Ergodicity of stochastic Boussinesq equations driven by Lévy processes

We consider a class of stochastic Boussinesq equations driven by Lévy processes and establish the uniqueness of its invariant measure. The proof is based on the progressive stopping time technique.     

Test for autocorrelation change in discretely observed Ornstein-Uhlenbeck processes driven by Lévy processes

In this paper,we consider the problem of testing for an autocorrelation change in discretely observed Ornstein-Uhlenbeck processes driven by Lévy processes.For a test,we propose a class of test statistics constructed by an iterated cumulative sums of squares of the difference between two adjacent observations.It is shown that each of the test statistics weakly converges to the supremum of the square of a Brownian bridge.The test statistics are evaluated by some empirical results.     

An L 2-theory for a class of SPDEs driven by Lévy processes

In this paper we present an L 2-theory for a class of stochastic partial differential equations driven by Lévy processes.The coefficients of the equations are random functions depending on time and space variables,and no smoothness assumption of the coefficients is assumed.     

Games of singular control and stopping driven by spectrally one-sided Lévy processes

We study a zero-sum game where the evolution of a spectrally one-sided Lévy process is modified by a singular controller and is terminated by the stopper. The singular controller minimizes the expected values of running, controlling and terminal costs while the stopper maximizes them. Using fluctuation theory and scale functions, we derive a saddle point and the value function of the game. Numerical examples under phase-type Lévy processes are also given.     

Remark on pathwise uniqueness of stochastic differential equations driven by Lévy processes

Consider real-valued processes determined by stochastic differential equations driven by Lévy processes. The jump parts of the driving Lévy process are not always α-stable ones, nor symmetric ones. In the present article, we shall study the pathwise uniqueness of the solutions to the stochastic differential equations under the conditions on the coefficients that the diffusion and the jump terms are Hölder continuous, while the drift one is monotonic. Our approach is based on Gronwall’s inequality.     

The first hitting time of the integers by symmetric Lévy processes

For one-dimensional Brownian motion, the exit time from an interval has finite exponential moments and its probability density is expanded in exponential terms. In this note we establish its counterpart for certain symmetric Lévy processes. Applying the theory of Pick functions, we study properties of the Laplace transform of the first hitting time of the integer lattice as a meromorphic function in detail. Its density is expanded in exponential terms and the poles and the zeros of a Pick function play a crucial role.Intermediate results concerning finite exponential moments are also obtained for a class of nonsymmetric Lévy processes.     

Successful couplings for a class of stochastic differential equations driven by Lévy processes

By constructing proper coupling operators for the integro-differential type Markov generator, we establish the existence of a successful coupling for a class of stochastic differential equations driven by Lévy processes. Our result implies a new Liouville theorem for space-time bounded harmonic functions with respect to the underlying Markov semigroups, and it is sharp for Ornstein-Uhlenbeck processes driven by ??-stable Lévy processes.     

Random attractors for stochastic differential equations driven by two-sided Lévy processes

Abstract

In this paper, the asymptotic behavior of solutions for a nonlinear Marcus stochastic differential equation with multiplicative two-sided Lévy noise is studied. We plan to consider this equation as a random dynamical system. Thus, we have to interpret a Lévy noise as a two-sided metric dynamical system. For that, we have to introduce some fundamental properties of such a noise. So far most studies have only discussed two-sided Lévy processes which are defined by combining two-independent Lévy processes. In this paper, we use another definition of two-sided Lévy process by expanding the probability space. Having this metric dynamical system we will show that the Marcus stochastic differential equation with a particular drift coefficient and multiplicative noise generates a random dynamical system which has a random attractor.     

Transformation of measures generated by jump Lévy processes

Let ξ(t), t ∈ [0, 1], be a jump Lévy process. We denote by the law of ξ in the Skorokhod space [0, 1]. Under some nondegeneracy condition on the Lévy measure Λ of the process, we construct a group of -preserving transformations of the space [0, 1]. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 175–188.     

Coupling and exponential ergodicity for stochastic differential equations driven by Lévy processes

We present a novel idea for a coupling of solutions of stochastic differential equations driven by Lévy noise, inspired by some results from the optimal transportation theory. Then we use this coupling to obtain exponential contractivity of the semigroups associated with these solutions with respect to an appropriately chosen Kantorovich distance. As a corollary, we obtain exponential convergence rates in the total variation and standard $L^1$-Wasserstein distances.     

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.     

Extremes of Lévy driven mixed MA processes with convolution equivalent distributions

We investigate the extremal behavior of stationary mixed MA processes for t ≥ 0, where f is a deterministic function and Λ is an infinitely divisible and independently scattered random measure. Particular examples of mixed MA processes are superpositions of Ornstein-Uhlenbeck processes, applied as stochastic volatility models in Barndorff-Nielsen and Shephard (2001a). We assume that the finite dimensional distributions of Λ are in the class of convolution equivalent tails and in the maximum domain of attraction of the Gumbel distribution. On the one hand, we compute the tail behavior of Y(0) and sup _{t ∈ [0,1]} Y(t). On the other hand, we study the extremes of Y by means of marked point processes based on maxima of Y in random intervals. A complementary result guarantees the convergence of the running maxima of Y to the Gumbel distribution. Financial support from the Deutsche Forschungsgemeinschaft through a research grant is gratefully acknowledged.     

Abrupt Lévy processes

Among Lévy processes with unbounded variation, we distinguish the abrupt ones, which are characterised by infinitely sharp extrema. Stable processes with parameter α>1 and creeping Lévy processes are abrupt. We give a characterisation of abrupt processes and study their Dini derivatives at all points of their trajectories.     

Asymptotic behaviour of first passage time distributions for Lévy processes

Let $X$ be a real valued Lévy process that is in the domain of attraction of a stable law without centering with norming function $c.$ As an analogue of the random walk results in Vatutin and Wachtel (Probab Theory Relat Fields 143(1–2):177–217, 2009) and Doney (Probab Theory Relat Fields 152(3–4):559–588, 2012), we study the local behaviour of the distribution of the lifetime $\zeta $ under the characteristic measure $\underline{n}$ of excursions away from $0$ of the process $X$ reflected in its past infimum, and of the first passage time of $X$ below $0,$ $T_{0}=\inf \{t>0:X_{t}<0\},$ under $\mathbb{P }_{x}(\cdot ),$ for $x>0,$ in two different regimes for $x,$ viz. $x=o(c(\cdot ))$ and $x>D c(\cdot ),$ for some $D>0.$ We sharpen our estimates by distinguishing between two types of path behaviour, viz. continuous passage at $T_{0}$ and discontinuous passage. In order to prove our main results we establish some sharp local estimates for the entrance law of the excursion process associated to $X$ reflected in its past infimum.     

A note on convergence of option prices and their Greeks for Lévy models

We study the robustness of option prices to model variation after a change of measure where the measure depends on the model choice. We consider geometric Lévy models in which the infinite activity of the small jumps is approximated by a scaled Brownian motion. For the Esscher transform, the minimal entropy martingale measure, the minimal martingale measure and the mean variance martingale measure, we show that the option prices and their corresponding deltas converge as the scaling of the Brownian motion part tends to zero. We give some examples illustrating our results.     

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.     

Martingale solutions for the three-dimensional stochastic nonhomogeneous incompressible Navier–Stokes equations driven by Lévy processes

In this paper, the three-dimensional stochastic nonhomogeneous incompressible Navier–Stokes equations driven by Lévy processes consisting of the Brownian motion, the compensated Poisson random measure and the Poisson random measure are considered in a bounded domain. We obtain the existence of martingale solutions. The construction of the solution is based on the classical Galerkin approximation method, the stopping times, the stochastic compactness method and the Jakubowski–Skorokhod theorem.     

Simulation of Student–Lévy processes using series representations

Lévy processes have become very popular in many applications in finance, physics and beyond. The Student–Lévy process is one interesting special case where increments are heavy-tailed and, for 1-increments, Student t distributed. Although theoretically available, there is a lack of path simulation techniques in the literature due to its complicated form. In this paper we address this issue using series representations with the inverse Lévy measure method and the rejection method and prove upper bounds for the mean squared approximation error. In the numerical section we discuss a numerical inversion scheme to find the inverse Lévy measure efficiently. We extend the existing numerical inverse Lévy measure method to incorporate explosive Lévy tail measures. Monte Carlo studies verify the error bounds and the effectiveness of the simulation routine. As a side result we obtain series representations of the so called inverse gamma subordinator which are used to generate paths in this model.     

Structural properties of reflected Lévy processes

This paper considers a number of structural properties of reflected Lévy processes, where both one-sided reflection (at 0) and two-sided reflection (at both 0 and K>0) are examined. With V _t being the position of the reflected process at time t, we focus on the analysis of $\zeta(t):=\mathbb{E}V_{t}$ and $\xi(t):=\mathbb{V}\mathrm{ar}V_{t}$ . We prove that for the one- and two-sided reflection, ζ(t) is increasing and concave, whereas for the one-sided reflection, ξ(t) is increasing. In most proofs we first establish the claim for the discrete-time counterpart (that is, a reflected random walk), and then use a limiting argument. A key step in our proofs for the two-sided reflection is a new representation of the position of the reflected process in terms of the driving Lévy process.