scholarly journals Regularity of Local Times Associated with Volterra–Lévy Processes and Path-Wise Regularization of Stochastic Differential Equations

2021 ◽  
Author(s):  
Fabian A. Harang ◽  
Chengcheng Ling
Keyword(s):  
Differential Equations ◽  
Local Time ◽  
Lévy Process ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process ◽  
Local Times ◽  
Volterra Kernel ◽  
Spatial Regularity ◽  
Regularizing Effects

AbstractWe investigate the space-time regularity of the local time associated with Volterra–Lévy processes, including Volterra processes driven by $$\alpha $$ α -stable processes for $$\alpha \in (0,2]$$ α ∈ ( 0 , 2 ] . We show that the spatial regularity of the local time for Volterra–Lévy process is $${\mathbb {P}}$$ P -a.s. inverse proportional to the singularity of the associated Volterra kernel. We apply our results to the investigation of path-wise regularizing effects obtained by perturbation of ordinary differential equations by a Volterra–Lévy process which has sufficiently regular local time. Following along the lines of Harang and Perkowski (2020), we show existence, uniqueness and differentiability of the flow associated with such equations.

An Euler–Poisson scheme for Lévy driven stochastic differential equations

2016 ◽  
Vol 53 (1) ◽  
pp. 262-278 ◽  
Author(s):  
A. Ferreiro-Castilla ◽  
A. E. Kyprianou ◽  
R. Scheichl
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Lévy Process ◽  
Lévy Processes ◽  
Approximate Solutions ◽  
Levy Processes ◽  
Levy Process ◽  
Arrival Times ◽  
Grid Points ◽  
Interesting Alternative

Abstract We describe an Euler scheme to approximate solutions of Lévy driven stochastic differential equations (SDEs) where the grid points are given by the arrival times of a Poisson process and thus are random. This result extends the previous work of Ferreiro-Castilla et al. (2014). We provide a complete numerical analysis of the algorithm to approximate the terminal value of the SDE and prove that the mean-square error converges with rate O(n-1/2). The only requirement of the methodology is to have exact samples from the resolvent of the Lévy process driving the SDE. Classical examples, such as stable processes, subclasses of spectrally one-sided Lévy processes, and new families, such as meromorphic Lévy processes (Kuznetsov et al. (2012), are examples for which our algorithm provides an interesting alternative to existing methods, due to its straightforward implementation and its robustness with respect to the jump structure of the driving Lévy process.

scholarly journals Oscillation of modes of some semi-stable Lévy processes

1993 ◽  
Vol 132 ◽  
pp. 141-153 ◽  
Author(s):  
Toshiro Watanabe
Keyword(s):  
Lévy Process ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process ◽  
Stable Processes ◽  
Unimodal Lévy Process ◽  
Oscillating Mode

In this paper it is shown that there is a unimodal Levy process with oscillating mode. After the author first constructed an example of such a self-decomposable process, Sato pointed out that it belongs to the class of semi-stable processes with β < 0. We prove that all non-symmetric semi-stable self-decomposable processes with β < 0 have oscillating modes.

scholarly journals Applications of factorization embeddings for Lévy processes

2006 ◽  
Vol 38 (03) ◽  
pp. 768-791 ◽  
Author(s):  
A. B. Dieker
Keyword(s):  
Lévy Process ◽  
Joint Distribution ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process ◽  
Tail Asymptotics ◽  
Lévy Measure ◽  
Risk Models ◽  
General Properties ◽  
Phase Type

We give three applications of the Pecherskii-Rogozin-Spitzer identity for Lévy processes. First, we find the joint distribution of the supremum and the epoch at which it is ‘attained’ if a Lévy process has phase-type upward jumps. We also find the characteristics of the ladder process. Second, we establish general properties of perturbed risk models, and obtain explicit fluctuation identities in the case that the Lévy process is spectrally positive. Third, we study the tail asymptotics for the supremum of a Lévy process under different assumptions on the tail of the Lévy measure.

Variance Optimal Stopping for Geometric Lévy Processes

2015 ◽  
Vol 47 (01) ◽  
pp. 128-145 ◽  
Author(s):  
Kamille Sofie Tågholt Gad ◽  
Jesper Lund Pedersen
Keyword(s):  
Optimal Stopping ◽  
Lévy Process ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process ◽  
Stopping Times ◽  
Optimal Stopping Problem ◽  
Maximum Variance ◽  
Geometric Levy Process

The main result of this paper is the solution to the optimal stopping problem of maximizing the variance of a geometric Lévy process. We call this problem the variance problem. We show that, for some geometric Lévy processes, we achieve higher variances by allowing randomized stopping. Furthermore, for some geometric Lévy processes, the problem has a solution only if randomized stopping is allowed. When randomized stopping is allowed, we give a solution to the variance problem. We identify the Lévy processes for which the allowance of randomized stopping times increases the maximum variance. When it does, we also solve the variance problem without randomized stopping.

Multiple points for symmetric Lévy processes

1978 ◽  
Vol 83 (1) ◽  
pp. 83-90 ◽  
Author(s):  
John Hawkes
Keyword(s):  
Markov Process ◽  
Euclidean Space ◽  
Lévy Process ◽  
Lévy Processes ◽  
Transition Function ◽  
Levy Processes ◽  
Levy Process ◽  
Dimensional Euclidean Space ◽  
Multiple Points ◽  
Image Position

Let Xt be a Lévy process in Rd, d-dimensional euclidean space. That is X is a Markov process whose transition function satisfies

scholarly journals Necessity of weak subordination for some strongly subordinated Lévy processes

2021 ◽  
Vol 58 (4) ◽  
pp. 868-879
Author(s):  
Boris Buchmann ◽  
Kevin W. Lu
Keyword(s):  
Lévy Process ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process

AbstractConsider the strong subordination of a multivariate Lévy process with a multivariate subordinator. If the subordinate is a stack of independent Lévy processes and the components of the subordinator are indistinguishable within each stack, then strong subordination produces a Lévy process; otherwise it may not. Weak subordination was introduced to extend strong subordination, always producing a Lévy process even when strong subordination does not. Here we prove that strong and weak subordination are equal in law under the aforementioned condition. In addition, we prove that if strong subordination is a Lévy process then it is necessarily equal in law to weak subordination in two cases: firstly when the subordinator is deterministic, and secondly when it is pure-jump with finite activity.

scholarly journals A Liouville theorem for Lévy generators

Positivity ◽  
2020 ◽  
Author(s):  
Franziska Kühn
Keyword(s):  
Weak Solution ◽  
Laplace Equation ◽  
Lévy Process ◽  
Lévy Processes ◽  
Infinitesimal Generator ◽  
Polynomial Growth ◽  
Liouville Theorem ◽  
Levy Processes ◽  
Levy Process ◽  
Regularity Estimates

AbstractUnder mild assumptions, we establish a Liouville theorem for the “Laplace” equation $$Au=0$$ A u = 0 associated with the infinitesimal generator A of a Lévy process: If u is a weak solution to $$Au=0$$ A u = 0 which is at most of (suitable) polynomial growth, then u is a polynomial. As a by-product, we obtain new regularity estimates for semigroups associated with Lévy processes.

scholarly journals Notes on energy for space-time processes over Lévy processes

1991 ◽  
Vol 122 ◽  
pp. 63-74 ◽  
Author(s):  
Mamoru Kanda
Keyword(s):  
Euclidean Space ◽  
Probability Measure ◽  
Lévy Process ◽  
Lévy Processes ◽  
Levy Processes ◽  
Space Time ◽  
Levy Process ◽  
Independent Increments

Let X = (Xt, 0 ≤ t < ∞) be a Lévy process on the Euclidean space Rd, that is, a process on Rd with stationary independent increments which has right continuous paths with left limits. We denote by Px the probability measure such that PX(X0 = x) = 1 and by Ex the expectation relative to Px. The process is characterized by the exponent Ψ through.

ORTHOGONAL DECOMPOSITIONS FOR LÉVY PROCESSES WITH AN APPLICATION TO THE GAMMA, PASCAL, AND MEIXNER PROCESSES

2003 ◽  
Vol 06 (01) ◽  
pp. 73-102 ◽  
Author(s):  
EUGENE LYTVYNOV
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Lévy Process ◽  
Lévy Processes ◽  
Random Variable ◽  
Levy Processes ◽  
Levy Process ◽  
Nuclear Space ◽  
Multiple Stochastic Integrals ◽  
Orthogonal Decompositions

It is well known that between all processes with independent increments, essentially only the Brownian motion and the Poisson process possess the chaotic representation property (CRP). Thus, a natural question appears: What is an appropriate analog of the CRP in the case of a general Lévy process. At least three approaches are possible here. The first one, due to Itô, uses the CRP of the Brownian motion and the Poisson process, as well as the representation of a Lévy process through those processes. The second approach, due to Nualart and Schoutens, consists of representing any square-integrable random variable as a sum of multiple stochastic integrals constructed with respect to a family of orthogonalized centered power jumps processes. The third approach, never applied before to the Lévy processes, uses the idea of orthogonalization of polynomials with respect to a probability measure defined on the dual of a nuclear space. The main aims of this paper are to develop the three approaches in the case of a general (ℝ-valued) Lévy process on a Riemannian manifold and (what is more important) to understand a relationship between these approaches. We apply the obtained results to the gamma, Pascal, and Meixner processes, in which case the analysis related to the orthogonalized polynomials becomes essentially simpler and richer than in the general case.

Variance Optimal Stopping for Geometric Lévy Processes

2015 ◽  
Vol 47 (1) ◽  
pp. 128-145 ◽  
Author(s):  
Kamille Sofie Tågholt Gad ◽  
Jesper Lund Pedersen
Keyword(s):  
Optimal Stopping ◽  
Lévy Process ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process ◽  
Stopping Times ◽  
Optimal Stopping Problem ◽  
Maximum Variance ◽  
Geometric Levy Process

The main result of this paper is the solution to the optimal stopping problem of maximizing the variance of a geometric Lévy process. We call this problem the variance problem. We show that, for some geometric Lévy processes, we achieve higher variances by allowing randomized stopping. Furthermore, for some geometric Lévy processes, the problem has a solution only if randomized stopping is allowed. When randomized stopping is allowed, we give a solution to the variance problem. We identify the Lévy processes for which the allowance of randomized stopping times increases the maximum variance. When it does, we also solve the variance problem without randomized stopping.

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