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Über dieses Buch

The series of advanced courses initiated in Séminaire de Probabilités XXXIII continues with a course by Ivan Nourdin on Gaussian approximations using Malliavin calculus. The Séminaire also occasionally publishes a series of contributions on a unifying subject; in this spirit, selected participants to the September 2011 Conference on Stochastic Filtrations, held in Strasbourg and organized by Michel Émery, have also contributed to the present volume. The rest of the work covers a wide range of topics, such as stochastic calculus and Markov processes, random matrices and free probability, and combinatorial optimization.

Inhaltsverzeichnis

Frontmatter

Specialized Course

Frontmatter

Lectures on Gaussian Approximations with Malliavin Calculus

In a seminal paper of 2005, Nualart and Peccati [40] discovered a surprising central limit theorem (called the “Fourth Moment Theorem” in the sequel) for sequences of multiple stochastic integrals of a fixed order: in this context, convergence in distribution to the standard normal law is equivalent to convergence of just the fourth moment. Shortly afterwards, Peccati and Tudor [46] gave a multidimensional version of this characterization.

Ivan Nourdin

Other Contributions

Frontmatter

Some Sufficient Conditions for the Ergodicity of the Lévy Transformation

We propose a possible way of attacking the question posed originally by Daniel Revuz and Marc Yor in their book published in 1991. They asked whether the Lévy transformation of the Wiener-space is ergodic. Our main results are formulated in terms of a strongly stationary sequence of random variables obtained by evaluating the iterated paths at time one. Roughly speaking, this sequence has to approach zero “sufficiently fast”. For example, one of our results states that if the expected hitting time of small neighborhoods of the origin do not grow faster than the inverse of the size of these sets then the Lévy transformation is strongly mixing, hence ergodic.

Vilmos Prokaj

Vershik’s Intermediate Level Standardness Criterion and the Scale of an Automorphism

In the case of

r

n

-adic filtrations, Vershik’s standardness criterion takes a particular form, hereafter called Vershik’s intermediate level criterion. This criterion, whose nature is combinatorial, has been intensively used in the ergodic-theoretic literature, but it is not easily applicable by probabilists because it is stated in a language specific to the theory of measurable partitions and has not been translated into probabilistic terms. We aim to provide an easily applicable probabilistic statement of this criterion. Finally, Vershik’s intermediate level criterion is illustrated by revisiting Vershik’s definition of the scale of an invertible measure-preserving transformation.

Stéphane Laurent

Filtrations Indexed by Ordinals; Application to a Conjecture of S. Laurent

The following fact has been conjectured by Stéphane Laurent (Conjecture 3.18, page 160 of Séminaire de Probabilités XLIII):

Let

$$\,\mathcal{F} = (\mathcal{F}_{t})$$

and

$$\,\mathcal{G} = (\mathcal{G}_{t})$$

be two filtrations on some probability space, and suppose that every

$$\,\mathcal{F}$$

-martingale

is also a

$$\,\mathcal{G}$$

-martingale. For

s

<

t

,

if

$$\,\mathcal{G}_{t}$$

is generated by

$$\,\mathcal{G}_{s}$$

and by countably

many events, then

$$\,\mathcal{F}_{t}$$

is generated by

$$\,\mathcal{F}_{s}$$

and by countably many events.

In this statement, “and by countably many events” can equivalently be replaced with “and by some separable

σ

-algebra”, or with “and by some random variable valued in some Polish space”. We propose a rather intuitive proof of this conjecture, based on the following necessary and sufficient condition:

Given a probability space, let

$$\,\mathcal{D}$$

be a σ -algebra of measurable sets and

$$\,\mathcal{C}$$

a sub

-

σ

-

algebra of

$$\,\mathcal{D}$$

.

Then

$$\,\mathcal{D}$$

is generated by

$$\,\mathcal{C}$$

and by countably many events if and only if there exists no strictly increasing filtration

$$\mathcal{F} = (\mathcal{F}_{\alpha })_{\alpha <\boldsymbol\aleph _{1}}$$

,

indexed by the set

$$\,\lfloor \lceil 0,\boldsymbol\aleph _{1}\lfloor \lceil $$

of all countable ordinals, and satisfying

$$\,\mathcal{C}\subseteq \mathcal{F}_{\alpha }\subseteq \mathcal{D}\,$$

for each α

. Another question then arises: can the martingale hypothesis on

$$\mathcal{F}$$

and

$$\mathcal{G}$$

be replaced by a more general condition involving the null events but not the values of the probability? We propose such a weaker hypothesis, but we are no longer able to derive the conclusion from it; so the question is left open.

Claude Dellacherie, Michel Émery

A Planar Borel Set Which Divides Every Non-negligible Borel Product

In the unit square [0, 1] ×[0, 1] endowed with the Lebesgue measure

λ

, we construct a Borel subset

A

with the following property: if

U

and

V

are any two non-negligible Borel subsets of [0, 1], then

$$0 <\lambda {\bigl ( A \cap (U \times V )\bigr )} <\lambda (U \times V )$$

.

Michel Émery

Characterising Ocone Local Martingales with Reflections

Let

M

= (

M

t

)

t

≥ 0

be any continuous real-valued stochastic process such that

M

0

= 0. Chaumont and Vostrikova proved that if there exists a sequence (

a

n

)

n

≥ 1

of positive real numbers converging to 0 such that

M

satisfies the reflection principle at levels 0,

a

n

and 2

a

n

, for each

n

≥ 1, then

M

is an Ocone local martingale. They also asked whether the reflection principle at levels 0 and

a

n

only (for each

n

≥ 1) is sufficient to ensure that

M

is an Ocone local martingale. We give a positive answer to this question, using a slightly different approach, which provides the following intermediate result. Let

a

and

b

be two positive real numbers such that

$$a/(a + b)$$

is not dyadic. If

M

satisfies the reflection principle at the level 0 and at the first passage-time in { −

a

,

b

}, then

M

is close to a local martingale in the following sense: |

e

[

M

S

M

] | ≤

a

+

b

for every stopping time

S

in the canonical filtration of

$$\mathbf{w} =\{ w \in \mathcal{C}(\mathbf{r}_{+},\mathbf{r}): w(0) = 0\}$$

such that the stopped process

M

⋅ ∧ (

S

M

)

is uniformly bounded.

Jean Brossard, Christophe Leuridan

Approximation and Stability of Solutions of SDEs Driven by a Symmetric α Stable Process with Non-Lipschitz Coefficients

Firstly, we investigate Euler–Maruyama approximation for solutions of stochastic differential equations (SDEs) driven by a symmetric

α

stable process under Komatsu condition for coefficients. The approximation implies naturally the existence of strong solutions. Secondly, we study the stability of solutions under Komatsu condition, and also discuss it under Belfadli–Ouknine condition.

Hiroya Hashimoto

Path Properties and Regularity of Affine Processes on General State Spaces

We provide a new proof for regularity of affine processes on general state spaces by methods from the theory of Markovian semimartingales. On the way to this result we also show that the definition of an affine process, namely as stochastically continuous time-homogeneous Markov process with exponential affine Fourier–Laplace transform, already implies the existence of a càdlàg version. This was one of the last open issues in the fundaments of affine processes.

Christa Cuchiero, Josef Teichmann

Langevin Process Reflected on a Partially Elastic Boundary II

A particle subject to a white noise external forcing moves according to a Langevin process. Consider now that the particle is reflected at a boundary which restores a portion

c

of the incoming speed at each bounce. For

c

strictly smaller than the critical value

$$c_{\mathit{crit}} =\exp (-\pi /\sqrt{3})$$

, the bounces of the reflected process accumulate in a finite time, yielding a very different behavior from the most studied cases of perfectly elastic reflection—c = 1—and totally inelastic reflection—

c

= 0. We show that nonetheless the particle is not necessarily absorbed after this accumulation of bounces. We define a “resurrected” reflected process as a recurrent extension of the absorbed process, and study some of its properties. We also prove that this resurrected reflected process is the unique solution to the stochastic partial differential equation describing the model, for which well-posedness is nothing obvious. Our approach consists in defining the process conditioned on never being absorbed, via an

h

-transform, and then giving the Itō excursion measure of the recurrent extension thanks to a formula fairly similar to Imhof’s relation.

Emmanuel Jacob

Windings of Planar Stable Processes

Using a generalization of the skew-product representation of planar Brownian motion and the analogue of Spitzer’s celebrated asymptotic Theorem for stable processes due to Bertoin and Werner, for which we provide a new easy proof, we obtain some limit Theorems for the exit time from a cone of stable processes of index

α

∈ (0, 2). We also study the case

t

→ 0 and we prove some Laws of the Iterated Logarithm (LIL) for the (well-defined) winding process associated to our planar stable process.

R. A. Doney, S. Vakeroudis

An Elementary Proof that the First Hitting Time of an Open Set by a Jump Process is a Stopping Time

We give a short and elementary proof that the first hitting time of an open set by the jump process of a càdlàg adapted process is a stopping time.

Alexander Sokol

Catalytic Branching Processes via Spine Techniques and Renewal Theory

In this article we contribute to the moment analysis of branching processes in catalytic media. The many-to-few lemma based on the spine technique is used to derive a system of (discrete space) partial differential equations for the number of particles in a variation of constants formulation. The long-time behaviour is then deduced from renewal theorems and induction.

Leif Döring, Matthew I. Roberts

Malliavin Calculus and Self Normalized Sums

We study the self-normalized sums of independent random variables from the perspective of the Malliavin calculus. We give the chaotic expansion for them and we prove a Berry–Esséen bound with respect to several distances.

Solesne Bourguin, Ciprian A. Tudor

A Note on Stochastic Calculus in Vector Bundles

The aim of these notes is to relate covariant stochastic integration in a vector bundle

E

[as in Norris (

Séminaire de Probabilités, XXVI

, vol. 1526, Springer, Berlin, 1992, pp. 189–209)] with the usual Stratonovich calculus via the connector

$$\mathcal{K}_{\nabla }: \mathit{TE} \rightarrow E$$

[cf. e.g. Paterson (Canad. J. Math. 27(4):766–791, 1975) or Poor (

Differential Geometric Structures

, McGraw-Hill, New York, 1981)] which carries the connection dependence.

Pedro J. Catuogno, Diego S. Ledesma, Paulo R. Ruffino

Functional Co-monotony of Processes with Applications to Peacocks and Barrier Options

We show that several general classes of stochastic processes satisfy a functional co-monotony principle, including processes with independent increments, Brownian bridge, Brownian diffusions, Liouville processes, fractional Brownian motion. As a first application, we recover and extend some recent results about peacock processes obtained by Hirsch et al. in (

Peacocks and Associated Martingales, with Explicit Constructions

, Bocconi & Springer, 2011, 430p) [see also (Peacocks sous l’hypothèse de monotonie conditionnelle et caractérisation des 2-martingales en termes de peacoks, thèse de l’Université de Lorraine, 2012, 169p)] which were themselves motivated by a former work of Carr et al. in (Finance Res. Lett. 5:162–171, 2008) about the sensitivities of Asian options with respect to their volatility and residual maturity (seniority). We also derive semi-universal bounds for various barrier options.

Gilles Pagès

Fluctuations of the Traces of Complex-Valued Random Matrices

The aim of this paper is to provide a central limit theorem for complex random matrices

$$(X_{i,j})_{i,j\geq 1}$$

with i.i.d. entries having moments of any order. Tao and Vu (Ann. Probab. 38(5):2023–2065, 2010) showed that for large renormalized random matrices, the spectral measure converges to a circular law. Rider and Silverstein (Ann. Probab. 34(6):2118–2143, 2006) studied the fluctuations around this circular law in the case where the imaginary part and the real part of the random variable

X

i

,

j

have densities with respect to Lebesgue measure which have an upper bound, and their moments of order

k

do not grow faster than

$${k}^{\alpha k}$$

, with

α

> 0. Their result does not cover the case of real random matrices. Nourdin and Peccati (ALEA 7:341–375, 2008) established a central limit theorem for real random matrices using a probabilistic approach. The main contribution of this paper is to use the same probabilistic approach to generalize the central limit theorem to complex random matrices.

Salim Noreddine

Functionals of the Brownian Bridge

We discuss the distributions of three functionals of the free Brownian bridge: its L

2

-norm, the second component of its signature and its Lévy area. All of these are freely infinitely divisible. Two representations of the free Brownian bridge as series of free semicircular random variables are introduced and used. These are analogous to the Fourier representations of the classical Brownian bridge due to Lévy and Kac and the latter extends to all semicircular processes.

Janosch Ortmann

Étude spectrale minutieuse de processus moins indécis que les autres

On cherche ici à quantifier la convergence à l’équilibre de processus de Markov non réversibles, en particulier en temps court. La simplicité des modèles considérés nous permet de donner une expression assez explicite de l’évolution temporelle de l’erreur

L

2

en norme opérateur et de la comparer avec celle des cas réversibles correspondants.

Laurent Miclo, Pierre Monmarché

Combinatorial Optimization Over Two Random Point Sets

Let

$$(\mathcal{X},\mathcal{Y})$$

be a pair of random point sets in

$${\mathbb{R}}^{d}$$

of equal cardinal obtained by sampling independently 2

n

points from a common probability distribution

μ

. In this paper, we are interested by functions

L

of

$$(\mathcal{X},\mathcal{Y})$$

which appear in combinatorial optimization. Typical examples include the minimal length of a matching of

$$\mathcal{X}$$

and

$$\mathcal{Y}$$

, the length of a traveling salesperson tour constrained to alternate between points of each set, or the minimal length of a connected bipartite

r

-regular graph with vertex set

$$(\mathcal{X},\mathcal{Y})$$

. As the size

n

of the point sets goes to infinity, we give sufficient conditions on the function

L

and the probability measure

μ

which guarantee the convergence of

$$L(\mathcal{X},\mathcal{Y})$$

under a suitable scaling. In the case of the minimal length matching, we extend results of Dobrić and Yukich, and Boutet de Monvel and Martin.

Franck Barthe, Charles Bordenave

A Simple Proof of Duquesne’s Theorem on Contour Processes of Conditioned Galton–Watson Trees

We give a simple new proof of a theorem of Duquesne, stating that the properly rescaled contour function of a critical aperiodic Galton–Watson tree, whose offspring distribution is in the domain of attraction of a stable law of index

θ

∈ (1, 2], conditioned on having total progeny

n

, converges in the functional sense to the normalized excursion of the continuous-time height function of a strictly stable spectrally positive Lévy process of index

θ

. To this end, we generalize an idea of Le Gall which consists in using an absolute continuity relation between the conditional probability of having total progeny exactly

n

and the conditional probability of having total progeny at least

n

. This new method is robust and can be adapted to establish invariance theorems for Galton–Watson trees having

n

vertices whose degrees are prescribed to belong to a fixed subset of the positive integers.

Igor Kortchemski

Backmatter

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