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UID:556@lincs.fr
DTSTART;TZID=Europe/Paris:20200914T140000
DTEND;TZID=Europe/Paris:20200914T170000
DTSTAMP:20200910T135958Z
URL:https://www.lincs.fr/events/thesis-defense-malliavin-calculus-and-diri
 chlet-structures-for-independent-random-variables/
SUMMARY:Thesis defense : "Malliavin calculus and Dirichlet structures for
 independent random variables"
DESCRIPTION: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.\nWe 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.\nThe 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.\nThe
 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.\n\nZoom
 link:\n\n[&nbsp\;https://telecom-paris.zoom.us/j/93395747795?pwd=VzBYRmE0T0
 taVXFnZTU1WVBPRXV5UT09&nbsp\;|&nbsp\;https://telecom-paris.zoom.us/j/933957
 47795?pwd=VzBYRmE0T0taVXFnZTU1WVBPRXV5UT09&nbsp\;]\n\nID de réunion : 933
 9574 7795\nCode secret : 887763
CATEGORIES:PhD Defense
LOCATION:LINCS Seminars room\, 23\, avenue d'Italie\, Paris\, 75013\,
 France
GEO:48.828400;2.356897
X-APPLE-STRUCTURED-LOCATION;VALUE=URI;X-ADDRESS=23\, avenue d'Italie\,
 Paris\, 75013\, France;X-APPLE-RADIUS=100;X-TITLE=LINCS Seminars
 room:geo:48.828400,2.356897
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TZID:Europe/Paris
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DTSTART:20200329T030000
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TZOFFSETTO:+0200
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