Numerical Analysis of Additive, Lévy and Feller Processes with Applications to Option Pricing

Reichmann, Oleg; Schwab, Christoph

doi:10.1007/978-3-642-14007-5_3

Cited by 4 publications

(2 citation statements)

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“…These assumptions can be expressed in terms of marginals of the process and a Lévy copula. We refer to [36,Section 2.3], [32,Section 4.3] and [14,31] for details and the definition of a Lévy copula. Note that subordinators are excluded by condition (iii).…”

Section: Lévy Processesmentioning

confidence: 99%

“…The reason for considering Discontinuous Galerkin discretizations lies in the structure of the equations: Continuous Galerkin Finite Element Methods (CGFEM for short) which are based on continuous, piecewise polynomial functions on simplicial partitions, cf. [27,32], are not applicable in general as they are well-known to become unstable for operators with dominating drift. The Discontinuous Galerkin (DG for short) Finite Element discretizations allow to accurately discretize drift-dominated operators via a judicious choice of the numerical flux to account for dominating drift.…”

Section: Introductionmentioning

confidence: 99%

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hp-DGFEM FOR KOLMOGOROV–FOKKER–PLANCK EQUATIONS OF MULTIVARIATE LÉVY PROCESSES

Marazzina

Reichmann

Schwab

2012

Math. Models Methods Appl. Sci.

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We analyze the discretization of nonlocal degenerate integrodifferential equations arising as so-called forward equations for jump-diffusion processes. Such equations arise in option pricing problems when the stochastic dynamics of the markets is modeled by Lévy driven stochastic volatility models. Well-posedness of the arising equations is addressed. We develop and analyze stable discretization schemes, in particular the discontinuous Galerkin Finite Element Methods (DG-FEM). In the DG-FEM, a new regularization of hypersingular integrals in the Dirichlet form of the pure jump part of infinite variation processes is proposed, allowing in particular a stable DG discretization of hypersingular integral operators. Robustness of the stabilized discretization with respect to various degeneracies in the characteristic triple of the stochastic process is proved. We provide in particular an hp-error analysis of the DG-FEM. Numerical experiments for model equations confirm the theoretical results.

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Section: Lévy Processesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%