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Providing a broad overview of the current state of the art in probability theory and its applications, and featuring an article coauthored by Mark Yor, this volume contains contributions on branching processes, Lévy processes, random walks and martingales and their connection with, among other topics, rough paths, semi-groups, heat kernel asymptotics and mathematical finance.

Inhaltsverzeichnis

Frontmatter

Branching Random Walk in an Inhomogeneous Breeding Potential

Abstract
We consider a continuous-time branching random walk in the inhomogeneous breeding potential β | ⋅ |  p , where β > 0, p ≥ 0. We prove that the population almost surely explodes in finite time if p > 1 and doesn’t explode if p ≤ 1. In the non-explosive cases, we determine the asymptotic behaviour of the rightmost particle.
Sergey Bocharov, Simon C. Harris

The Backbone Decomposition for Spatially Dependent Supercritical Superprocesses

Abstract
Consider any supercritical Galton-Watson process which may become extinct with positive probability. It is a well-understood and intuitively obvious phenomenon that, on the survival set, the process may be pathwise decomposed into a stochastically ‘thinner’ Galton-Watson process, which almost surely survives and which is decorated with immigrants, at every time step, initiating independent copies of the original Galton-Watson process conditioned to become extinct. The thinner process is known as the backbone and characterizes the genealogical lines of descent of prolific individuals in the original process. Here, prolific means individuals who have at least one descendant in every subsequent generation to their ownn.
Starting with Evans and O’Connell (Can Math Bull 37:187–196, 1994), there exists a cluster of literature, (Engländer and Pinsky, Ann Probab 27:684–730, 1999; Salisbury and Verzani, Probab Theory Relat Fields 115:237–285, 1999; Duquesne and Winkel, Probab Theory Relat Fields 139:313–371, 2007; Berestycki, Kyprianou and Murillo-Salas, Stoch Proc Appl 121:1315–1331, 2011; Kyprianou and Ren, Stat Probab Lett 82:139–144, 2012),describing the analogue of this decomposition (the so-called backbone decomposition) for a variety of different classes of superprocesses and continuous-state branching processes. Note that the latter family of stochastic processes may be seen as the total mass process of superprocesses with non-spatially dependent branching mechanism.In this article we consolidate the aforementioned collection of results concerning backbone decompositions and describe a result for a general class of supercritical superprocesses with spatially dependent branching mechanisms. Our approach exposes the commonality and robustness of many of the existing arguments in the literature.
A. E. Kyprianou, J-L. Pérez, Y.-X. Ren

On Bochner-Kolmogorov Theorem

Abstract
We prove the Bochner-Kolmogorov theorem on the existence of the limit of projective systems of second countable Hausdorff (non-metrizable) spaces with tight probabilities, such that the projection mappings are merely measurable functions. Our direct and transparent approach (using Lusin’s theorem) should be compared with the previous work where the spaces are assumed metrizable and the main idea was to reduce the general context to a regular one via some isomorphisms. The motivation of the revisit of this classical result is an application to the construction of the continuous time fragmentation processes and related branching processes, based on a measurable identification between the space of all fragmentation sizes considered by J. Bertoin and the limit of a projective system of spaces of finite configurations.
Lucian Beznea, Iulian Cîmpean

Small Time Asymptotics for an Example of Strictly Hypoelliptic Heat Kernel

Abstract
A small time asymptotics of the density is established for a simplified (non-Gaussian, strictly hypoelliptic) second chaos process tangent to the Dudley relativistic diffusion.
Jacques Franchi

Onsager-Machlup Functional for Uniformly Elliptic Time-Inhomogeneous Diffusion

Abstract
In this paper, we will compute the Onsager-Machlup functional of an inhomogeneous uniformly elliptic diffusion process. This functional is very similar to the corresponding functional for homogeneous diffusions; indeed, the only difference come from the infinitesimal variation of the volume. We will also use the Onsager-Machlup functional to study small ball probability for weighted sup-norm of some inhomogeneous diffusion.
Koléhè A. Coulibaly-Pasquier

G-Brownian Motion as Rough Paths and Differential Equations Driven by G-Brownian Motion

Abstract
The present article is devoted to the study of sample paths of G-Brownian motion and stochastic differential equations (SDEs) driven by G-Brownian motion from the viewpoint of rough path theory. As the starting point, by using techniques in rough path theory, we show that quasi-surely, sample paths of G-Brownian motion can be enhanced to the second level in a canonical way so that they become geometric rough paths of roughness 2 < p < 3. This result enables us to introduce the notion of rough differential equations (RDEs) driven by G-Brownian motion in the pathwise sense under the general framework of rough paths. Next we establish the fundamental relation between SDEs and RDEs driven by G-Brownian motion. As an application, we introduce the notion of SDEs on a differentiable manifold driven by G-Brownian motion and construct solutions from the RDE point of view by using pathwise localization technique. This is the starting point of developing G-Brownian motion on a Riemannian manifold, based on the idea of Eells-Elworthy-Malliavin. The last part of this article is devoted to such construction for a wide and interesting class of G-functions whose invariant group is the orthogonal group. In particular, we establish the generating nonlinear heat equation for such G-Brownian motion on a Riemannian manifold. We also develop the Euler-Maruyama approximation for SDEs driven by G-Brownian motion of independent interest.
Xi Geng, Zhongmin Qian, Danyu Yang

Flows Driven by Banach Space-Valued Rough Paths

Abstract
We show in this note how the machinery of \(\mathcal{C}^{1}\)-approximate flows devised in the work Flows driven by rough paths, and applied there to reprove and extend most of the results on Banach space-valued rough differential equations driven by a finite dimensional rough path can be used to deal with rough differential equations driven by an infinite dimensional Banach space-valued weak geometric Hölder p-rough paths, for any p > 2, giving back Lyons’ theory in its full force in a simple way.
Ismaël Bailleul

Some Properties of Path Measures

Abstract
We call any measure on a path space, a path measure. Some notions about path measures which appear naturally when solving the Schrödinger problem are presented and worked out in detail.
Christian Léonard

Semi Log-Concave Markov Diffusions

Abstract
In this paper we intend to give a comprehensive approach of functional inequalities for diffusion processes under various “curvature” assumptions. One of them coincides with the usual Γ 2 curvature of Bakry and Emery in the case of a (reversible) drifted Brownian motion, but differs for more general diffusion processes. Our approach using simple coupling arguments together with classical stochastic tools, allows us to obtain new results, to recover and to extend already known results, giving in many situations explicit (though non optimal) bounds. In particular, we show new results for gradient/semigroup commutation in the log concave case. Some new convergence to equilibrium in the granular media equation is also exhibited.
P. Cattiaux, A. Guillin

On Maximal Inequalities for Purely Discontinuous Martingales in Infinite Dimensions

Abstract
The purpose of this paper is to give a survey of a class of maximal inequalities for purely discontinuous martingales, as well as for stochastic integral and convolutions with respect to Poisson measures, in infinite dimensional spaces. Such maximal inequalities are important in the study of stochastic partial differential equations with noise of jump type.
Carlo Marinelli, Michael Röckner

Admissible Trading Strategies Under Transaction Costs

Abstract
A well known result in stochastic analysis reads as follows: for an \(\mathbb{R}\)-valued super-martingale X = (X t )0 ≤ t ≤ T such that the terminal value X T is non-negative, we have that the entire process X is non-negative. An analogous result holds true in the no arbitrage theory of mathematical finance: under the assumption of no arbitrage, an admissible portfolio process x + (H ⋅ S) verifying x + (H ⋅ S) T  ≥ 0 also satisfies x + (H ⋅ S) t  ≥ 0, for all 0 ≤ t ≤ T. In the present paper we derive an analogous result in the presence of transaction costs. In fact, we give two versions: one with a numéraire-based, and one with a numéraire-free notion of admissibility. It turns out that this distinction on the primal side perfectly corresponds to the difference between local martingales and true martingales on the dual side. A counter-example reveals that the consideration of transaction costs makes things more delicate than in the frictionless setting.
Walter Schachermayer

Potentials of Stable Processes

Abstract
For a stable process, we give an explicit formula for the potential measure of the process killed outside a bounded interval and the joint law of the overshoot, undershoot and undershoot from the maximum at exit from a bounded interval. We obtain the equivalent quantities for a stable process reflected in its infimum. The results are obtained by exploiting a simple connection with the Lamperti representation and exit problems of stable processes.
A. E. Kyprianou, A. R. Watson

Unimodality of Hitting Times for Stable Processes

Abstract
We show that the hitting times for points of real α-stable Lévy processes (1 < α ≤ 2) are unimodal random variables. The argument relies on strong unimodality and several recent multiplicative identities in law. In the symmetric case we use a factorization of Yano et al. (Sémin Probab XLII:187–227, 2009), whereas in the completely asymmetric case we apply an identity of the second author (Simon, Stochastics 83(2):203–214, 2011). The method extends to the general case thanks to a fractional moment evaluation due to Kuznetsov et al. (Electr. J. Probab. 19:30, 1–26, 2014), for which we also provide a short independent proof.
Julien Letemplier, Thomas Simon

On the Law of a Triplet Associated with the Pseudo-Brownian Bridge

Abstract
We identify the distribution of a natural triplet associated with the pseudo-Brownian bridge. In particular, for B a Brownian motion and T 1 its first hitting time of the level one, this remarkable law allows us to understand some properties of the process \((B_{\mathit{uT}_{1}}/\sqrt{T_{1}},\ u \leq 1)\) under uniform random sampling, a study started in (Elie, Rosenbaum, and Yor, On the expectation of normalized Brownian functionals up to first hitting times, Preprint, arXiv:1310.1181, 2013).
Mathieu Rosenbaum, Marc Yor

Skew-Product Decomposition of Planar Brownian Motion and Complementability

Abstract
Let Z be a complex Brownian motion starting at 0 and W the complex Brownian motion defined by
$$\displaystyle{W_{t} =\int _{ 0}^{t} \frac{\,\overline{\!Z_{s}\!\!}\,\,} {\vert Z_{s}\vert }\,\mathrm{d}Z_{s}\;.}$$
The natural filtration \(\mathcal{F}^{W}\) of W is the filtration generated by Z up to an arbitrary rotation. We show that given any two different matrices Q 1 and Q 2 in O 2(R), there exists an \(\,\mathcal{F}^{Z}\)-previsible process H taking values in {Q 1, Q 2} such that the Brownian motion ∫ H ⋅ dW generates the whole filtration \(\,\mathcal{F}^{Z}\). As a consequence, for all a and b in R such that \(\,a^{2} + b^{2} = 1\), the Brownian motion a(W) + b(W) is complementable in \(\,\mathcal{F}^{Z}\).
Jean Brossard, Michel Émery, Christophe Leuridan

On the Exactness of the Lévy-Transformation

Abstract
In a recent paper we gave a sufficient condition for the strong mixing property of the Lévy-transformation. In this note we show that it actually implies a much stronger property, namely exactness.
Vilmos Prokaj

Multi-Occupation Field Generates the Borel-Sigma-Field of Loops

Abstract
In this article, we consider the space of càdlàg loops on a Polish space S. The loop space can be equipped with a “Skorokhod” metric. Moreover, it is Polish under this metric. Our main result is to prove that the Borel-σ-field on the space of loops is generated by a class of loop functionals: the multi-occupation field. This result generalizes the result in the discrete case, see (Le Jan, Markov Paths, Loops and Fields, vol. 2026, Springer, Heidelberg, 2011).
Yinshan Chang

Ergodicity, Decisions, and Partial Information

Abstract
In the simplest sequential decision problem for an ergodic stochastic process X, at each time n a decision u n is made as a function of past observations \(X_{0},\ldots,X_{n-1}\), and a loss l(u n , X n ) is incurred. In this setting, it is known that one may choose (under a mild integrability assumption) a decision strategy whose pathwise time-average loss is asymptotically smaller than that of any other strategy. The corresponding problem in the case of partial information proves to be much more delicate, however: if the process X is not observable, but decisions must be based on the observation of a different process Y, the existence of pathwise optimal strategies is not guaranteed. The aim of this paper is to exhibit connections between pathwise optimal strategies and notions from ergodic theory. The sequential decision problem is developed in the general setting of an ergodic dynamical system \((\varOmega,\mathcal{B},\mathbf{P},T)\) with partial information \(\mathcal{Y}\subseteq \mathcal{B}\). The existence of pathwise optimal strategies grounded in two basic properties: the conditional ergodic theory of the dynamical system, and the complexity of the loss function. When the loss function is not too complex, a general sufficient condition for the existence of pathwise optimal strategies is that the dynamical system is a conditional K-automorphism relative to the past observations \(\bigvee _{n\geq 0}T^{n}\mathcal{Y}\). If the conditional ergodicity assumption is strengthened, the complexity assumption can be weakened. Several examples demonstrate the interplay between complexity and ergodicity, which does not arise in the case of full information. Our results also yield a decision-theoretic characterization of weak mixing in ergodic theory, and establish pathwise optimality of ergodic nonlinear filters.
Ramon van Handel

Invariance Principle for the Random Walk Conditioned to Have Few Zeros

Abstract
We consider a nearest neighbor random walk on \(\mathbb{Z}\) starting at zero, conditioned to return at zero at time 2n and to have a number z n of zeros on (0, 2n]. As \(n \rightarrow +\infty \), if \(z_{n} = o(\sqrt{n})\), we show that the rescaled random walk converges toward the Brownian excursion normalized to have unit duration. This generalizes the classical result for the case z n  ≡ 1.
Laurent Serlet

A Short Proof of Stein’s Universal Multiplier Theorem

Abstract
We give a short proof of Stein’s universal multiplier theorem, purely by probabilistic methods, thus avoiding any use of harmonic analysis techniques (complex interpolation or transference methods).
Dario Trevisan

On a Flow of Operators Associated to Virtual Permutations

Abstract
In (Comptes Rend Acad Sci Paris 316:773–778, 1993), Kerov, Olshanski and Vershik introduce the so-called virtual permutations, defined as families of permutations \((\sigma _{N})_{N\geq 1}\), σ N in the symmetric group of order N, such that the cycle structure of σ N can be deduced from the structure of σ N+1 simply by removing the element N + 1. The virtual permutations, and in particular the probability measures on the corresponding space which are invariant by conjugation, have been studied in a more detailed way by Tsilevich in (J Math Sci 87(6):4072–4081, 1997) and (Theory Probab Appl 44(1):60–74, 1999). In the present article, we prove that for a large class of such invariant measures (containing in particular the Ewens measure of any parameter θ ≥ 0), it is possible to associate a flow \((T^{\alpha })_{\alpha \in \mathbb{R}}\) of random operators on a suitable function space. Moreover, if \((\sigma _{N})_{N\geq 1}\) is a random virtual permutation following a distribution in the class described above, the operator T α can be interpreted as the limit, in a sense which has to be made precise, of the permutation \(\sigma _{N}^{\alpha _{N}}\), where N goes to infinity and α N is equivalent to α N. In relation with this interpretation, we prove that the eigenvalues of the infinitesimal generator of \((T^{\alpha })_{\alpha \in \mathbb{R}}\) are equal to the limit of the rescaled eigenangles of the permutation matrix associated to σ N .
Joseph Najnudel, Ashkan Nikeghbali

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