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

Abstract

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.

Optimal impulse control of a portfolio with a fixed transaction cost

“…We discretize the PDE LV = 0 considering a finite element technique with polynomial of degree 1, and a Crank-Nicholson scheme. For details on the implementation of the PSOR algorithm see, for example, Wilmott et al (1995), while for applications of the finite element technique to financial problems see, as examples, Achdou and Pironneau (2005), Barucci and Marazzina (2012), Marazzina et al (2012).…”

A Flexible Galerkin Scheme for Option Pricing in Lévy Models

“…Wavelet-Galerkin methods for PIDEs related to a class generalizing tempered stable Lévy processes are derived in Matache et al (2005) for American options and see e.g. Marazzina et al (2012) for a high-dimensional extension. Radial basis for the Merton and Kou model, American and European options are provided by Chan and Hubbert (2014) and further developed for CGMY models by Brummelhuis and Chan (2014).…”

High order splitting schemes with complex timesteps and their application in mathematical finance

“…In particular, if the first order hyperbolic drift part is solved by a method such as streamline diffusion finite elements or a discontinuous Galerkin method, we can expect that the method is robust for vanishing diffusivity. The need for such methods, notably for applications in mathematical finance, was recently observed in [30]. Other methods yielding such robustness are streamline diffusion methods, see [31,16], and discontinuous Galerkin methods, see, e.g., [7,24].…”