On Ornstein–Uhlenbeck driven by Ornstein–Uhlenbeck processes

We investigate the asymptotic behavior of the maximum likelihood estimators of the unknown parameters of positive recurrent Ornstein–Uhlenbeck processes driven by Ornstein–Uhlenbeck processes.     

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.     

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.     

On the pathwise uniqueness of solutions of stochastic differential equations driven by symmetric stable Lévy processes1

We investigate a sufficient condition for pathwise uniqueness property for 1D stochastic differential equation driven by symmetric α-stable Lévy process, where α ∈ (1, 2).     

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.     

On convergence of random walks generated by compound Cox processes to Lévy processes

A functional limit theorem is proved establishing weak convergence of random walks generated by compound doubly stochastic Poisson processes to Lévy processes in the Skorokhod space. As corollaries, theorems are proved on convergence of random walks with jumps having finite variances to Lévy processes with mixed normal distributions, and in particular, to stable Lévy processes.     

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.     

L p-approximation by Kantorovič operators

Оператор Канторович а дляf∈L _p(I), I=[0,1], определяе тся соотношением $$P_n (f,x) = (n + 1)\sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)} x^k (1 - x)^{n - 1} \int\limits_{I_k } {f(t)dt,} $$ гдеI _k=[k/(n}+1),(k+1)/(n+ 1)],n∈N. Доказывается, что есл ир>1 иf∈W _p ² (I), т.е.f абсол ютно непрерывна наI иf″∈L _p(I), то $$\left\| {P_n f - f} \right\|_p = O(n^{ - 1} ).$$ Далее, установлено, чт о еслиf∈L _p(I),p>1 и ∥P _n f-f∥_р=О(n ^?1), тоf∈S, гдеS={f∶f аб-солютно непрерывна наI, x(1?x)f′(x)=∝ ₀ ^x h(t)dt, гдеh∈L _p(I) и ∝ ₀ ¹ h(t)dt=0}. Если жеf∈L_p(I),p>1, то из условия ∥P _n(f)?f∥_pL=o(n^?1) вытекает, чтоf постоянна почти всюду.     

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.     

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.     

Path dependent equations driven by Hölder processes

This article investigates existence results for path-dependent differential equations driven by a Hölder function where the integrals are understood in the Young sense. The two main results are proved via an application of Schauder theorem and the vector field is allowed to be unbounded. The Hölder function is typically the trajectory of a stochastic process.     

On transience of Lévy-type processes

In this paper, we study weak and strong transience of a class of Feller processes associated with pseudo-differential operators, the so-called Lévy-type processes. As a main result, we derive Chung-Fuchs type conditions (in terms of the symbol of the corresponding pseudo-differential operator) for these properties, which are sharp for Lévy processes. Also, as a consequence, we discuss the weak and strong transience with respect to the dimension of the state space and Pruitt indices, thus generalizing some well-known results related to elliptic diffusion and stable Lévy processes. Finally, in the case when the symbol is radial (in the co-variable) we provide conditions for the weak and strong transience in terms of the Lévy measures.     

On subexponential tails for the maxima of negatively driven compound renewal and Lévy processes

We study subexponential tail asymptotics for the distribution of the maximum $M_t?{sup}_{u\in [0,t]}{X}_u$ of a process $X_t$ with negative drift for the entire range of $t>0$. We consider compound renewal processes with linear drift and Lévy processes. For both processes we also formulate and prove the principle of a single big jump for their maxima. The class of compound renewal processes with drift particularly includes the Cramér–Lundberg renewal risk process.     

Successful couplings for a class of stochastic differential equations driven by Lvy 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 L’evy 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 α-stable L’evy processes.     

Computation of Greeks in LIBOR models driven by time–inhomogeneous Lévy processes

The aim of this article is to compute Greeks, i.e. price sensitivities in the framework of the Lévy LIBOR model. Two approaches are discussed. The first approach is based on the integration-by-parts formula, which lies at the core of the application of the Malliavin calculus to finance. The second approach consists of using Fourier-based methods for pricing derivatives. We illustrate the result by applying the formula to a caplet price where the jump part of the driving process of the underlying model is given by a time–inhomogeneous Gamma process and alternatively by a Variance Gamma process.     

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.     

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).