Feller Processes Research Articles

On Exponential Functionals of Lévy Processes

Exponential functionals of Levy processes appear as stationary distributions of generalized Ornstein–Uhlenbeck (GOU) processes. In this paper we obtain the infinitesimal generator of the GOU process and show that it is a Feller process. Further, we use these results to investigate properties of the mapping $$\Phi $$ , which maps two independent Levy processes to their corresponding exponential functional, where one of the processes is assumed to be fixed. We show that in many cases this mapping is injective, and give the inverse mapping in terms of (Levy) characteristics. Also, continuity of $$\Phi $$ is treated, and some results on its range are obtained.

Long-time behavior of stable-like processes

In this paper, we consider a long-time behavior of stable-like processes. A stable-like process is a Feller process given by the symbol p(x,ξ)=−iβ(x)ξ+γ(x)|ξ|α(x), where α(x)∈(0,2), β(x)∈R and γ(x)∈(0,∞). More precisely, we give sufficient conditions for recurrence, transience and ergodicity of stable-like processes in terms of the stability function α(x), the drift function β(x) and the scaling function γ(x). Further, as a special case of these results we give a new proof for the recurrence and transience property of one-dimensional symmetric stable Lévy processes with the index of stability α≠1.

Publisher's Note: First-passage and escape problems in the Feller process [Phys. Rev. E86, 041116 (2012)

Publisher's Note: First-passage and escape problems in the Feller process [Phys. Rev. E<b>86</b>, 041116 (2012)

First-passage and escape problems in the Feller process

The Feller process is an one-dimensional diffusion process with linear drift and state-dependent diffusion coefficient vanishing at the origin. The process is positive definite and it is this property along with its linear character that have made Feller process a convenient candidate for the modeling of a number of phenomena ranging from single-neuron firing to volatility of financial assets. While general properties of the process have long been well known, less known are properties related to level crossing such as the first-passage and the escape problems. In this work we thoroughly address these questions.

Construction of ${\mathcal L}^{{p}}$ -strong Feller Processes via Dirichlet Forms and Applications to Elliptic Diffusions

We provide a general construction scheme for $\mathcal L^p$-strong Feller processes on locally compact separable metric spaces. Starting from a regular Dirichlet form and specified regularity assumptions, we construct an associated semigroup and resolvents of kernels having the $\mathcal L^p$-strong Feller property. They allow us to construct a process which solves the corresponding martingale problem for all starting points from a known set, namely the set where the regularity assumptions hold. We apply this result to construct elliptic diffusions having locally Lipschitz matrix coefficients and singular drifts on general open sets with absorption at the boundary. In this application elliptic regularity results imply the desired regularity assumptions.

LAGRANGIAN AND HAMILTONIAN FEYNMAN FORMULAE FOR SOME FELLER SEMIGROUPS AND THEIR PERTURBATIONS

A Feynman formula is a representation of a solution of an initial (or initial-boundary) value problem for an evolution equation (or, equivalently, a representation of the semigroup resolving the problem) by a limit of n-fold iterated integrals of some elementary functions as n → ∞. In this note we obtain some Feynman formulae for a class of semigroups associated with Feller processes. Finite-dimensional integrals in the Feynman formulae give approximations for functional integrals in some Feynman–Kac formulae corresponding to the underlying processes. Hence, these Feynman formulae give an effective tool to calculate functional integrals with respect to probability measures generated by these Feller processes and, in particular, to obtain simulations of Feller processes.

Some theorems on Feller processes: Transience, local times and ultracontractivity

We present sufficient conditions for the transience and the existence of local times of a Feller process, and the ultracontractivity of the associated Feller semigroup; these conditions are sharp for Lévy processes. The proof uses a local symmetrization technique and a uniform upper bound for the characteristic function of a Feller process. As a by-product, we obtain for stable-like processes (in the sense of R. Bass) on R d \mathbb {R}^d with smooth variable index α ( x ) ∈ ( 0 , 2 ) \alpha (x)\in (0,2) a transience criterion in terms of the exponent α ( x ) \alpha (x) ; if d = 1 d=1 and inf x ∈ R α ( x ) ∈ ( 1 , 2 ) \inf _{x\in \mathbb {R}} \alpha (x)\in (1,2) , then the stable-like process has local times.

Optimal stopping problems for some Markov processes

In this paper, we solve explicitly the optimal stopping problem with random discounting and an additive functional as cost of observations for a regular linear diffusion. We also extend the results to the class of one-sided regular Feller processes. This generalizes the result of Beibel and Lerche [Statist. Sinica 7 (1997) 93-108] and [Teor. Veroyatn. Primen. 45 (2000) 657-669] and Irles and Paulsen [Sequential Anal. 23 (2004) 297-316]. Our approach relies on a combination of techniques borrowed from potential theory and stochastic calculus. We illustrate our results by detailing some new examples ranging from linear diffusions to Markov processes of the spectrally negative type.

STRONG FELLER CONTINUITY OF FELLER PROCESSES AND SEMIGROUPS

We study two equivalent characterizations of the strong Feller property for a Markov process and of the associated sub-Markovian semigroup. One is described in terms of locally uniform absolute continuity, whereas the other uses local Orlicz-ultracontractivity. These criteria generalize many existing results on strong Feller continuity and seem to be more natural for Feller processes. By establishing the estimates of the first exit time from balls, we also investigate the continuity of harmonic functions for Feller processes which enjoy the strong Feller property.

On the semimartingale nature of Feller processes with killing

Let U be an open set in Rd. We show that under a mild assumption on the richness of the generator, a Feller process in (U,B(U)) with (predictable) killing is a semimartingale. To this end, we generalize the notion of semimartingales in a natural way to those ‘with killing’. Furthermore we calculate the semimartingale characteristics of the Feller process explicitly and analyze their connections to the symbol. Finally we derive a probabilistic formula for calculating the symbol of the process.

Asymptotic approximations for stationary distributions of many-server queues with abandonment

A many-server queueing system is considered in which customers arrive according to a renewal process and have service and patience times that are drawn from two independent sequences of independent, identically distributed random variables. Customers enter service in the order of arrival and are assumed to abandon the queue if the waiting time in queue exceeds the patience time. The state of the system with $N$ servers is represented by a four-component process that consists of the forward recurrence time of the arrival process, a pair of measure-valued processes, one that keeps track of the waiting times of customers in queue and the other that keeps track of the amounts of time customers present in the system have been in service and a real-valued process that represents the total number of customers in the system. Under general assumptions, it is shown that the state process is a Feller process, admits a stationary distribution and is ergodic. It is also shown that the associated sequence of scaled stationary distributions is tight, and that any subsequence converges to an invariant state for the fluid limit. In particular, this implies that when the associated fluid limit has a unique invariant state, then the sequence of stationary distributions converges, as $N\rightarrow \infty$, to the invariant state. In addition, a simple example is given to illustrate that, both in the presence and absence of abandonments, the $N\rightarrow \infty$ and $t\rightarrow \infty$ limits cannot always be interchanged.

Felling of individual freestanding nanoobjects using focused-ion-beam milling for investigations of structural and transport properties

We report that, to enable studies of their compositional, structural and electrical properties, freestanding individual nanoobjects can be selectively felled in a controllable way by the technique of low-current focused-ion-beam (FIB) milling with the ion beam at a chosen angle of incidence to the nanoobject. To demonstrate the suitability of the technique, we report results for zigzag/straight tungsten nanowires grown vertically on support substrates and then felled for characterization. We also describe a systematic investigation of the effect of the experimental geometry and parameters on the felling process and on the induced wire-bending phenomenon. The method of felling freestanding nanoobjects using FIB is an advantageous new technique enabling investigations of the properties of selected individual nanoobjects.

Delta–Gamma hedging of mortality and interest rate risk

One of the major concerns of life insurers and pension funds is the increasing longevity of their beneficiaries. This paper studies the hedging problem of annuity cash flows when mortality and interest rates are stochastic. We first propose a Delta–Gamma hedging technique for mortality risk. The risk factor against which to hedge is the difference between the actual mortality intensity in the future and its “forecast” today, the forward intensity. We specialize the hedging technique first to the case in which mortality intensities are affine, then to Ornstein–Uhlenbeck and Feller processes, providing actuarial justifications for this selection. We show that, without imposing no arbitrage, we can get equivalent probability measures under which the HJM condition for no arbitrage is satisfied. Last, we extend our results to the presence of both interest rate and mortality risk. We provide a UK calibrated example of Delta–Gamma hedging of both mortality and interest rate risk.

Detecting the Maximum of a Scalar Diffusion with Negative Drift

Let $X$ be a scalar diffusion process with drift coefficient pointing towards the origin, i.e. $X$ is mean-reverting. We denote by $X^*$ the corresponding running maximum, $T_0$ the first time $X$ hits the level zero. Given an increasing and convex loss function $\ell$, we consider the following optimal stopping problem: $\inf_{0\leq\theta\leq T_0}\mathbb{E}[\ell(X^*_{T_0}-X_\theta)],$ over all stopping times $\theta$ with values in $[0,T_0]$. For the quadratic loss function and under mild conditions, we prove that an optimal stopping time exists and is defined by: $\theta^*=T_0\wedge\inf\{t\geq 0;~X^*_t\geq \gamma(X_t)\},$ where the boundary $\gamma$ is explicitly characterized as the concatenation of the solutions of two equations. We investigate some examples such as the Ornstein-Uhlenbeck process, the CIR--Feller process, as well as the standard and drifted Brownian motions.

Semigroups of operators that describe a Feller process on the line, which is the result of pasting together two diffusion processes

Semigroups of operators that describe a Feller process on the line, which is the result of pasting together two diffusion processes

Study of dependence for some stochastic processes: Symbolic Markov copulae

We study dependence between components of multivariate (nice Feller) Markov processes: what conditions need to be satisfied by a multivariate Markov process so that its components are Markovian with respect to the filtration of the entire process and such that they follow prescribed laws? To answer this question, we introduce a symbolic approach, which is rooted in the concept of pseudo-differential operator (PDO). We investigate connections between dependence, in the sense described above, and the PDOs. In particular, we study the problem of constructing a multivariate nice Feller process with given marginal laws in terms of symbols of the related PDOs. This approach leads to relatively simple conditions, which provide solutions to this problem.

The Euler Scheme for Feller Processes

We consider the Euler scheme for stochastic differential equations with jumps, whose intensity might be infinite and the jump structure may depend on the position. This general type of SDE is explicitly given for Feller processes and a general convergence condition is presented. In particular, the characteristic functions of the increments of the Euler scheme are calculated in terms of the symbol of the Feller process in a closed form. These increments are increments of Lévy processes and, thus, the Euler scheme can be used for simulation by applying standard techniques from Lévy processes.

On the coupling property of Lévy processes

We give necessary and sufficient conditions guaranteeing that the coupling for L\'evy processes (with non-degenerate jump part) is successful. Our method relies on explicit formulae for the transition semigroup of a compound Poisson process and earlier results by Mineka and Lindvall-Rogers on couplings of random walks. In particular, we obtain that a L\'{e}vy process admits a successful coupling, if it is a strong Feller process or if the L\'evy (jump) measure has an absolutely continuous component.

On a Burgers type nonlinear equation perturbed by a pure jump Lévy noise in Rd

In this paper, we study a stochastic fractional Burgers type nonlinear equation driven by a pure jump Lévy space–time white noise with d-dimensional spatial variables x∈Rd. Our equation involves a Markovian generator of a stable-like Feller process with variable order α(x). Under certain polynomial growth conditions, we establish the existence and uniqueness of an Lp(Rd)-valued (local) solution for the initial value problem to our equation. Our approaches are essentially based on the estimates of the fundamental solution to the stable-like Markovian generator and the Lp-theory of the stochastic integral with respect to the pure jump Lévy space–time white noise.

Pseudo-differential operators and Markov semigroups on compact Lie groups

We extend the Ruzhansky–Turunen theory of pseudo-differential operators on compact Lie groups into a tool that can be used to investigate group-valued Markov processes in the spirit of the work in Euclidean spaces of N. Jacob and collaborators. Feller semigroups, their generators and resolvents are exhibited as pseudo-differential operators and the symbols of the operators forming the semigroup are expressed in terms of the Fourier transform of the transition kernel. The symbols are explicitly computed for some examples including the Feller processes associated to stochastic flows arising from solutions of stochastic differential equations on the group driven by Lévy processes. We study a family of Lévy-type linear operators on general Lie groups that are pseudo-differential operators when the group is compact and find conditions for them to give rise to symmetric Dirichlet forms.