Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation

We develop a notion of nonlinear expectation–G$G$-expectation–generated by a nonlinear heat equation with infinitesimal generator G$G$. We first study multi-dimensional G$G$-normal distributions. With this nonlinear distribution we can introduce our G$G$-expectation under which the canonical process is a multi-dimensional G$G$-Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Itô’s type with respect to our G$G$-Brownian motion, and derive the related Itô’s formula. We have also obtained the existence and uniqueness of stochastic differential equations under our G$G$-expectation.     

Quasi-stationary distributions and Yaglom limits of self-similar Markov processes

We discuss the existence and characterization of quasi-stationary distributions and Yaglom limits of self-similar Markov processes that reach 0 in finite time. By Yaglom limit, we mean the existence of a deterministic function g$g$ and a non-trivial probability measure ν$\nu$ such that the process rescaled by g$g$ and conditioned on non-extinction converges in distribution towards ν$\nu$. We will see that a Yaglom limit exists if and only if the extinction time at 0$0$ of the process is in the domain of attraction of an extreme law and we will then treat separately three cases, according to whether the extinction time is in the domain of attraction of a Gumbel, Weibull or Fréchet law. In each of these cases, necessary and sufficient conditions on the parameters of the underlying Lévy process are given for the extinction time to be in the required domain of attraction. The limit of the process conditioned to be positive is then characterized by a multiplicative equation which is connected to a factorization of the exponential distribution in the Gumbel case, a factorization of a Beta distribution in the Weibull case and a factorization of a Pareto distribution in the Fréchet case.     

A combinatorial proof of the skew K-saturation theorem

We give a combinatorial proof of the skew version of the K-saturation theorem. More precisely, for any positive integer k$k$, we give an explicit injection from the set of skew semistandard Young tableaux with skew shape kλ/kμ$k\lambda /k\mu$ and type kν$k\nu$, to the set of skew semistandard Young tableaux of shape λ/μ$\lambda /\mu$ and type ν$\nu$.     

Oscillation of harmonic functions for subordinate Brownian motion and its applications

In this paper, we establish an oscillation estimate of nonnegative harmonic functions for a pure-jump subordinate Brownian motion. The infinitesimal generator of such subordinate Brownian motion is an integro-differential operator. As an application, we give a probabilistic proof of the following form of relative Fatou theorem for such subordinate Brownian motion X$X$ in a bounded κ$\kappa$-fat open set; if u$u$ is a positive harmonic function with respect to X$X$ in a bounded κ$\kappa$-fat open set D$D$ and h$h$ is a positive harmonic function in D$D$ vanishing on D^c$D^c$, then the non-tangential limit of u/h$u/h$ exists almost everywhere with respect to the Martin-representing measure of h$h$.     

Subconstituents of orthogonal graphs of odd characteristic – continued

The subconstituents of the orthogonal graph O(2ν+δ,q)$O(2\nu +\delta ,q)$, where ν?2$\nu ?2$ and δ∈{1,2}$\delta \in \{1,2\}$, over a finite field of odd characteristic are shown to be quasi-strongly regular. Furthermore, the first subconstituent is shown to be co-edge regular, and when ν?3$\nu ?3$ its automorphism group is determined. The second subconstituent is shown to be edge regular, and when ν?2$\nu ?2$ its automorphism group is determined. Their parameters and chromatic numbers are also determined.     

Second order backward stochastic differential equations under a monotonicity condition

In a recent paper, Soner, Touzi and Zhang (2012) [19] have introduced a notion of second order backward stochastic differential equations (2BSDEs), which are naturally linked to a class of fully non-linear PDEs. They proved existence and uniqueness for a generator which is uniformly Lipschitz in the variables y$y$ and z$z$. The aim of this paper is to extend these results to the case of a generator satisfying a monotonicity condition in y$y$. More precisely, we prove existence and uniqueness for 2BSDEs with a generator which is Lipschitz in z$z$ and uniformly continuous with linear growth in y$y$. Moreover, we emphasize throughout the paper the major difficulties and differences due to the 2BSDE framework.     

Weak convergence of censored and reflected stable processes

It is shown that if a sequence of open n$n$-sets _Dk$D_k$ increases to an open n$n$-set D$D$ then reflected stable processes in _Dk$D_k$ converge weakly to the reflected stable process in D$D$ for every starting point x$x$ in D$D$. The same result holds for censored α$\alpha$-stable processes for every x$x$ in D$D$ if D$D$ and _Dk$D_k$ satisfy the uniform Hardy inequality. Using the method in the proof of the above results, we also prove the weak convergence of reflected Brownian motions in unbounded domains.     

Approximate martingale estimating functions for stochastic differential equations with small noises

An approximate martingale estimating function with an eigenfunction is proposed for an estimation problem about an unknown drift parameter for a one-dimensional diffusion process with small perturbed parameter ε$\epsilon$ from discrete time observations at n$n$ regularly spaced time points k/n$k/n$, k=0,1,…,n$k=0,1,\dots ,n$. We show asymptotic efficiency of an M$M$-estimator derived from the approximate martingale estimating function as ε→0$\epsilon \to 0$ and n→∞$n\to \infty$ simultaneously.     

Simulation of the continuous time random walk of the space-fractional diffusion equations

In this article, we discuss the solution of the space-fractional diffusion equation with and without central linear drift in the Fourier domain and show the strong connection between it and the α$\alpha$-stable Lévy distribution, 0<α<2$0<\alpha <2$. We use some relevant transformations of the independent variables x$x$ and t$t$, to find the solution of the space-fractional diffusion equation with central linear drift which is a special form of the space-fractional Fokker–Planck equation which is useful in studying the dynamic behaviour of stochastic differential equations driven by the non-Gaussian (Lévy) noises. We simulate the continuous time random walk of these models by using the Monte Carlo method.     

A short proof of the Doob–Meyer theorem

Every submartingale S$S$ of class D$D$ has a unique Doob–Meyer decomposition S=M+A$S=M+A$, where M$M$ is a martingale and A$A$ is a predictable increasing process starting at 0.     

Backward stochastic differential equations associated to jump Markov processes and applications

In this paper we study backward stochastic differential equations (BSDEs) driven by the compensated random measure associated to a given pure jump Markov process X$X$ on a general state space K$K$. We apply these results to prove well-posedness of a class of nonlinear parabolic differential equations on K$K$, that generalize the Kolmogorov equation of X$X$. Finally we formulate and solve optimal control problems for Markov jump processes, relating the value function and the optimal control law to an appropriate BSDE that also allows to construct probabilistically the unique solution to the Hamilton–Jacobi–Bellman equation and to identify it with the value function.     

On prime ideals and radicals of polynomial rings and graded rings

We extend some known results on radicals and prime ideals from polynomial rings and Laurent polynomial rings to Z$\mathbb{Z}$-graded rings, i.e, rings graded by the additive group of integers. The main of them concerns the Brown–McCoy radical G$G$ and the radical S$S$, which for a given ring A$A$ is defined as the intersection of prime ideals I$I$ of A$A$ such that A/I$A/I$ is a ring with a large center. The studies are related to some open problems on the radicals G$G$ and S$S$ of polynomial rings and situated in the context of Koethe’s problem.     

Some results on general quadratic reflected BSDEs driven by a continuous martingale

We study the well-posedness of general reflected BSDEs driven by a continuous martingale, when the coefficient f$f$ of the driver has at most quadratic growth in the control variable Z$Z$, with a bounded terminal condition and a lower obstacle which is bounded above. We obtain the basic results in this setting: comparison and uniqueness, existence, stability. For the comparison theorem and the special comparison theorem for reflected BSDEs (which allows one to compare the increasing processes of two solutions), we give intrinsic proofs which do not rely on the comparison theorem for standard BSDEs. This allows to obtain the special comparison theorem under minimal assumptions. We obtain existence by using the fixed point theorem and then a series of perturbations, first in the case where f$f$ is Lipschitz in the primary variable Y$Y$, and then in the case where f$f$ can have slightly-superlinear growth and the case where f$f$ is monotonous in Y$Y$ with arbitrary growth. We also obtain a local Lipschitz estimate in BMO$BMO$ for the martingale part of the solution.     

The Quicksort process

Quicksort on the fly returns the input of n$n$ reals in increasing natural order during the sorting process. Correctly normalized the running time up to returning the l$l$-th smallest out of n$n$ seen as a process in l$l$ converges weakly to a limiting process with path in the space of cadlag functions.     

Hopf subalgebras and tensor powers of generalized permutation modules

By means of a certain module V$V$ and its tensor powers in a finite tensor category, we study a question of whether the depth of a Hopf subalgebra R$R$ of a finite-dimensional Hopf algebra H$H$ is finite. The module V$V$ is the counit representation induced from R$R$ to H$H$, which is then a generalized permutation module, as well as a module coalgebra. We show that if in the subalgebra pair either Hopf algebra has finite representation type, or V$V$ is either semisimple with R^∗$R^{\ast }$ pointed, projective, or its tensor powers satisfy a Burnside ring formula over a finite set of Hopf subalgebras including R$R$, then the depth of R$R$ in H$H$ is finite. One assigns a nonnegative integer depth to V$V$, or any other H$H$-module, by comparing the truncated tensor algebras of V$V$ in a finite tensor category and so obtains upper and lower bounds for depth of a Hopf subalgebra. For example, a relative Hopf restricted module has depth 1, and a permutation module of a corefree subgroup has depth less than the number of values of its character.     

Large scale properties of the IIIC for 2D percolation

We reinvestigate the 2D problem of the inhomogeneous incipient infinite cluster   where, in an independent percolation model, the density decays to _pc$p_c$ with an inverse power, λ$\lambda$, of the distance to the origin. Assuming the existence of critical exponents (as is known in the case of the triangular site lattice) if the power is less than 1/ν$1/\nu$, with ν$\nu$ the correlation length exponent, we demonstrate an infinite cluster with scale dimension given by _DH=2−βλ$D_H=2-\beta \lambda$. Further, we investigate the critical case _λc=1/ν${\lambda }_c=1/\nu$ and show that iterated logarithmic corrections will tip the balance between the possibility and impossibility of an infinite cluster.     

Strongly quasi-Hamiltonian-connected semicomplete multipartite digraphs

A semicomplete multipartite or semicomplete c$c$-partite digraph D$D$ is a biorientation of a c$c$-partite graph. A semicomplete multipartite digraph D$D$ is called strongly quasi-Hamiltonian-connected, if for any two distinct vertices x$x$ and y$y$ of D$D$, there is a path P$P$ from x$x$ to y$y$ such that P$P$ contains at least one vertex from each partite set of D$D$.     

Measurability of semimartingale characteristics with respect to the probability law

Given a càdlàg process X$X$ on a filtered measurable space, we construct a version of its semimartingale characteristics which is measurable with respect to the underlying probability law. More precisely, let _Psem${\mathfrak{P}}_{sem}$ be the set of all probability measures P$P$ under which X$X$ is a semimartingale. We construct processes (B^P,C,ν^P)$(B^P,C,{\nu }^P)$ which are jointly measurable in time, space, and the probability law P$P$, and are versions of the semimartingale characteristics of X$X$ under P$P$ for each P∈_Psem$P\in {\mathfrak{P}}_{sem}$. This result gives a general and unifying answer to measurability questions that arise in the context of quasi-sure analysis and stochastic control under the weak formulation.     

Conditional residual lifetimes of coherent systems

In this paper, we derive mixture representations for the reliability function of the conditional residual lifetime of a coherent system with n$n$ independent and identically distributed (i.i.d.) components under the condition that at least j$j$ and at most k−1$k-1$ (j<k$j<k$) components have failed by time t$t$. Based on these mixture representations, we then discuss stochastic comparisons of the conditional residual lifetimes of two coherent systems with independent and identical components.