Representation of asymptotic values for nonexpansive stochastic control systems

In ergodic stochastic problems the limit of the value function $V_{\lambda }$ of the associated discounted cost functional with infinite time horizon is studied, when the discounted factor $\lambda$ tends to zero. These problems have been well studied in the literature and the used assumptions guarantee that the value function $\lambda V_{\lambda }$ converges uniformly to a constant as $\lambda \to 0$. The objective of this work consists in studying these problems under the assumption, namely, the nonexpansivity assumption, under which the limit function is not necessarily constant. Our discussion goes beyond the case of the stochastic control problem with infinite time horizon and discusses also $V_{\lambda }$ given by a Hamilton–Jacobi–Bellman equation of second order which is not necessarily associated with a stochastic control problem. On the other hand, the stochastic control case generalizes considerably earlier works by considering cost functionals defined through a backward stochastic differential equation with infinite time horizon and we give an explicit representation formula for the limit of $\lambda V_{\lambda }$, as $\lambda \to 0$.     

Absolute continuity of the law for the two dimensional stochastic Navier–Stokes equations

We consider the two dimensional Navier–Stokes equations in vorticity form with a stochastic forcing term given by a gaussian noise, white in time and colored in space. First, we prove existence and uniqueness of a weak (in the Walsh sense) solution process $\xi$ and we show that, if the initial vorticity ${\xi }_0$ is continuous in space, then there exists a space–time continuous version of the solution. In addition we show that the solution $\xi (t,x)$ (evaluated at fixed points in time and space) is locally differentiable in the Malliavin calculus sense and that its image law is absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}$.     

Fluctuation theory for level-dependent Lévy risk processes

A level-dependent Lévy process solves the stochastic differential equation $dU\left(t\right)=dX\left(t\right)??\left(U\left(t\right)\right)dt$, where $X$ is a spectrally negative Lévy process. A special case is a multi-refracted Lévy process with ${?}_k\left(x\right)={\sum }_{j=1}^k{\delta }_j{\mathbf{1}}_{\{x\ge b_j\}}$. A general rate function $?$ that is non-decreasing and locally Lipschitz continuous is also considered. We discuss solutions of the above stochastic differential equation and investigate the so-called scale functions, which are counterparts of the scale functions from the theory of Lévy processes. We show how fluctuation identities for $U$ can be expressed via these scale functions. We demonstrate that the derivatives of the scale functions are solutions of Volterra integral equations.     

On the centre of mass of a random walk

For a random walk $S_n$ on ${\mathbb{R}}^d$ we study the asymptotic behaviour of the associated centre of mass process $G_n=n^{?1}{\sum }_{i=1}^n{S}_i$. For lattice distributions we give conditions for a local limit theorem to hold. We prove that if the increments of the walk have zero mean and finite second moment, $G_n$ is recurrent if $d=1$ and transient if $d\ge 2$. In the transient case we show that $G_n$ has a diffusive rate of escape. These results extend work of Grill, who considered simple symmetric random walk. We also give a class of random walks with symmetric heavy-tailed increments for which $G_n$ is transient in $d=1$.     

Properties of G-martingales with finite variation and the application to G-Sobolev spaces

As is known, if $B={\left(B_t\right)}_{t\in [0,T]}$ is a $G$-Brownian motion, a process of form ${\int }_0^t{\eta }_{s}d{〈B〉}_s?{\int }_0^{t}2G\left({\eta }_s\right)ds$, $\eta \in M_G^1(0,T)$, is a non-increasing $G$-martingale. In this paper, we shall show that a non-increasing $G$-martingale cannot be form of ${\int }_0^t{\eta }_{s}ds$ or ${\int }_0^t{\gamma }_{s}d{〈B〉}_s$, $\eta ,\gamma \in M_G^1(0,T)$, which implies that the decomposition for generalized $G$-Itô processes is unique: For arbitrary $\zeta \in H_G^1(0,T)$, $\eta \in M_G^1(0,T)$ and non-increasing $G$-martingales $K,L$, if ${\int }_0^t{\zeta }_{s}d{B}_s+{\int }_0^t{\eta }_{s}ds+K_t=L_t,t\in [0,T],$then we have $\eta \equiv 0$, $\zeta \equiv 0$ and$K_t=L_t$. As an application, we give a characterization to the $G$-Sobolev spaces introduced in Peng and Song (2015).     

Cube-complements of generalized Fibonacci cubes

The generalized Fibonacci cube $Q_h\left(f\right)$ is the graph obtained from the $h$-cube $Q_h$ by removing all vertices that contain a given binary string $f$ as a substring. If $G$ is an induced subgraph of $Q_h$, then the cube-complement of $G$ is the graph induced by the vertices of $Q_h$ which are not in $G$. In particular, the cube-complement of a generalized Fibonacci cube $Q_h\left(f\right)$ is the subgraph of $Q_h$ induced by the set of all vertices that contain $f$ as a substring. The questions whether a cube-complement of a generalized Fibonacci cube is (i) connected, (ii) an isometric subgraph of a hypercube or (iii) a median graph are studied. Questions (ii) and (iii) are completely solved, i.e. the sets of binary strings that allow a graph of this class to be an isometric subgraph of a hypercube or a median graph are given. The cube-complement of a daisy cube is also studied.     

A scaling analysis of a star network with logarithmic weights

The paper investigates the properties of a class of resource allocation algorithms for communication networks: if a node of this network has $L$ requests to transmit and is idle, it tries to access the channel at a rate proportional to $log(1+L)$. A stochastic model of such an algorithm is investigated in the case of the star network, in which $J$ nodes can transmit simultaneously, but interfere with a central node 0 in such a way that node 0 cannot transmit while one of the other nodes does. One studies the impact of the log policy on these $J+1$ interacting communication nodes. A fluid scaling analysis of the network is derived with the scaling parameter $N$ being the norm of the initial state. It is shown that the asymptotic fluid behavior of the system is a consequence of the evolution of the state of the network on a specific time scale $(N^t,t\in (0,1))$. The main result is that, on this time scale and under appropriate conditions, the state of a node with index $j\ge 1$ is of the order of $N^{a_j\left(t\right)}$, with $0\le a_j\left(t\right)<1$, where $t?a_j\left(t\right)$ is a piecewise linear function. Convergence results on the fluid time scale and a stability property are derived as a consequence of this study.     

Ergodic aspects of some Ornstein–Uhlenbeck type processes related to Lévy processes

This work concerns the Ornstein–Uhlenbeck type process associated to a positive self-similar Markov process ${\left(X\left(t\right)\right)}_{t\ge 0}$ which drifts to $\infty$, namely $U\left(t\right)?{\mathrm{e}}^{?t}X({\mathrm{e}}^t?1)$. We point out that $U$ is always a (topologically) recurrent ergodic Markov process. We identify its invariant measure in terms of the law of the exponential functional $\stackrel{?}{I}?{\int }_0^{\infty }exp\left({\stackrel{?}{\xi }}_s\right)\mathrm{d}s$, where $\stackrel{?}{\xi }$ is the dual of the real-valued Lévy process $\xi$ related to $X$ by the Lamperti transformation. This invariant measure is infinite (i.e. $U$ is null-recurrent) if and only if ${\xi }_1?L^1\left(\mathbb{P}\right)$. In that case, we determine the family of Lévy processes $\xi$ for which $U$ fulfills the conclusions of the Darling–Kac theorem. Our approach relies crucially on a remarkable connection due to Patie (Patie, 2008) with another generalized Ornstein–Uhlenbeck process that can be associated to the Lévy process $\xi$, and properties of time-substitutions based on additive functionals.     

Small time convergence of subordinators with regularly or slowly varying canonical measure

We consider subordinators $X_{\alpha }={\left(X_{\alpha }\left(t\right)\right)}_{t\ge 0}$ in the domain of attraction at 0 of a stable subordinator ${\left(S_{\alpha }\left(t\right)\right)}_{t\ge 0}$ (where $\alpha \in (0,1)$); thus, with the property that ${\overline{\Pi }}_{\alpha }$, the tail function of the canonical measure of $X_{\alpha }$, is regularly varying of index $?\alpha \in (?1,0)$ as $x\downarrow 0$. We also analyse the boundary case, $\alpha =0$, when ${\overline{\Pi }}_{\alpha }$ is slowly varying at 0. When $\alpha \in (0,1)$, we show that ${\left(t{\overline{\Pi }}_{\alpha }\left(X_{\alpha }\left(t\right)\right)\right)}^{?1}$ converges in distribution, as $t\downarrow 0$, to the random variable ${\left(S_{\alpha }\left(1\right)\right)}^{\alpha }$. This latter random variable, as a function of $\alpha$, converges in distribution as $\alpha \downarrow 0$ to the inverse of an exponential random variable. We prove these convergences, also generalised to functional versions (convergence in $\mathbb{D}[0,1]$), and to trimmed versions, whereby a fixed number of its largest jumps up to a specified time are subtracted from the process. The $\alpha =0$ case produces convergence to an extremal process constructed from ordered jumps of a Cauchy subordinator. Our results generalise random walk and stable process results of Darling, Cressie, Kasahara, Kotani and Watanabe.     

Nonlinear stochastic time-fractional slow and fast diffusion equations on Rd

This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables: $\left({?}^{\beta }+\frac{\nu }{2}{(?\Delta )}^{\alpha ∕2}\right)u(t,x)=I_t^{\gamma }\left[\rho \left(u(t,x)\right)\stackrel{?}{W}(t,x)\right],t>0,x\in {\mathbb{R}}^d,$where $\stackrel{?}{W}$ is the space–time white noise, $\alpha \in (0,2]$, $\beta \in (0,2)$, $\gamma \ge 0$ and $\nu >0$. Fundamental solutions and their properties, in particular the nonnegativity, are derived. The existence and uniqueness of solution together with the moment bounds of the solution are obtained under Dalang’s condition: $d<2\alpha +\frac{\alpha }{\beta }min(2\gamma ?1,0)$. In some cases, the initial data can be measures. When $\beta \in (0,1]$, we prove the sample path regularity of the solution.     

The survival probability of critical and subcritical branching processes in finite state space Markovian environment

Let ${\left(Z_n\right)}_{n\ge 0}$ be a branching process in a random environment defined by a Markov chain ${\left(X_n\right)}_{n\ge 0}$ with values in a finite state space $\mathbb{X}$. Let ${\mathbb{P}}_i$ be the probability law generated by the trajectories of ${\left(X_n\right)}_{n\ge 0}$ starting at $X_0=i\in \mathbb{X}.$ We study the asymptotic behaviour of the joint survival probability ${\mathbb{P}}_i\left(Z_n>0,X_n=j\right)$, $j\in \mathbb{X}$ as $n\to +\infty$ in the critical and strongly, intermediate and weakly subcritical cases.     

Strong approximation and a central limit theorem for St. Petersburg sums

The St. Petersburg paradox (Bernoulli, 1738) concerns the fair entry fee in a game where the winnings are distributed as $P(X=2^k)=2^{?k},k=1,2,\dots$. The tails of $X$ are not regularly varying and the sequence $S_n$ of accumulated gains has, suitably centered and normalized, a class of semistable laws as subsequential limit distributions (Martin-Löf, 1985; Csörg? and Dodunekova, 1991). This has led to a clarification of the paradox and an interesting and unusual asymptotic theory in past decades. In this paper we prove that $S_n$ can be approximated by a semistable Lévy process $\{L\left(n\right),n\ge 1\}$ with a.s. error $O\left(\sqrt{n}{(logn)}^{1+\epsilon }\right)$ and, surprisingly, the error term is asymptotically normal, exhibiting an unexpected central limit theorem in St. Petersburg theory.     

Stable-like fluctuations of Biggins’ martingales

Let ${\left(W_n\left(\theta \right)\right)}_{n\in {\mathbb{N}}_0}$ be Biggins’ martingale associated with a supercritical branching random walk, and let $W\left(\theta \right)$ be its almost sure limit. Under a natural condition for the offspring point process in the branching random walk, we show that if the law of $W_1\left(\theta \right)$ belongs to the domain of normal attraction of an $\alpha$-stable distribution for some $\alpha \in (1,2)$, then, as $n\to \infty$, there is weak convergence of the tail process ${(W\left(\theta \right)?W_{n?k}\left(\theta \right))}_{k\in {\mathbb{N}}_0}$, properly normalized, to a random scale multiple of a stationary autoregressive process of order one with $\alpha$-stable marginals.     

The tamed unadjusted Langevin algorithm

In this article, we consider the problem of sampling from a probability measure $\pi$ having a density on ${\mathbb{R}}^d$ proportional to $x?{\mathrm{e}}^{?U\left(x\right)}$. The Euler discretization of the Langevin stochastic differential equation (SDE) is known to be unstable, when the potential $U$ is superlinear. Based on previous works on the taming of superlinear drift coefficients for SDEs, we introduce the Tamed Unadjusted Langevin Algorithm (TULA) and obtain non-asymptotic bounds in $V$-total variation norm and Wasserstein distance of order 2 between the iterates of TULA and $\pi$, as well as weak error bounds. Numerical experiments are presented which support our findings.