341 research outputs found
Poisson discretizations of Wiener functionals and Malliavin operators with Wasserstein estimates
No full textThis article proposes a global, chaos-based procedure for the discretization of functionals of Brownian motion into functionals of a Poisson process with intensity λ>0. Under this discretization we study the weak convergence, as the intensity of the underlying Poisson process goes to infinity, of Poisson functionals and their corresponding Malliavin-type derivatives to their Wiener counterparts. In addition, we derive a convergence rate of O(λ ) for the Poisson discretization of Wiener functionals by combining the multivariate Chen–Stein method with the Malliavin calculus. Our proposed sufficient condition for establishing the mentioned convergence rate involves the kernel functions in the Wiener chaos, yet we provide examples, especially the discretization of some common path dependent Wiener functionals, to which our results apply without committing the explicit computations of such kernels. To the best our knowledge, these are the first results in the literature on the universal convergence rate of a global discretization of general Wiener functionals.Ministry of Education (MOE)Accepted versionThe first author-Nicolas Privault acknowledges the financial support from the Singapore MOE Tier 2 Grant MOE2016-T2-1-036. The first author also expresses his gratitude to the hospitality of CUHK when he first discussed with the other two authors on the possibility of working out the present novel topic. The second author— Phillip Yam acknowledges the financial supports from HKGRF-14300717 with the project title: New kinds of Forward–backward Stochastic Systems with Applications, HKSAR-GRF-14301015 with title: Advance in Mean Field Theory, Direct Grant for Research 2014/15 with project code: 4053141 offered by CUHK, and the International Partnerships Development Programme 2013–14, OAL, CUHK, Hong Kong with which Nicolas and Phillip can sit together to work effectively out the present article. The last author–Zheng Zhang acknowledges the financial support from Renmin University of China with the project code 297517501221 together with the project title “Applications of Nonparametric Method in Missing Data”, and the fund for building world-class universities (disciplines) of Renmin University of China; for the purpose of his official grant acknowledgment, the last author, with the consensus of the two other authors, likes to formally declare that the present work is completed by even contribution of each of us with our authorships listed in alphabetical order of our surnames
Hypothesis testing and Skorokhod
No full textstochastic integration Nicolas Privault D'epartement de Math'ematiques, Universit'e de La Rochell
Contributions à l'étude des marchés discontinus par le calcul de Malliavin
No full textJURY: Ying HU, Professeur, Université de Rennes I, Rapporteur. Monique JEANBLANC, Professeur, Université d'Evry Val d'Essonne, Examinateur. Nicolas PRIVAULT, Professeur, Université de La Rochelle, Directeur de thèse. Marek RUTKOWSKI, Professeur Warsaw University of Technology, Président du jury. Nizar TOUZI, Professeur CREST et Université de Paris I, Rapporteur. Josep VIVES, Professor titular, Universitat Autònoma de Barcelona, Rapporteur.We consider markets driven by normal martingales which have the chaotic representation property, e.g.: martingales satisfying a deterministic structure equation, Azéma martingales. Replicating hedging strategies for European, Asian and Lookback options are explicitly computed using either the Clark-Ocone formula or an extension of the Delta-hedging method, depending on which is most appropriate. Using the Malliavin calculus on Poisson space we compute Greeks for Asian options in a market driven by a Poisson process. We also consider a stochastic volatility model with jumps where the underlying asset price is driven by process sum of a 2-dimensional Brownian motion and Poisson process. The market is incomplete and there exists an infinity of equivalent martingale measures. We minimize the entropy to choose such a measure, under which we determine the strategy minimizing the variance.La constatation que les prix des actifs boursiers sautent brusquement a conduit à étudier des modèles de marchés avec sauts. Cette thèse va dans cette direction. On y considère des marchés dirigés par des martingales normales qui ont la propriété de représentation chaotique: les martingales vérifiant une équation de structure déterministe, la martingale d'Azéma, etc. On trouve des stratégies de couverture pour les options européennes, asiatiques et Lookback soit par la formule d'Itô, soit par la formule de Clark-Ocone selon la plus appropriée. L'application du calcul de Malliavin au calcul des Greeks est traitée pour les options asiatiques dans le cas d'un marché dirigé par un processus de Poisson. On traite aussi de couverture dans un modèle à volatilité stochastique avec sauts où le prix de l'actif risqué est dirigé par un processus somme d'un mouvement brownien et d'un processus de Poisson 2-dimensionnels. Le marché est incomplet et il existe une infinité de mesures martingales équivalentes. On minimise l'entropie pour choisir telle mesure. Sous celle-ci on calcule la stratégie minimisant la variance
Contributions à l'étude des marchés discontinus par le calcul de Malliavin
No full textJURY: Ying HU, Professeur, Université de Rennes I, Rapporteur. Monique JEANBLANC, Professeur, Université d'Evry Val d'Essonne, Examinateur. Nicolas PRIVAULT, Professeur, Université de La Rochelle, Directeur de thèse. Marek RUTKOWSKI, Professeur Warsaw University of Technology, Président du jury. Nizar TOUZI, Professeur CREST et Université de Paris I, Rapporteur. Josep VIVES, Professor titular, Universitat Autònoma de Barcelona, Rapporteur.We consider markets driven by normal martingales which have the chaotic representation property, e.g.: martingales satisfying a deterministic structure equation, Azéma martingales. Replicating hedging strategies for European, Asian and Lookback options are explicitly computed using either the Clark-Ocone formula or an extension of the Delta-hedging method, depending on which is most appropriate. Using the Malliavin calculus on Poisson space we compute Greeks for Asian options in a market driven by a Poisson process. We also consider a stochastic volatility model with jumps where the underlying asset price is driven by process sum of a 2-dimensional Brownian motion and Poisson process. The market is incomplete and there exists an infinity of equivalent martingale measures. We minimize the entropy to choose such a measure, under which we determine the strategy minimizing the variance.La constatation que les prix des actifs boursiers sautent brusquement a conduit à étudier des modèles de marchés avec sauts. Cette thèse va dans cette direction. On y considère des marchés dirigés par des martingales normales qui ont la propriété de représentation chaotique: les martingales vérifiant une équation de structure déterministe, la martingale d'Azéma, etc. On trouve des stratégies de couverture pour les options européennes, asiatiques et Lookback soit par la formule d'Itô, soit par la formule de Clark-Ocone selon la plus appropriée. L'application du calcul de Malliavin au calcul des Greeks est traitée pour les options asiatiques dans le cas d'un marché dirigé par un processus de Poisson. On traite aussi de couverture dans un modèle à volatilité stochastique avec sauts où le prix de l'actif risqué est dirigé par un processus somme d'un mouvement brownien et d'un processus de Poisson 2-dimensionnels. Le marché est incomplet et il existe une infinité de mesures martingales équivalentes. On minimise l'entropie pour choisir telle mesure. Sous celle-ci on calcule la stratégie minimisant la variance
Cardinality estimation for random stopping sets based on Poisson point processes
No full textWe construct unbiased estimators for the distribution of the number of points inside random stopping sets based on a Poisson point process. Our approach is based on moment identities for stopping sets, showing that the random count of points inside the complement of a stopping set S has a Poisson distribution conditionally to S. The proofs do not require the use of set-indexed martingales, and our estimators have a lower variance when compared to standard sampling. Numerical simulations are presented for examples such as the convex hull and the Voronoi flower of a Poisson point process, and their complements
Third Cumulant Stein Approximation for Poisson Stochastic Integrals
No full textWe derive Edgeworth-type expansions for Poisson stochastic integrals, based on cumulant operators defined by the Malliavin calculus. As a consequence we obtain Stein approximation bounds for stochastic integrals, which are based on third cumulants instead of the L -norm term found in the literature. The use of the third cumulant results in a convergence rate faster than the classical Berry–Esseen rate for certain examples.Ministry of Education (MOE)Accepted versionThis research was supported by the Singapore MOE Tier 2 Grant MOE2016-T2-1- 036
Poisson sphere counting processes with random radii
No full textWe consider a random sphere covering model made of random balls with interacting random
radii of the product form R(r,ω) =
rG(ω), based on a Poisson random
measure ω(dy,dr) on
Rd ×
R+. We provide sufficient conditions under which the
corresponding random ball counting processes are well-defined, and we study the fractional
behavior of the associated random fields. The main results rely on moment formulas for
Poisson stochastic integrals with random integrands
A calculus on Fock space and its probabilistic interpretations
No full textAbstractWe introduce on the Fock space Φ(L2(R+)) two operators ∇⊖ and ∇⊗ expressing the infinitesimal perturbations of random variables by time changes in the Wiener and Poisson probabilistic interpretations of Φ(L2(R+)). These operators have close connections with stochastic integration, regularity of laws, chaotic expansions and complement the annihilation and creation operators ∇- and ∇+ that are related to perturbations by shifts of trajectories
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