Seminaire De Probabilites XXXVIII

403 Pages·2005·2.252 MB·English-French

by  Ronald A. Doney (auth.), Michel Émery, Michel Ledoux, Marc Yor (eds.)

1857
Lecture Notes in Mathematics
Editors:
J.--M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
Subseries:
InstitutdeMathe´matiques,Universite´deStrasbourg
Adviser:J.-L.Loday

´
M. Emery M. Ledoux M. Yor (Eds.)
Se´minaire
de Probabilite´s XXXVIII
123
Editors
MichelE´mery MarcYor
InstitutdeRecherche LaboratoiredeProbabilite´s
Mathe´matiqueAvance´e etMode`lesAle´atoires
Universite´LouisPasteur Universite´PierreetMarieCurie
7,rueRene´Descartes Boˆıtecourrier188
67084StrasbourgCedex,France 4placeJussieu
75252ParisCedex05,France
e-mail:[email protected]
MichelLedoux
LaboratoiredeStatistiques
etProbabilite´s
Universite´PaulSabatier
118,routedeNarbonne
31601ToulouseCedex,France
e-mail:[email protected]
LibraryofCongressControlNumber:2004115715
MathematicsSubjectClassification(2000):60Gxx,60Hxx,60Jxx
ISSN0075-8434LectureNotesinMathematics
ISSN0720-8766Se´minairedeProbabilite´s
ISBN3-540-23973-1SpringerBerlinHeidelbergNewYork
DOI:10.1007/b104072
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C’est avec gratitude et admiration que nous d´edions ce volume `a
Jacques Az´ema,
`a l’occasion de son 65e anniversaire. Ses travaux, parmi lesquels ceux sur
le retournement du temps, le balayage, les ferm´es al´eatoires et bien suˆr la
martingale d’Az´ema, ont prolong´e, toujours avec originalit´e et ´el´egance, la
th´eorie g´en´eraledes processus.
Sonapparente d´econtraction,sa r´eellerigueur et ses incessantesquestions
(“hishealthyskepticism”,commel’´ecrivaitJ.WalshdansTempsLocaux),ont
´et´eindissociablesduS´eminairedeProbabilit´espendantdenombreusesann´ees.
We are also indebted and grateful to Anthony Phan, whose patient and
time-consuming work behind the scene, up to minute details, on typography,
formatting and TEXnicalities, was a key ingredient in the production of the
present volume.
Volume XXXIX, whichconsists ofcontributions dedicatedto the memory
of P. A. Meyer, is being prepared at the same time as this one and should
appear soon, also in the Springer LNM series. It may be considered as a
companion to the special issue, also in memory of Meyer, of the Annales de
l’Institut Henri Poincar´e.
Finally, the R´edaction of the S´eminaire is thoroughly modified: J. Az´ema
retiredfromourteamafterS´eminaireXXXVIIwascompleted;now,following
his steps, two of us—M. Ledoux and M. Yor—are also leaving the board.
From volume XL onwards, the new R´edaction will consist of Catherine
Donati-Martin(Paris),MichelE´mery(Strasbourg),AlainRouault(Versailles)
andChristopheStricker(Besan¸con).Thecombinedexpertiseofthenewmem-
bers of the board will be an important asset to blend the themes which are
traditionally studied in the S´eminaire together with the newer developments
in Probability Theory in general and Stochastic Processes in particular.
M. E´mery, M. Ledoux, M. Yor

Table des Mati`eres
Processus de L´evy
Tanaka’s construction for random walks and L´evy processes
Ronald A. Doney ................................................ 1
Some excursion calculations for spectrally one-sided L´evy
processes
Ronald A. Doney ................................................ 5
A martingale review of some fluctuation theory for spectrally
negative L´evy processes
Andreas E. Kyprianou, Zbigniew Palmowski ......................... 16
A potential-theoretical review of some exit problems of
spectrally negative L´evy processes
Martijn R. Pistorius ............................................. 30
Some martingales associated to reflected L´evy processes
Laurent Nguyen-Ngoc, Marc Yor ................................... 42
Generalised Ornstein–Uhlenbeck processes and the
convergence of L´evy integrals
K. Bruce Erickson, Ross A. Maller................................. 70
Autres expos´es
Spectral gap for log-concave probability measures on the real
line
Pierre Foug`eres.................................................. 95
VIII Table des Mati`eres
Propri´et´e de Choquet–Deny et fonctions harmoniques sur les
hypergroupes commutatifs
Laurent Godefroy ................................................124
Exponential decay parameters associated with excessive
measures
Mioara Buiculescu ...............................................135
Positive bilinear mappings associated with stochastic
processes
Valentin Grecea..................................................145
An almost sure approximation for the predictable process
in the Doob–Meyer decomposition theorem
Adam Jakubowski ................................................158
On stochastic integrals up to infinity and predictable criteria
for integrability
Alexander Cherny, Albert Shiryaev .................................165
Remarks on the true no-arbitrage property
Yuri Kabanov, Christophe Stricker .................................186
Information-equivalence: On filtrations created by
independent increments
Hans Bu¨hler.....................................................195
Rotations and tangent processes on Wiener space
Moshe Zakai ....................................................205
Lp multiplier theorem for the Hodge–Kodaira operator
Ichiro Shigekawa.................................................226
Gaussian limits for vector-valued multiple stochastic integrals
Giovanni Peccati, Ciprian A. Tudor ................................247
Derivatives of self-intersection local times
Jay Rosen ......................................................263
On squared fractional Brownian motions
Nathalie Eisenbaum, Ciprian A. Tudor .............................282
Regularity and identification of generalized multifractional
Gaussian processes
Antoine Ayache, Albert Benassi, Serge Cohen, Jacques L´evy V´ehel .....290
Failure of the Raikov theorem for free random variables
Florent Benaych-Georges..........................................313
Table des Mati`eres IX
A sharp small deviation inequality for the largest eigenvalue
of a random matrix
Guillaume Aubrun ...............................................320
The tangent space to a hypoelliptic diffusion and applications
Fabrice Baudoin .................................................338
Homogenizationof a diffusionwithlocallyperiodiccoefficients
Abdellatif Bench´erif-Madani, E´tienne Pardoux .......................363
Tanaka’s Construction for Random Walks and
L´evy Processes
Ronald A. Doney
Department of Mathematics, University of Manchester
Oxford Road,Manchester, UK M13 9PL
e-mail: [email protected]
Summary. Tanaka’s construction gives a pathwise construction of “random walk
conditioned to stay positive”, and has recently been used in [3] and [8] to establish
other results about this process. In this note we give a simpler proof of Tanaka’s
construction using a method which also extendsto thecase of L´evy processes.
1 The random walk case
If S is any rw starting at zero which does not drift to −∞, we write S∗ for
S killed at time σ := min(n (cid:1) 1 : S (cid:2) 0), and S↑ for the for the harmonic
n
transformofS∗ whichcorrespondsto “conditioningS to staypositive”.Thus
for x>0,y >0, and x=0 when n=0
P(S↑ ∈dy|S↑ =x)= V(y) P(S ∈dy|S =x)= V(y) P(S ∈dy−x),
n+1 n V(x) n+1 n V(x) 1
(1)
where V is the renewalfunction in the weak increasing ladder process of−S.
In [10], Tanaka showed that a process R got by time-reversing one by one
the excursions below the maximum of S has the same distribution as S↑;
specifically if {(T ,H ),k (cid:1)0} denotes the strict increasing ladder process of
k k
S (with T =H ≡0) then R is defined by
0 0
T(cid:1)k+1
R =0, R =H + Y , T <n(cid:2)T , k (cid:1)0. (2)
0 n k i k k+1
i=Tk+1+Tk+1−n
If S drifts to +∞, then it is well known (see [9]) that the post-minimum
process
(cid:2) (cid:3)
−→
S :=(S −S , n(cid:1)0), where J =max n:S =minS (3)
J+n J n r
r(cid:1)n
alsohas the distributionof S↑. Inthis casea very simple argumentwas given
−→
in[7]toshowthatthedistributionsofRand S agree,thusyieldingaproofof
M.E´mery,M.Ledoux,andM.Yor(Eds.):LNM1857, pp.1–4,2005.
(cid:1)c Springer-VerlagBerlinHeidelberg2005