Persistence and extinction of a stochastic non-autonomous Gilpin–Ayala system driven by Lévy noise

In this paper, we consider the persistence and extinction of a stochastic non-autonomous Gilpin–Ayala system driven by Lévy noise. Sufficient criteria for extinction, non-persistence in the mean and weak persistence of the system are established. The threshold between weak persistence and extinction is obtained. From the results we can see that both persistence and extinction have close relationships with Lévy noise. Some simulation figures are introduced to demonstrate the analytical findings.     

Convergence of the all-time supremum of a Lévy process in the heavy-traffic regime

In this paper we derive a technique for obtaining limit theorems for suprema of Lévy processes from their random walk counterparts. For each a>0, let $\{Y^{(a)}_{n}:n\ge1\}In this paper we derive a technique for obtaining limit theorems for suprema of Lévy processes from their random walk counterparts. For each a>0, let {Y^(a)_n:n 3 1}\{Y^{(a)}_{n}:n\ge1\} be a sequence of independent and identically distributed random variables and {X^(a)_t:t 3 0}\{X^{(a)}_{t}:t\ge0\} be a Lévy process such that X₁^(a)=^dY₁^(a)X_{1}^{(a)}\stackrel{d}{=}Y_{1}^{(a)}, \mathbbEX₁^(a) < 0\mathbb{E}X_{1}^{(a)}<0 and \mathbbEX₁^(a)-0\mathbb{E}X_{1}^{(a)}\uparrow0 as a↓0. Let S^(a)_n=?_k=1ⁿ Y^(a)_kS^{(a)}_{n}=\sum _{k=1}^{n} Y^{(a)}_{k}. Then, under some mild assumptions, , for some random variable and some function Δ(⋅). We utilize this result to present a number of limit theorems for suprema of Lévy processes in the heavy-traffic regime.     

On a class of singular stochastic control problems driven by Lévy noise

A class of singular stochastic control problems whose value functions satisfy an invariance property was studied by Lasry and Lions (2000). They have shown that, within this class, any singular control problem is equivalent to the corresponding standard stochastic control problem. The equivalence is in the sense that their value functions are equal. In this work, we clarify their idea and extend their work to allow Lévy type noise. In addition, for the purpose of application, we apply our result to an optimal trade execution problem studied by Lasry and Lions (2007).     

Stochastic resonance in a genetic toggle model with harmonic excitation and Lévy noise

Stochastic resonance is investigated to explain the beneficial effect of Lévy noise on gene expression of genetic toggle model with harmonic excitation. The dynamic change of protein concentration of genetic toggle model under combined drives of harmonic excitation and Lévy noise is obtained numerically. Stochastic resonance is presented through the classical measure of signal-to-noise-ratio. Then from two aspects of combined drives on the protein at high or low concentration, the changes of protein concentration and signal-to-noise-ratio are discussed, respectively. When combined drives are within the protein at high concentration, the increasing Lévy noise intensity can promote the transition between the high and low concentrations, and the low protein concentration hardly fluctuates under the small noise intensity. It is also shown that the increase of stability index, skewness parameter of Lévy noise and amplitude of harmonic excitation can suppress the optimum collaboration of stochastic resonance. On the other hand, when combined drives are within the protein at low concentration, the increasing noise intensity can enhance the transition between the high and low concentrations, and the increase of stability index, skewness parameter and amplitude can strengthen the optimum collaboration of stochastic resonance. By the synergic actions of stochastic resonance, it is demonstrated that combined effect of harmonic excitation and Lévy stimuli can be utilized to promote the gene expression of proteins in genetic toggle model.     

Stability analysis of a stochastic Gilpin–Ayala model driven by Lévy noise

A stochastic one-dimensional Gilpin–Ayala model driven by Lévy noise is presented in this paper. Firstly, we show that this model has a unique global positive solution under certain conditions. Then sufficient conditions for the almost sure exponential stability and moment exponential stability of the trivial solution are established. Results show that the jump noise can make the trivial solution stable under some conditions. Numerical example is introduced to illustrate the results.     

Stochastic population dynamics driven by Lévy noise

This paper considers stochastic population dynamics driven by Lévy noise. The contributions of this paper lie in that: (a) Using the Khasminskii–Mao theorem, we show that the stochastic differential equation associated with our model has a unique global positive solution; (b) Applying an exponential martingale inequality with jumps, we discuss the asymptotic pathwise estimation of such a model.     

Branching Processes in a Lévy Random Environment

In this paper, we introduce branching processes in a Lévy random environment. In order to define this class of processes, we study a particular class of non-negative stochastic differential equations driven by a white noise and Poisson random measures which are mutually independent. Following similar techniques as in Dawson and Li (Ann. Probab. 40:813–857, 2012) and Li and Pu (Electron. Commun. Probab. 17(33):1–13, 2012), we obtain existence and uniqueness of strong local solutions of such stochastic equations. We use the latter result to construct continuous state branching processes with immigration and competition in a Lévy random environment as a strong solution of a stochastic differential equation. We also study the long term behaviour of two interesting examples: the case with no immigration and no competition and the case with linear growth and logistic competition.     

On the Price of Risk of the Underlying Markov Chain in a Regime-Switching Exponential Lévy Model

Regime-switching models (RSM) have been recently used in the literature as alternatives to the Black-Scholes model. Several authors favor RSM as being more realistic since, by construction, they model those exogenous macroeconomic cycles against which asset prices evolve. In the context of derivatives pricing, these models lead to incomplete markets and therefore there exist multiple Equivalent Martingale Measures (EMM) yielding different pricing rules. A fair amount of literature (Buffington and Elliott, Int J Theor Appl Finance 40:267–282, 2002; Elliott et al., Ann Finance 1(4):23–432, 2005) focuses on conveniently choosing a family of EMM leading to closed-form formulas for option prices. These studies often make the assumption that the risk associated with the Markov chain is not priced. Recently, Siu and Yang (Acta Math Appl Sin Engl Ser 25(3):339–388, 2009), proposed an EMM kernel that takes into account all risk components of a regime-switching Black-Scholes model. In this paper, we extend the results and observations made in Siu and Yang (Acta Math Appl Sin Engl Ser 25(3):339–388, 2009) in order to include more general Lévy regime-switching models that allow us to assess the influence of jumps on the price of risk. In particular, numerical results are given for Regime-switching Jump-Diffusion and Variance-Gamma models. Also, we carry out a comparative analysis of the resulting option price formulas with existing regime-switching models such as Naik (J Financ 48:1969–1984, 1993) and Boyle and Draviam (Insur Math Econ 40:267–282, 2007).     

Stochastic Processes Associated with a Sum of the Lévy Laplacians

Abstract

In this paper, we give a stochastic expression of a semigroup generated by a sum of the Lévy Laplacians acting on a class of S-transforms of white noise distributions in terms of an infinite sequence of independent Brownian motions.     

Minimax detection of a signal in ℓp metrics

The problem of the minimax detection of a signal is considered in a Gaussian white noise of intensity 0. The set of the signals represents an ellipsoid in the _p-metric with lengths of semiaxes a_k ^–1 0 as k , from which a ball of radius 0 as 0 in the _p-metric has been removed. Asymptotically minimax tests have been constructed for the cases 1 < p and asymptotically sharp estimates of the minimax efficiency have been obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 152–168, 1990.     

The Jacobi Field of a Lévy Process

We derive an explicit formula for the Jacobi field that is acting in an extended Fock space and corresponds to an ( -valued) Lévy process on a Riemannian manifold. The support of the measure of jumps in the Lévy–Khintchine representation for the Lévy process is supposed to have an infinite number of points. We characterize the gamma, Pascal, and Meixner processes as the only Lévy process whose Jacobi field leaves the set of finite continuous elements of the extended Fock space invariant.     

Well-posedness for the stochastic 2D primitive equations with Lévy noise

The two-dimensional primitive equations with Lévy noise are studied in this paper. We proved the existence and uniqueness of the solutions in a fixed probability space which based on a priori estimates, weak convergence method and monotonicity arguments.     

Large deviations for the optimal filter of nonlinear dynamical systems driven by Lévy noise

In this paper, we focus on the asymptotic behavior of the optimal filter where both signal and observation processes are driven by Lévy noises. Indeed, we study large deviations for the case where the signal-to-noise ratio is small by considering weak convergence arguments. To that end, we first prove the uniqueness of the solution of the controlled Zakai and Kushner–Stratonovich equations. For this, we employ a method which transforms the associated equations into SDEs in an appropriate Hilbert space. Next, taking into account the controlled analogue of Zakai and Kushner–Stratonovich equations, respectively, the large deviation principle follows by employing the existence, uniqueness and tightness of the solutions.     

On occupation times in the red of Lévy risk models

In this paper, we obtain analytical expression for the distribution of the occupation time in the red (below level 0) up to an (independent) exponential horizon for spectrally negative Lévy risk processes and refracted spectrally negative Lévy risk processes. This result improves the existing literature in which only the Laplace transforms are known. Due to the close connection between occupation time and many other quantities, we provide a few applications of our results including future drawdown, inverse occupation time, Parisian ruin with exponential delay, and the last time at running maximum.     

Inference on the Lévy measure in case of noisy observations

We discuss inference on the Lévy measure in case of noisy observations. An extension of the pre-averaging method allows for a consistent estimation of the associated spectral function. The asymptotic behaviour of the novel estimator is the same as without noise.     

Transportation distances and noise sensitivity of multiplicative Lévy SDE with applications

This article assesses the distance between the laws of stochastic differential equations with multiplicative Lévy noise on path space in terms of their characteristics. The notion of transportation distance on the set of Lévy kernels introduced by Kosenkova and Kulik yields a natural and statistically tractable upper bound on the noise sensitivity. This extends recent results for the additive case in terms of coupling distances to the multiplicative case. The strength of this notion is shown in a statistical implementation for simulations and the example of a benchmark time series in paleoclimate.     

Global solutions of stochastic 2D Navier-Stokes equations with Lévy noise

In this paper, we prove the global existence and uniqueness of the strong and weak solutions for 2D Navier-Stokes equations on the torus perturbed by a Lévy process. The existence of invariant measure of the solutions are proved also. This work was supported by National Basic Research Program of China (Grant No. 2006CB8059000), Science Fund for Creative Research Groups (Grant No. 10721101), National Natural Science Foundation of China (Grant Nos. 10671197, 10671168), Science Foundation of Jiangsu Province (Grant Nos. BK2006032, 06-A-038, 07-333) and Key Lab of Random Complex Structures and Data Science, Chinese Academy of Sciences     

A geometric interpretation of the transition density of a symmetric Lévy process

We study for a class of symmetric Lévy processes with state space R n the transition density pt（x） in terms of two one-parameter families of metrics, （dt）t>0 and （δt）t>0. The first family of metrics describes the diagonal term pt（0）; it is induced by the characteristic exponent ψ of the Lévy process by dt（x, y） = ^1/2tψ（x-y）. The second and new family of metrics δt relates to ^1/2tψ through the formulawhere F denotes the Fourier transform. Thus we obtain the following "Gaussian" representation of the transition density: pt（x） = pt（0）e- δ2t （x,0） where pt（0） corresponds to a volume term related to tψ and where an "exponential" decay is governed by δ2t . This gives a complete and new geometric, intrinsic interpretation of pt（x）.