Asymmetric non-Gaussian effects in a tumor growth model with immunization

The dynamical evolution of a tumor growth model, under immune surveillance and subject to asymmetric non-Gaussian α-stable Lévy noise, is explored. The lifetime of a tumor staying in the range between the tumor-free state and the stable tumor state, and the likelihood of noise-induced tumor extinction, are characterized by the mean residence time and the escape probability, respectively. For various initial densities of tumor cells, the mean residence time and the escape probability are computed with different noise parameters. It is observed that unlike the Gaussian noise or symmetric non-Gaussian noise, the asymmetric non-Gaussian noise plays a constructive role in the tumor evolution in this simple model. By adjusting the noise parameters, the mean residence time can be shortened and the escape probability can be increased, simultaneously. This suggests that a tumor may be mitigated with higher probability in a shorter time, under certain external environmental stimuli.     

Mortality modelling with Lévy processes

This paper addresses the modelling of human mortality by the aid of doubly stochastic processes with an intensity driven by a positive Lévy process. We focus on intensities having a mean reverting stochastic component. Furthermore, driving Lévy processes are pure jump processes belonging to the class of α-stable subordinators. In this setting, expressions of survival probabilities are inferred, the pricing is discussed and numerical applications to actuarial valuations are proposed.     

Elementary bifurcations for a simple dynamical system under non-Gaussian Lévy noises

Nonlinear dynamical systems are sometimes under the influence of random fluctuations. It is desirable to examine possible bifurcations for stochastic dynamical systems when a parameter varies.     

Lévy white noise measures on infinite-dimensional spaces: Existence and characterization of the measurable support

It is shown that a Lévy white noise measure Λ always exists as a Borel measure on the dual K^′ of the space K of C^∞ functions on R with compact support. Then a characterization theorem that ensures that the measurable support of Λ is contained in S^′ is proved. In the course of the proofs, a representation of the Lévy process as a function on K^′ is obtained and stochastic Lévy integrals are studied.     

Lévy random bridges and the modelling of financial information

The information-based asset-pricing framework of Brody-Hughston-Macrina (BHM) is extended to include a wider class of models for market information. To model the information flow, we introduce a class of processes called Lévy random bridges (LRBs), generalising the Brownian bridge and gamma bridge information processes of BHM. Given its terminal value at T, an LRB has the law of a Lévy bridge. We consider an asset that generates a cash-flow X_T at T. The information about X_T is modelled by an LRB with terminal value X_T. The price process of the asset is worked out, along with the prices of options.     

Gated Polling Systems with Lévy Inflow and Inter-Dependent Switchover Times: A Dynamical-Systems Approach

We study asymmetric polling systems where: (i) the incoming workflow processes follow general Lévy-subordinator statistics; and, (ii) the server attends the channels according to the gated service regime, and incurs random inter-dependentswitchover times when moving from one channel to the other. The analysis follows a dynamical-systems approach: a stochastic Poincaré map, governing the one-cycle dynamics of the polling system is introduced, and its statistical characteristics are studied. Explicit formulae regarding the evolution of the mean, covariance, and Laplace transform of the Poincaré map are derived. The forward orbit of the maps transform – a nonlinear deterministic dynamical system in Laplace space – fully characterizes the stochastic dynamics of the polling system. This enables us to explore the long-term behavior of the system: we prove convergence to a (unique) steady-state equilibrium, prove the equilibrium is stationary, and compute its statistical characteristics.     

Limit Processes for Age-Dependent Branching Particle Systems

We consider systems of spatially distributed branching particles in R ^d . The particle lifelengths are of general form, hence the time propagation of the system is typically not Markov. A natural time-space-mass scaling is applied to a sequence of particle systems and we derive limit results for the corresponding sequence of measure-valued processes. The limit is identified as the projection on R ^d of a superprocess in R ₊×R ^d . The additive functional characterizing the superprocess is the scaling limit of certain point processes, which count generations along a line of descent for the branching particles.     

A Stochastic Process Generated by the Lévy Laplacian

In this paper we give a stochastic process generated by the Lévy Laplacian in the white noise analysis with a characterization of the Laplacian.     

Synchronization of systems of Marcus canonical equations driven by α-stable noises

Dynamical systems driven by Gaussian noises have been considered extensively in modeling, simulation and theory. However, complex systems in engineering and science are often subject to non-Gaussian fluctuations or uncertainties. A coupled dynamical system under non-Gaussian Lévy noise is considered. After discussing cocycle property, stationary orbits and random attractors, a synchronization phenomenon is shown to occur, when the drift terms of the coupled system satisfy certain dissipativity conditions. The synchronization result implies that coupled dynamical systems share a dynamical feature in certain asymptotic sense.     

The Segal-Bargmann transform for Lévy white noise functionals associated with non-integrable Lévy processes

By using a method of truncation, we derive the closed form of the Segal-Bargmann transform of Lévy white noise functionals associated with a Lévy process with the Lévy spectrum without the moment condition. Besides, a sufficient and necessary condition to the existence of Lévy stochastic integrals is obtained.     

Optimal control and dependence modeling of insurance portfolios with Lévy dynamics

In this paper we are interested in optimizing proportional reinsurance and investment policies in a multidimensional Lévy-driven insurance model. The criterion is that of maximizing exponential utility. Solving the classical Hamilton-Jacobi-Bellman equation yields that the optimal retention level keeps a constant amount of claims regardless of time and the company’s wealth level.A special feature of our construction is to allow for dependencies of the risk reserves in different business lines. Dependence is modeled via an Archimedean Lévy copula. We derive a sufficient and necessary condition for an Archimedean Lévy generator to create a multidimensional positive Lévy copula in arbitrary dimension.Based on these results we identify structure conditions for the generator and the Lévy measure of an Archimedean Lévy copula under which an insurance company reinsures a larger fraction of claims from one business line than from another.     

Ergodicity of stochastic 2D Navier–Stokes equation with Lévy noise

In this paper we deal with the 2D Navier-Stokes equation perturbed by a Lévy noise force whose white noise part is non-degenerate and that the intensity measure of Poisson measure is σ-finite. Existence and uniqueness of invariant measure for this equation is obtained, two main properties of the Markov semigroup associated with this equation are proved. In other words, strong Feller property and irreducibility hold in the same space.     

Stochastic Lotka–Volterra systems with Lévy noise

This paper is concerned with stochastic Lotka–Volterra models perturbed by Lévy noise. Firstly, stochastic logistic models with Lévy noise are investigated. Sufficient and necessary conditions for stochastic permanence and extinction are obtained. Then three stochastic Lotka–Volterra models of two interacting species perturbed by Lévy noise (i.e., predator–prey system, competition system and cooperation system) are studied. For each system, sufficient and necessary conditions for persistence in the mean and extinction of each population are established. The results reveal that firstly, both persistence and extinction have close relationships with Lévy noise; Secondly, the interaction rates play very important roles in determining the persistence and extinction of the species.     

On the Stability of One Characterization of Stable Distributions

In 1923, G. Polya proved that if X ₁ and X ₂ are independent identically distributed random variables (i.i.d.r.v.) with finite variance, then the distributions of X ₁ and (X ₁+X ₂)/ $\sqrt 2$ are coincidental iff X ₁ has the normal distribution with zero mean. Is an analogous theorem possible for an couple of statistics X ₁ and (X ₁+X ₂)/2^1/α if α<2? P. Lévy constructed an example that denies that hypothesis. However, having supplemented the condition of coincidence of the distributions of X ₁ and (X ₁+X ₂)/2^1/α with a similar condition, namely, requiring, in addition, for the distributions of X ₁ and (X ₁+X ₂+X ₃)/3^1/α to be coincident (here X ₁,X ₂ and X ₃ are i.i.d.r.v.), P. Lévy has proved that X ₁ and X ₂ have a strictly stable distribution. The stability of this characterization in a metric λ₀ (that is defined in the class of characteristic functions by analogy with a uniform metric defined in the class of distributions) without an additional symmetry assumption as well as the stability in a Lévy metric L are analizied in this paper.     

Numerical analysis and applications of Fokker-Planck equations for stochastic dynamical systems with multiplicative α-stable noises

In this paper, we study the nonlocal Fokker-Planck equations (FPEs) associated with Lévy-driven scalar stochastic dynamical systems. We first derive the Fokker-Planck equation for the case of multiplicative symmetric α-stable noises, by the adjoint operator method. Then we construct a finite difference scheme to simulate the nonlocal FPE on either bounded or infinite domain. It is shown that the semi-discrete scheme satisfies the discrete maximum principle and converges. Some experiments are conducted to validate the numerical method. Finally, we extend the results to the asymmetric case and present an application to the nonlinear filtering problem.     

Harnack inequalities for stochastic equations driven by Lévy noise

By using coupling argument and regularization approximations of the underlying subordinator, dimension-free Harnack inequalities are established for a class of stochastic equations driven by a Lévy noise containing a subordinate Brownian motion. The Harnack inequalities are new even for linear equations driven by Lévy noise, and the gradient estimate implied by our log-Harnack inequality considerably generalizes some recent results on gradient estimates and coupling properties derived for Lévy processes or linear equations driven by Lévy noise. The main results are also extended to semilinear stochastic equations in Hilbert spaces.     

Simulated annealing for Lévy-driven jump-diffusions

We consider a one-dimensional dynamical system driven by a vector field −U^′$-U^{\prime }$, where U$U$ is a multi-well potential satisfying some regularity conditions. We perturb this dynamical system by a stable symmetric non-Gaussian Lévy process whose scale decreases as a power function of time. It turns out that the limiting behaviour of the perturbed dynamical system is different for slow and fast decrease rates of the noise intensity. As opposed to the well-studied Gaussian case, the support of the limiting law is not located in the set of global minima of U$U$.     

Limit set of inhomogeneous Ornstein-Uhlenbeck processes,destabilization and annealing

We consider the Diffusion Process obtained by perturbing a dynamical system having a single equilibrium point x, by a fixed time-inhomogeneous Gaussian process whose intensity tends to 0 at infinity. We establish criteria for the exit time from a neighborhood of x to be a.s. finite by linking this fact with the structure of the limit set at infinity. We are also able to compute this limit set for inhomogeneous Ornstein-Uhlenbeck processes associated to linear systems. An application is given to simulated annealing.     

Synchronization for stochastic hybrid coupled controlled systems with Lévy noise

In this paper, synchronization for stochastic hybrid-delayed coupled systems with Lévy noise on a network (SHDCLN) is investigated via aperiodically intermittent control. Here time delays, Markovian switching and Lévy noise are considered on a network simultaneously for the first time. After that, by means of Lyapunov method, graph theory, and some techniques of inequality, some sufficient conditions are derived to guarantee the synchronization for SHDCLN. In addition, the designed range of aperiodically intermittent controller parameters is shown. Meanwhile, the coupling strength and the perturbed intensity of noise have a great impact on the intensity of control. Then, we investigate synchronization for stochastic hybrid delayed Chua's circuits with Lévy noise on a network as a practical application of our theoretical results. Finally, a numerical example is given to illustrate the effectiveness of the theoretical results.