Thesis defense : “Malliavin calculus and Dirichlet structures for independent random variables”

Speaker : Hélène Halconruy
Date: 14/09/2020
Time: 2:00 pm - 5:00 pm
Location: LINCS Seminars room


In a pioneer work from 1976, Paul Malliavin lays the foundations for an infinite-dimensional stochastic calculus of variations. Initially developed on the Wiener space, the Malliavin calculus was later extended to other families of processes such as Poisson, Rademacher or Lévy processes. It seems possible to identify a terminology common to all these formalisms; this lies on the notions of Malliavin operators (gradient, divergence, Laplacian / Ornstein-Uhlenbeck operator) and the fundamental relation between the gradient operator and the divergence (defined as the adjoint of the gradient): the integration by parts formula.
We develop in this thesis a Malliavin calculus for two classes of discrete processes: the sequences of independent random variables (not necessarily identically distributed) and the compound geometric processes. These constructions were motivated by several applications; two of them will be developed during the presentation.
The first one deals with the estimation of distances between two probability laws using Stein’s method. One of the step of this method is performed using advantageously Malliavin integration by parts formula. In order to use it to get convergence results for functionals of independent random variables, we equip any countable product of probability spaces with a discrete Dirichlet-Malliavin structure built on a family of Malliavin operators (gradient discrete, divergence, operator number), an integration by parts formula, and the forms of Dirichlet naturally induced in this context. In this framework, we establish a Normal and Gamma approximation criteria by functionals of independent random variables in terms of Malliavin operators.
The motivation for the second work was to use our tools to deal with portfolio management problems in a discrete financial market model. It turned out to be impossible to state an hedging formula within our precedent formalism. We replace thus the initial model with a ternary model, underlied by a compound geometric process for which we develop a pseudo-decomposition chaotic and define Malliavin operators. By plugging this formalism to portfolio optimization, we show that the minimizing quadratic risk strategy can be expressed in terms of the newly introduced Malliavin operators.

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ID de réunion : 933 9574 7795
Code secret : 887763