by means of probabilistic tools such as the Snell envelop of processes and reflected backward stochastic differential equations. A viscosity solutions approach is employed to carry out a fine analysis on the associated system of m variational inequalities with inter-connected obstacles. We show that the vector of value functions of the optimal problem is the unique viscosity solution to the system. Finally in the third part, we deal the problem of existence and uniqueness of a solution for à backward stochastic differential equation (BSDE for short) with two strictly separated continuous reflecting barriers in the case when the terminal value, the generator and the obstacle process are Lр-integrable with р Є (1, 2). The main idea is to use the concept of local solution to construct the global one. As applications, we obtain new results in zerosum Dynkin games and in double obstacle variational inequalities theories.
Author: Hao Wang
Publisher:
ISBN:
Category :
Languages : en
Pages : 153
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The main objective of the thesis is to study the existence and uniqueness of solutions of reflected backward stochastic differential equations and to relate this notion to the study of the problems such as the reversible investment or so-called optimal switching problem, the mixed zero-sum stochastic differential games and the probabilistic interpretation of the weak solution of partial differential equations, either in viscosity sense or in Sobolev space under different framework.
Author: Xuzhe Zhao
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Category :
Languages : en
Pages :
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There are three main results in this thesis. The first is existence and uniqueness of the solution in viscosity sense for a system of nonlinear m variational integral-partial differential equations with interconnected obstacles. From the probabilistic point of view, this system is related to optimal stochastic switching problem when the noise is driven by a Lévy process. As a by-product we obtain that the value function of the switching problem is continuous and unique solution of its associated Hamilton-Jacobi-Bellman system of equations. Next, we study a general class of min-max and max-min nonlinear second-order integral-partial variational inequalities with interconnected bilateralobstacles, related to a multiple modes zero-sum switching game with jumps. Using Perron's method and by the help of systems of penalized unilateral reflected backward SDEs with jumps, we construct a continuous with polynomial growth viscosity solution, and a comparison result yields the uniqueness of the solution. At last, we deal with the solutions of systems of PDEs with bilateral inter-connected obstacles of min-max and max-min types in the Brownian framework. These systems arise naturally in stochastic switching zero-sum game problems. We show that when the switching costs of one side are smooth, the solutions of the min-max and max-min systems coincide. Furthermore, this solution is identified as the value function of the zero-sum switching game.
Author: Situ Rong
Publisher: CRC Press
ISBN: 9781584881254
Category : Mathematics
Languages : en
Pages : 228
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Many important physical variables satisfy certain dynamic evolution systems and can take only non-negative values. Therefore, one can study such variables by studying these dynamic systems. One can put some conditions on the coefficients to ensure non-negative values in deterministic cases. However, as a random process disturbs the system, the components of solutions to stochastic differential equations (SDE) can keep changing between arbitrary large positive and negative values-even in the simplest case. To overcome this difficulty, the author examines the reflecting stochastic differential equation (RSDE) with the coordinate planes as its boundary-or with a more general boundary. Reflecting Stochastic Differential Equations with Jumps and Applications systematically studies the general theory and applications of these equations. In particular, the author examines the existence, uniqueness, comparison, convergence, and stability of strong solutions to cases where the RSDE has discontinuous coefficients-with greater than linear growth-that may include jump reflection. He derives the nonlinear filtering and Zakai equations, the Maximum Principle for stochastic optimal control, and the necessary and sufficient conditions for the existence of optimal control. Most of the material presented in this book is new, including much new work by the author concerning SDEs both with and without reflection. Much of it appears here for the first time. With the application of RSDEs to various real-life problems, such as the stochastic population and neurophysiological control problems-both addressed in the text-scientists dealing with stochastic dynamic systems will find this an interesting and useful work.
Author: Anis Matoussi
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Category :
Languages : fr
Pages : 0
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Author: Anis Matoussi
Publisher:
ISBN:
Category :
Languages : fr
Pages : 204
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Cette thèse a pour objet, d'une part, l'étude des équations différentielles stochastiques rétrogrades réfléchies (EDDSR) et d'autre part, la preuve de l'existence et l'unicité des solutions d'équations aux dérivées partielles stochastiques quasi-linéaires (EDPS), formulées dans un sens faible ; en utilisant des solutions généralisées des équations différentielles doublement stochastiques rétrogrades (EDDSR). dans la première partie, on s'attache à montrer l'existence d'une solution pour l'EDSR réfléchie sur une ou deux barrières à coefficient non Lipschitz. on s'interroge en effet sur les hypothèses minimales à inclure pour obtenir ce résultat. dans la seconde partie, on s'intéresse à l'EDPS quasi-lineaire suivante : U/T = LU (T, X) + F(T, X, U(T, X), (*U)(T, X))DT + H(T, X, U(T, X), (*U)(T, X))B/T(T), U(T, X) = G(X) ou G est une distribution. Compte tenu des résultats déjà connus sur ce sujet, nous répondons aux questions suivantes: - dans le cas ou les coefficients F(S, X, Y, Z) et H(S, X, Y, Z) sont linéaires en (Y, Z) et appartiennent à un espace de type Sobolev en X, existe-t-il une formulation faible des EDDSR pour donner une formule de Feynman-Kac pour la solution d'EDPS ? - dans le cas ou les coefficients sont non-linéaires, peut-on montrer l'existence et l'unicite d'une solution de l'EDPS et ainsi généraliser les résultats obtenus par Barles et Lesigne (1997) dans le cadre des EDP standards ?
Author: Anne Gégout-Petit
Publisher:
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Category :
Languages : fr
Pages : 0
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Author: David Chevance
Publisher:
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Category :
Languages : fr
Pages : 134
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LA PREMIERE PARTIE DE CETTE THESE A POUR OBJET LA CONSTRUCTION D'UN ALGORITHME PROBABILISTE POUR RESOUDRE NUMERIQUEMENT DES EQUATIONS DIFFERENTIELLES STOCHASTIQUES RETROGRADES (EDSR) DANS LE CAS MARKOVIEN, OU L'EQUATION EST ASSOCIEE A UN PROCESSUS FORWARD SOLUTION D'UNE EDS. NOUS DECRIVONS UN PREMIER ALGORITHME QUI REPOSE SUR UNE DOUBLE DISCRETISATION DE L'EQUATION, EN TEMPS ET EN ESPACE, ET UTILISE DES SIMULATIONS DE TRAJECTOIRES DU PROCESSUS FORWARD. LA DISCRETISATION EN TEMPS EST UNE EXTENSION DU SCHEMA D'EULER POUR LES EDS, OU L'ON A REMPLACE LE MOUVEMENT BROWNIEN PAR UNE MARCHE ALEATOIRE. ON INTRODUIT ENSUITE UNE APPROXIMATION SUPPLEMENTAIRE EN PROJETANT A CHAQUE INSTANT DE DISCRETISATION LE PROCESSUS FORWARD SUR L'ENSEMBLE DES TRAJECTOIRES SIMULEES. ON EVITE AINSI UNE COMPLEXITE ALGORITHMIQUE QUI SERAIT EXPONENTIELLE. NOUS MONTRONS UNE VITESSE DE CONVERGENCE POUR CET ALGORITHME DANS LE CADRE DE LA DIMENSION 1. NOUS PRESENTONS AUSSI UNE VARIANTE DE CE ALGORITHME, ADAPTEE A DES EDSR DONT LES PARAMETRES SONT MOINS REGULIERS, EN REMPLACANT NOTAMMENT LE SCHEMA D'EULER DANS LA DISCRETISATION DU PROCESSUS FORWARD PAR LE SCHEMA DE MILSHTEIN. CELA NOUS PERMET ENSUITE D'ECRIRE UN ALGORITHME DE DISCRETISATION D'EDSR REFLECHIES. DANS UNE SECONDE PARTIE, NOUS ANALYSONS L'APPROXIMATION DE MACMILLAN, ET BARONE-ADESI ET WHALEY, UTILISEE EN FINANCE POUR ESTIMER LE PRIX D'UNE OPTION AMERICAINE. EN ECRIVANT LE PRIX DE L'OPTION AMERICAINE COMME LA SOLUTION D'UNE CERTAINE EQUATION DIFFERENTIELLE STOCHASTIQUE RETROGRADE REFLECHIE, NOUS OBTENONS UNE BORNE GENERALE POUR L'ERREUR DE L'APPROXIMATION ET NOUS MONTRONS QUE L'APPROXIMATION CONVERGE VERS LE PRIX EXACT QUAND LA VOLATILITE DU SOUS-JACENT TEND VERS ZERO. NOUS PROPOSONS ENSUITE UNE DEUXIEME DEMONSTRATION, PLUS ELEMENTAIRE, DE CE RESULTAT ASYMPTOTIQUE, EN FAISANT INTERVENIR LE PRIX D'UN PUT PERPETUEL.
Author: Mingyu Xu
Publisher:
ISBN:
Category :
Languages : en
Pages : 218
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In the first chapter, we consider the reflected backward stochastic differential equation (BSDEsin short) with one or two right continuous and left limited (RCLL in short) barriers. Using the Picarditeration method, we obtained the existence and uniqueness of the solution of the reflected BSDEwith two RCLL barriers. Then we use the penalization method to the case of one RCLL barrier.Considering the solutions (Y n,Zn,Kn) of penalized equations as solutions of reflected BSDEs,we prove that the limit (Y,Z,K) is the solution of equation, by properties of Snell envelope andmonotonic limit theorem (Peng S., 1999). In the case of equation with two RCLL barriers, by theanalogue method, we prove the limit (Y,Z,K) of penalized equation is the solution of problem,by the representation of solutions via Dynkin game. Here we need a generalized monotonic limittheorem, which permit us to pass the limit for penalized equations.In a second work, we have generalized this type of result to the case where barriers are just inL2, by the method of penalization and the theory of g-supersolution.In the second chapter, we consider the reflected BSDEs with one continuous barrier, associatedto (_, f,L), when _ 2 L2(FT ), f(t, !, y, z) is continuous, satisfies monotonic and general increasingconditions on y, and Lipschitz condition on z, and when the barrier (Lt)0_t_T is a progressivelymeasurable continuous process, which verifies certain integrability condition.We have also notable prove the existence and uniqueness of solution in L2, for this reflectedequation with determinist terminal time. The proof of existence is effected by four steps. The firststep consists to prove the result under the boundness condition of _, f(t, 0) et L+. The second step(the most delicate) consists to relax the boundness condition of L+ ; the following two step permitus to obtain the general result, relaxing the boundness condition on _ and f(t, 0). The comparisontheorems play important roles, which help us to pass the limit in the equations. Then we study thecase when the terminal time is a stopping time. The existence and uniqueness of the solution arealso proved.In the third chapter, we have studied the reflected BSDEs with one barrier, whose generator fsatisfies the monotonic and general increasing condition on y, and quadratic and linear condition onz, when the barrier L is uniformly bounded. We prove the existence of a solution by approximation,under these conditions. We also find a necessary and sufficient condition for the case f(t, !, y, z) =|z|2, and construct its solution explicitly. For the case f(t, !, y, z) = |z|p, p 2 (1, 2), we prove asufficient condition.In the forth chapter, we treat the reflected BSDE with two barrier, when f satisfies the mono-tonic, continuous and general increasing conditions on y, and Lipschitz condition on z, like in thesecond chapter. For the barriers, we suppose that L and U are continuous, L
Author: N El Karoui
Publisher: CRC Press
ISBN: 9780582307339
Category : Mathematics
Languages : en
Pages : 236
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This book presents the texts of seminars presented during the years 1995 and 1996 at the Université Paris VI and is the first attempt to present a survey on this subject. Starting from the classical conditions for existence and unicity of a solution in the most simple case-which requires more than basic stochartic calculus-several refinements on the hypotheses are introduced to obtain more general results.
