Low-Dimensional Partial Integro-differential Equations for High-Dimensional Asian Options

Hepperger, Peter

doi:10.1007/978-3-319-02069-3_15

Cited by 1 publication

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“…The methods have been extended to multivariate models, see (Reich, Schwab, and Winter 2010), (Winter 2009) and the references therein. Also standard finite element methods are efficiently used for pricing basket options, even in high dimensional models using dimension reduction techniques, see (Hepperger 2010) and (Hepperger 2012). (Achdou 2008) provides a calibration procedure of a Lévy model based on PIDEs.…”

Section: Introductionmentioning

confidence: 99%

Sobolev index: A classification of Lévy processes

Glau

2012

Preprint

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We classify Lévy processes according to the solution spaces of the associated parabolic PIDEs. This classification reveals structural characteristics of the processes and is relevant for applications such as for solving PIDEs numerically for pricing options in Lévy models.The classification is done via the Fourier transform i.e. via the symbol of the process. We define the Sobolev index of a Lévy process by a certain growth condition on the symbol. It follows that for Lévy processes with Sobolev index α the corresponding evolution problem has a unique weak solution in the Sobolev-Slobodeckii space H α/2 . We show that this classification applies to a wide range of processes. Examples are the Brownian motion with or without drift, generalised hyperbolic (GH), CGMY and (semi) stable Lévy processes.A comparison of the Sobolev index with the Blumenthal-Getoor index sheds light on the structural implication of the classification. More precisely, we discuss the Sobolev index as an indicator of the smoothness of the distribution and of the variation of the paths of the process. This highlights the relation between the p-variation of the paths and the degree of smoothing effect that stems from the distribution.

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Section: Introductionmentioning

confidence: 99%

Sobolev index: A classification of Lévy processes

Glau

2012

Preprint

View full text Add to dashboard Cite

show abstract

Low-Dimensional Partial Integro-differential Equations for High-Dimensional Asian Options

Cited by 1 publication

References 16 publications

Sobolev index: A classification of Lévy processes

Sobolev index: A classification of Lévy processes

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