Stochastic calculus for convoluted Lévy processes

Bender, Christian; Marquardt, Tina

doi:10.3150/07-bej115

Cited by 30 publications

(38 citation statements)

References 15 publications

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“…• If f is sufficiently regular and has compact support, the above relation between the jumps of M and X has already been observed in several papers e.g. [3] and [13]. Without such regularity assumptions M may fail to be continuous, even if f vanishes on the diagonal.…”

Section: Remarkmentioning

confidence: 66%

Maximal Inequalities for Fractional Lévy and Related Processes

Bender

Knobloch

Oberacker

2015

Stochastic Analysis and Applications

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In this paper we study processes which are constructed by a convolution of a deterministic kernel with a martingale. A special emphasis is put on the case where the driving martingale is a centred Lévy process, which covers the popular class of fractional Lévy processes. As a main result we show that, under appropriate assumptions on the kernel and the martingale, the maximum process of the corresponding 'convoluted martingale' is p-integrable and we derive maximal inequalities in terms of the kernel and of the moments of the driving martingale.in terms of the kernel and the L p (P)-norm of the driving martingale. Let us illustrate our results for the case of a fractional Lévy process

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Section: Remarkmentioning

confidence: 66%

Maximal Inequalities for Fractional Lévy and Related Processes

Bender

Knobloch

Oberacker

2015

Stochastic Analysis and Applications

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“…Several approaches to the integration with respect to Lévy-driven Volterra processes are known. In [7], a Skorokhod type integral was considered. That construction followed S-transform approach, developed in [6] for fractional Brownian motion.…”

Section: Introductionmentioning

confidence: 99%

Fractional Calculus And Pathwise Integration for Volterra Processes Driven by Lévy and Martingale Noise

Nunno

Mishura

Ralchenko

2016

FCAA

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Abstract. We introduce a pathwise integration for Volterra processes driven by Lévy noise or martingale noise. These processes are widely used in applications to turbulence, signal processes, biology, and in environmental finance. Indeed they constitute a very flexible class of models, which include fractional Brownian and Lévy motions and it is part of the so-called ambit fields. A pathwise integration with respect of such Volterra processes aims at producing a framework where modelling is easily understandable from an information perspective. The techniques used are based on fractional calculus and in this there is a bridging of the stochastic and deterministic techniques. The present paper aims at setting the basis for a framework in which further computational rules can be devised. Our results are general in the choice of driving noise. Additionally we propose some further details in the relevant context subordinated Wiener processes.

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“…The representation follows directly from (3). As it requires one additional integration, a numerical evaluation of the fast Fourier transform methods is slower, but still feasible.…”

Section: Integrating Volatility Clustering Into Exponential Lévy Modelsmentioning

confidence: 99%

Integrating Volatility Clustering Into Exponential Lévy Models

Bender

Marquardt

2009

J. Appl. Probab.

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We introduce a class of stock models that interpolates between exponential Lévy models based on Brownian subordination and certain stochastic volatility models with Lévy-driven volatility, such as the Barndorff-Nielsen-Shephard model. The driving process in our model is a Brownian motion subordinated to a business time which is obtained by convolution of a Lévy subordinator with a deterministic kernel. We motivate several choices of the kernel that lead to volatility clusters while maintaining the sudden extreme movements of the stock. Moreover, we discuss some statistical and path properties of the models, prove absence of arbitrage and incompleteness, and explain how to price vanilla options by simulation and fast Fourier transform methods.

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Stochastic calculus for convoluted Lévy processes

Cited by 30 publications

References 15 publications

Maximal Inequalities for Fractional Lévy and Related Processes

Maximal Inequalities for Fractional Lévy and Related Processes

Fractional Calculus And Pathwise Integration for Volterra Processes Driven by Lévy and Martingale Noise

Integrating Volatility Clustering Into Exponential Lévy Models

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