Multilevel Monte Carlo simulation for Lévy processes based on the Wiener–Hopf factorisation

In Kuznetsov et al. (2011) a new Monte Carlo simulation technique was introduced for a large family of Lévy processes that is based on the Wiener–Hopf decomposition. We pursue this idea further by combining their technique with the recently introduced multilevel Monte Carlo methodology. Moreover, we provide here for the first time a theoretical analysis of the new Monte Carlo simulation technique in Kuznetsov et al. (2011) and of its multilevel variant for computing expectations of functions depending on the historical trajectory of a Lévy process. We derive rates of convergence for both methods and show that they are uniform with respect to the “jump activity” (e.g. characterised by the Blumenthal–Getoor index). We also present a modified version of the algorithm in Kuznetsov et al. (2011) which combined with the multilevel methodology obtains the optimal rate of convergence for general Lévy processes and Lipschitz functionals. This final result is only a theoretical one at present, since it requires independent sampling from a triple of distributions which is currently only possible for a limited number of processes.     

Exponential functional of Lévy processes: Generalized Weierstrass products and Wiener–Hopf factorization

In this note, we state a representation of the Mellin transform of the exponential functional of Lévy processes in terms of generalized Weierstrass products. As by-product, we obtain a multiplicative Wiener–Hopf factorization generalizing previous results obtained by Patie and Savov (2012) [14] as well as smoothness properties of its distribution.     

On the Inhomogeneous Conservative Wiener–Hopf Equation

We prove the existence of a solution to the inhomogeneous Wiener–Hopf equation whose kernel is a probability distribution generating a random walk drifting to ?∞. Asymptotic properties of a solution are found depending on the corresponding properties of the free term and the kernel of the equation.     

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.     

Approximations for the distributions of bounded variation Lévy processes

We propose a feasible method for approximating the marginal distributions and densities of a bounded variation Lévy process using polynomial expansions. We provide a fast recursive formula for approximating the coefficients of the expansions and estimating the order of the approximation error. Our expansions are shown to be the exact counterpart of successive approximations of the Lévy process by compound Poisson processes previously proposed by, for instance, Barndorff-Nielsen and Hubalek (2008) [Barndorff-Nielsen, O.E., Hubalek, F., 2008. Probability measures, Lévy measures, and analyticity in time. Bernoulli 3 (14), 764–790] and others, and hence, give an answer to an open problem raised therein.     

Extremum Problem for the Wiener–Hopf Equation

The extremum problem for the Wiener–Hopf equation obtained by replacing the condition u(x) = 0, x < 0, by the condition of the minimum of the quadratic functional of the function u(x)exp(–x), – < x < , is solved in closed form.     

A Wiener–Hopf based approach to numerical computations in fluctuation theory for Lévy processes

This paper focuses on numerical evaluation techniques related to fluctuation theory for Lévy processes; they can be applied in various domains, e.g., in finance in the pricing of so-called barrier options. More specifically, with $\bar{X}_t:= \sup _{0\le s\le t} X_s$ denoting the running maximum of the Lévy process $X_t$ , the aim is to evaluate $\mathbb{P }(\bar{X}_t \in \mathrm{d}x)$ for $t,x>0$ . The starting point is the Wiener–Hopf factorization, which yields an expression for the transform $\mathbb E e^{-\alpha \bar{X}_{e(\vartheta )}}$ of the running maximum at an exponential epoch (with $\vartheta ^{-1}$ the mean of this exponential random variable). This expression is first rewritten in a more convenient form, and then it is pointed out how to use Laplace inversion techniques to numerically evaluate $\mathbb{P }(\bar{X}_t\in \mathrm{d}x).$ In our experiments we rely on the efficient and accurate algorithm developed in den Iseger (Probab Eng Inf Sci 20:1–44, 2006). We illustrate the performance of the algorithm with various examples: Brownian motion (with drift), a compound Poisson process, and a jump diffusion process. In models with jumps, we are also able to compute the density of the first time a specific threshold is exceeded, jointly with the corresponding overshoot. The paper is concluded by pointing out how our algorithm can be used in order to analyze the Lévy process’ concave majorant.     

Local Wiener–Hopf factorization and indices over arbitrary fields

In this paper, a generalization to arbitrary fields of the usual Wiener–Hopf equivalence of complex valued rational matrix functions is given and the left local Wiener–Hopf factorization indices defined in our previous work [A. Amparan, S. Marcaida, I. Zaballa, Local realizations and local polynomial matrix representations of systems, Linear Algebra Appl. 425 (2007) 757–775] are proved to form a complete system of invariants for this equivalence relation. For the case when the field is algebraically closed a reduced form of a controllable matrix pair under the feedback equivalence is presented for which the controllability indices can be written as sums of the local controllability indices [A. Amparan, S. Marcaida, I. Zaballa, On the existence of linear systems with prescribed invariants for system similarity, Linear Algebra Appl. 413 (2006) 510–533].     

On the Volterra Factorization of the Wiener–Hopf Integral Operator

Mathematical Notes - The problem of the factorization of the Wiener–Hopf integral operator in the form of the product of the upper and lower Volterra operators is considered. Conditions for...     

Canonical Wiener–Hopf and spectral factorization of large-degree matrix polynomials

We present a novel Newton method for canonical Wiener–Hopf and spectral factorization of matrix polynomials. The initial vector results from solving a block Toeplitz-like system, and the Jacobi matrix governing the Newton iteration has nice structural and numerical properties. The local quadratic convergence of the method is proved and was tested numerically. For scalar polynomials of degree 20000, a superfast version of the method implemented on a laptop typically reqired about half a minute to produce an initial vector and then performed the Newton iteration within one second. In the matrix case, the method worked reproachless on a laptop with 8 Gigabyte RAM if the degree of the polynomial times the squared matrix dimension did not exceed 20000. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Leray–Hopf extension condition for the steady-state Navier–Stokes problem in multiply-connected bounded domains

Employing the approach of Takeshita (Pacific J Math 157:151–158, 1993), we give an elementary proof of the invalidity of the Leray–Hopf Extension Condition for certain multiply connected bounded domains of \({\mathbb {R}}^{n}\) , \(n=2,3\) , whenever the flow through the different components of the boundary is non-zero. Our proof is alternative to and, to an extent, more direct than the recent one proposed by Heywood (J Math Fluid Mech 13:449–457, 2011).     

Kendall random walk,Williamson transform,and the corresponding Wiener–Hopf factorization

We give some properties of hitting times and an analogue of the Wiener–Hopf factorization for the Kendall random walk. We also show that the Williamson transform is the best tool for problems connected with the Kendall convolution.     

Representation theory of finite groups in Wiener–Hopf factorization problem

We consider the Wiener–Hopf factorization problem for a matrix function that is completely defined by its first column: the succeeding columns are obtained from the first one by means of a finite group of permutations. The symmetry of this matrix function allows us to reduce the dimension of the problem. In particular, we find some relations between its partial indices and can compute some of the indices. In special cases, we can explicitly obtain the Wiener–Hopf factorization of the matrix function.     

Upper functions for lévy processes having only negative jumps

Let X_t,t ≥ 0 be a real valued process with stationary independentincrements having only negative jumps. We obtain b(t) such that lim sup X_t )/b(t) equals a finite positive constant with probability one as t → 0 and t → ∞ under extra condition. The hypotheses about the behavior of Lévy measure near zero and infinity are necessary to guarantee that the lim sup is positive     

On the refracted–reflected spectrally negative Lévy processes

We study a combination of the refracted and reflected Lévy processes. Given a spectrally negative Lévy process and two boundaries, it is reflected at the lower boundary while, whenever it is above the upper boundary, a linear drift at a constant rate is subtracted from the increments of the process. Using the scale functions, we compute the resolvent measure, the Laplace transform of the occupation times as well as other fluctuation identities that will be useful in applied probability including insurance, queues, and inventory management.     

Wiener–Hopf Equations Technique for Quasimonotone Variational Inequalities

In this paper, we use the Wiener–Hopf equations technique to suggest and analyze new iterative methods for solving general quasimonotone variational inequalities. These new methods differ from previous known methods for solving variational inequalities.     

On taxed spectrally negative Lévy processes with draw-down stopping

In this paper we consider a spectrally negative Lévy risk model with tax. With the ruin time replaced by a draw-down time with a linear draw-down function and for a constant tax rate, we find expressions for the present values of tax payments. They generalize previous results in Albrecher et al. (2008). Alternative proofs are given for the special case of Cramér–Lundberg risk models. Optimal barrier taxation policies are discussed.