Jump-type Processes Research Articles

Lévy processes in bounded domains: path-wise reflection scenarios and signatures of confinement

We discuss an impact of various (path-wise) reflection-from-the barrier scenarios upon confining properties of a paradigmatic family of symmetric α-stable Lévy processes, whose permanent residence in a finite interval on a line is secured by a two-sided reflection. Depending on the specific reflection ‘mechanism’, the inferred jump-type processes differ in their spectral and statistical characteristics, like e.g. relaxation properties, and functional shapes of invariant (equilibrium, or asymptotic near-equilibrium) probability density functions in the interval. The analysis is carried out in conjunction with attempts to give meaning to the notion of a reflecting Lévy process, in terms of the domain of its motion generator, to which an invariant pdf (actually an eigenfunction) does belong.

Geometric approximations to transition densities of Jump-type Markov processes

Abstract This paper is concerned with the transition functions of symmetric Levy-type processes generated by a pseudo-differential operator with variable coefficients. We first give the general estimates of heat kernels of jump diffusion semigroups, which leads to diagonal estimates of transition function and subordination in the context of two-dimensional Cauchy semigroup. Then off-diagonal estimates of special classes of Levy-type processes where transition function can be expressed using the diagonal estimation results and related metrics are derived. Furthermore, we show geometric approximation of the general two-dimensional Levy processes, and graphical experiments have been made by freezing the coefficients of the generators.

Dynamic Quantum Games

Quantum games represent the really twenty-first century branch of game theory, tightly linked to the modern development of quantum computing and quantum technologies. The main accent in these developments so far was made on stationary or repeated games. In this paper, we aim at initiating the truly dynamic theory with strategies chosen by players in real time. Since direct continuous observations are known to destroy quantum evolutions (so-called quantum Zeno paradox), the necessary new ingredient for quantum dynamic games must be the theory of non-direct observations and the corresponding quantum filtering. Apart from the technical problems in organizing feedback quantum control in real time, the difficulty in applying this theory for obtaining mathematically amenable control systems is due partially to the fact that it leads usually to rather non-trivial jump-type Markov processes and/or degenerate diffusions on manifolds, for which the corresponding control is very difficult to handle. The starting point for the present research is the remarkable discovery (quite unexpected, at least to the author) that there exists a very natural class of homodyne detections such that the diffusion processes on projective spaces resulting by filtering under such arrangements coincide exactly with the standard Brownian motions (BM) on these spaces. In some cases, one can even reduce the process to the plain BM on Euclidean spaces or tori. The theory of such motions is well studied making it possible to develop a tractable theory of related control and games, which can be at the same time practically implemented on quantum optical devices.

Quantum Mean-Field Games with the Observations of Counting Type

Quantum games and mean-field games (MFG) represent two important new branches of game theory. In a recent paper the author developed quantum MFGs merging these two branches. These quantum MFGs were based on the theory of continuous quantum observations and filtering of diffusive type. In the present paper we develop the analogous quantum MFG theory based on continuous quantum observations and filtering of counting type. However, proving existence and uniqueness of the solutions for resulting limiting forward-backward system based on jump-type processes on manifolds seems to be more complicated than for diffusions. In this paper we only prove that if a solution exists, then it gives an ϵ-Nash equilibrium for the corresponding N-player quantum game. The existence of solutions is suggested as an interesting open problem.

ВЫСОКОТЕХНОЛОГИЧНОЕ РАЗВИТИЕ ПРЕДПРИЯТИЯ: ФОРМИРОВАНИЕ ИНТЕГРАЦИОННО-БАЛАНСИРУЮЩЕГО МЕХАНИЗМА УПРАВЛЕНИЯ

The article substantiates the relevance of the formation of a management mechanism for high-tech development of an enterprise in the context of the transformation of organizational and technological structures and the uncertainty of their impact. It is proposed to solve the problems of imperfection of management methods for high-tech development of an enterprise on the basis of the methodology of integration-balancing management. A hypothesis for improving the efficiency of the results of short-term jump-type processes of transition to a stable high-tech type of development based on the results of analog-digital modeling is formulated. It implies the possibility of regulating processes based on the economic interpretation of a certain set of management functions for such development. The hypothesis is proved using the mathematical tools of analytical approximation of generalized stepwise functions. It has been established that the assumptions on the sustainability of the effective implementation of management processes for the innovative development of an enterprise are achievable in the formed complex of objects of complementary fields of activity. Such results are achieved and maintained in the target values by methods of improving the quality of management and expanding the space for integrating methods and resources in the spheres of education, science and production. New organizational approaches to the formation of a development management center are proposed, additional functions of regulating the quality of management based on the integration of resources are formulated. Integration is provided by a new tool set, represented by an analog-digital model of an integration-balancing type of control mechanism, including information management systems and controlling the sustainability of effective development. The simulation results have proved practical recommendations for improving management at a small innovation enterprise in the context of interaction with the facilities of the complex. This helps to implement the project of its transfer to the high-tech type of development according to the factors of the V–VI organizational and technological structures of the economy.

Fractional Laplacians in bounded domains: Killed, reflected, censored, and taboo Lévy flights.

The fractional Laplacian (-Δ)^{α/2}, α∈(0,2), has many equivalent (albeit formally different) realizations as a nonlocal generator of a family of α-stable stochastic processes in R^{n}. On the other hand, if the process is to be restricted to a bounded domain, there are many inequivalent proposals for what a boundary-data-respecting fractional Laplacian should actually be. This ambiguity not only holds true for each specific choice of the process behavior at the boundary (e.g., absorbtion, reflection, conditioning, or boundary taboos), but extends as well to its particular technical implementation (Dirichlet, Neumann, etc., problems). The inferred jump-type processes are inequivalent as well, differing in their spectral and statistical characteristics, which may strongly influence the ability of the formalism (if uncritically adopted) to provide an unambiguous description of real geometrically confined physical systems with disorder. Specifically that refers to their relaxation properties and the near-equilibrium asymptotic behavior. In the present paper we focus on Lévy flight-induced jump-type processes which are constrained to stay forever inside a finite domain. This refers to a concept of taboo processes (imported from Brownian to Lévy-stable contexts), to so-called censored processes, and to reflected Lévy flights whose status still remains to be unequivocally settled. As a by-product of our fractional spectral analysis, with reference to Neumann boundary conditions, we discuss disordered semiconducting heterojunctions as the bounded domain problem.

Finite system scheme for mutually catalytic branching with infinite branching rate

For many stochastic diffusion processes with mean field interaction, convergence of the rescaled total mass processes towards a diffusion process is known. Here, we show convergence of the so-called finite system scheme for interacting jump-type processes known as mutually catalytic branching processes with infinite branching rate. Due to the lack of second moments, the rescaling of time is different from the finite rate mutually catalytic case. The limit of rescaled total mass processes is identified as the finite rate mutually catalytic branching diffusion. The convergence of rescaled processes holds jointly with convergence of coordinate processes, where the latter converge at a different time scale.

A Malliavin–Skorohod calculus in L0 and L1 for additive and Volterra-type processes

In this paper we develop a Malliavin–Skorohod type calculus for additive processes in the and settings, extending the probabilistic interpretation of the Malliavin–Skorohod operators to this context. We prove calculus rules and obtain a generalization of the Clark–Hausmann–Ocone formula for random variables in . Our theory is then applied to extend the stochastic integration with respect to volatility modulated Lévy-driven Volterra processes recently introduced in the literature. Our work yields to substantially weaker conditions that permit to cover integration with respect to e.g. Volterra processes driven by -stable processes with . The presentation focuses on jump type processes.

Nonlocal Random Motions and the Trapping Problem

L\'evy stable (jump-type) processes are examples of intrinsically nonlocal random motions. This property becomes a serious obstacle if one attempts to model conditions under which a particular L\'evy process may be subject to physically implementable manipulations, whose ultimate goal is to confine the random motion in a spatially finite, possibly mesoscopic trap. We analyze thisissue for an exemplary case of the Cauchy process in a finiteinterval. Qualitatively, our observations extend to general jump-type processes that are driven by non-gaussian noises, classified by the integral part of the L\'evy-Khintchine formula.For clarity of arguments we discuss, as a reference model, the classic case of the Brownian motion in the interval.

Path-wise versus kinetic modeling for equilibrating non-Langevin jump-type processes

Abstract We discuss two independent methods of solution of a master equation whose biased jump transition rates account for long jumps of Lévy-stable type and admit a Boltzmannian (thermal) equilibrium to arise in the large time asymptotics of a probability density function ρ(x, t). Our main goal is to demonstrate a compatibility of a direct solution method (an explicit, albeit numerically assisted, integration of the master equation) with an indirect pathwise procedure, recently proposed in [Physica A 392, 3485, (2013)] as a valid tool for a dynamical analysis of non-Langevin jump-type processes. The path-wise method heavily relies on an accumulation of large sample path data, that are generated by means of a properly tailored Gillespie’s algorithm. Their statistical analysis in turn allows to infer the dynamics of ρ(x, t). However, no consistency check has been completed so far to demonstrate that both methods are fully compatible and indeed provide a solution of the same dynamical problem. Presently we remove this gap, with a focus on potential deficiencies (various cutoffs, including those upon the jump size) of approximations involved in simulation routines and solutions protocols.

Lévy flights in confining environments: Random paths and their statistics

We analyze a specific class of random systems that, while being driven by a symmetric Lévy stable noise, asymptotically set down at the Boltzmann-type equilibrium, represented by a probability density function (pdf) ρ∗(x)∼exp[−Φ(x)]. This behavior needs to be contrasted with the standard Langevin representation of Lévy jump-type processes. It is known that the choice of the drift function in the Newtonian form ∼−∇Φ excludes the existence of the Boltzmannian pdf ∼exp[−Φ(x)] (Eliazar–Klafter no go theorem). In view of this incompatibility statement, our main goal here is to establish the appropriate path-wise description of the equilibrating jump-type process. A priori given inputs are (i) jump transition rates entering the master equation for ρ(x,t) and (ii) the target (invariant) pdf ρ∗(x) of that equation, in the Boltzmannian form. We resort to numerical methods and construct a suitable modification of the Gillespie algorithm, originally invented in the chemical kinetics context. The generated sample trajectories show up a qualitative typicality, e.g. they display structural features of jumping paths (predominance of small vs large jumps) specific to particular stability indices μ∈(0,2). The obtained random paths statistical data allow us to infer an associated pdf ρ(x,t) dynamics which stands for a validity check of our procedure. The considered exemplary Boltzmannian equilibria ∼exp[−Φ(x)] refer to (i) harmonic potential Φ∼x2, (ii) logarithmic potential Φ∼nln(1+x2) with n=1,2 and (iii) locally periodic confining potential Φ∼sin2(2πx),|x|≤2, Φ∼(x2−4),|x|>2.

Trajectory Statistics of Confined L\'evy Flights and Boltzmann-type Equilibria

We analyze a specific class of random systems that are driven by a symmetric L\'{e}vy stable noise, where Langevin representation is absent. In view of the L\'{e}vy noise sensitivity to environmental inhomogeneities, the pertinent random motion asymptotically sets down at the Boltzmann-type equilibrium, represented by a probability density function (pdf) $\rho_*(x) \sim \exp [-\Phi (x)]$. Here, we infer pdf $\rho (x,t)$ based on numerical path-wise simulation of the underlying jump-type process. A priori given data are jump transition rates entering the master equation for $\rho (x,t)$ and its target pdf $\rho_*(x)$. To simulate the above processes, we construct a suitable modification of the Gillespie algorithm, originally invented in the chemical kinetics context. We exemplified our algorithm simulating different jump-type processes and discuss the dynamics of real physical systems where it can be useful.

Global heat kernel estimates for symmetric jump processes

In this paper, we study sharp heat kernel estimates for a large class of symmetric jump-type processes in R d for all t > 0. A prototype of the processes under consideration are symmetric jump processes on R d with jumping intensity

Cauchy flights in confining potentials

We analyze confining mechanisms for Lévy flights evolving under an influence of external potentials. Given a stationary probability density function (pdf), we address the reverse engineering problem: design a jump-type stochastic process whose target pdf (eventually asymptotic) equals the preselected one. To this end, dynamically distinct jump-type processes can be employed. We demonstrate that one “targeted stochasticity” scenario involves Langevin systems with a symmetric stable noise. Another derives from the Lévy–Schrödinger semigroup dynamics (closely linked with topologically induced super-diffusions), which has no standard Langevin representation. For computational and visualization purposes, the Cauchy driver is employed to exemplify our considerations.

On option pricing under a completely random measure via a generalized Esscher transform

In this paper, we develop an option valuation model when the price dynamics of the underlying risky asset is governed by the exponential of a pure jump process specified by a shifted kernel-biased completely random measure. The class of kernel-biased completely random measures is a rich class of jump-type processes introduced in [James, L.F., 2005. Bayesian Poisson process partition calculus with an application to Bayesian Lévy moving averages. Ann. Statist. 33, 1771–1799; James, L.F., 2006. Poisson calculus for spatial neutral to the right processes. Ann. Statist. 34, 416–440] and it provides a great deal of flexibility to incorporate both finite and infinite jump activities. It includes a general class of processes, namely, the generalized Gamma process, which in its turn includes the stable process, the Gamma process and the inverse Gaussian process as particular cases. The kernel-biased representation is a nice representation form and can describe different types of finite and infinite jump activities by choosing different mixing kernel functions. We employ a dynamic version of the Esscher transform, which resembles an exponential change of measures or a disintegration formula based on the Laplace functional used by James, to determine an equivalent martingale measure in the incomplete market. Closed-form option pricing formulae are obtained in some parametric cases, which provide practitioners with a convenient way to evaluate option prices.

Heat kernel estimates for jump processes of mixed types on metric measure spaces

In this paper, we investigate symmetric jump-type processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces. The class of metric measure spaces includes the Alfors d-regular sets, which is a class of fractal sets that contains geometrically self-similar sets. A typical example of our jump-type processes is the symmetric jump process with jumping intensity $$e^{-c_0 (x, y)|x-y|}\, \int_{\alpha_1}^{\alpha_2} \frac{c(\alpha, x, y)}{|x-y|^{d+\alpha}} \, \nu (d\alpha)$$ where ν is a probability measure on $$[\alpha_1, \alpha_2]\subset (0, 2)$$ , c(α, x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between two positive constants, and c 0(x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between γ1 and γ2, where either γ2 ≥ γ1 > 0 or γ1 = γ2 = 0. This example contains mixed symmetric stable processes on $${\mathbb{R}}^n$$ as well as mixed relativistic symmetric stable processes on $${\mathbb{R}}^n$$ . We establish parabolic Harnack principle and derive sharp two-sided heat kernel estimate for such jump-type processes.

Perturbations of Pseudodifferential Operators with Negative Definite Symbol

Pseudodifferential operators with negative definite symbols appear as generators of jump-type Markov processes. The purpose of this paper is to treat the large jumps of the process by a perturbation approach for the generator. This is of particular interest since in this way the generators are made accessible to a symbolic calculus of pseudodifferential operators. The main auxiliary result consists of a characterization of tightness of the jump measures in terms of the symbol.

Jump-type Fleming-Viot processes

In 1991 Perkins [7] showed that the normalized critical binary branching process is a time inhomogeneous Fleming-Viot process. In the present paper we extend this result to jump-type branching processes and we show that the normalized jump-type branching processes are in a new class of probability measure-valued processes which will be called ‘jump-type Fleming-Viot processes’. Furthermore we also show that by using these processes it is possible to introduce another new class of measure-valued processes which are obtained by the combination of jump-type branching processes and Fleming-Viot processes.

Jump-type Fleming-Viot processes

In 1991 Perkins [7] showed that the normalized critical binary branching process is a time inhomogeneous Fleming-Viot process. In the present paper we extend this result to jump-type branching processes and we show that the normalized jump-type branching processes are in a new class of probability measure-valued processes which will be called ‘jump-type Fleming-Viot processes’. Furthermore we also show that by using these processes it is possible to introduce another new class of measure-valued processes which are obtained by the combination of jump-type branching processes and Fleming-Viot processes.

Cauchy noise and affiliated stochastic processes

By departing from the previous attempt [Phys. Rev. E 51, 4114 (1995)] we give a detailed construction of conditional and perturbed Markov processes, under the assumption that the Cauchy law of probability replaces the Gaussian law (appropriate for the Wiener process) as the model of primordial noise. All considered processes are regarded as probabilistic solutions of the so-called Schrödinger interpolation problem, whose validity is thus extended to the jump-type processes and their step process approximants.