Existence of Lévy term structure models

Filipović, Damir; Tappe, Stefan

doi:10.1007/s00780-007-0054-4

Cited by 70 publications

(57 citation statements)

References 50 publications

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“…1Þ Filipović & Tappe (2006) in our finite activity case. The previous conditions are slightly more than standard for existence and uniqueness of mild solutions, i.e.…”

Section: Setting and Assumptionsmentioning

confidence: 71%

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Cubature on Wiener space in infinite dimension

Bayer

Teichmann

2008

Proc. R. Soc. A.

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We prove a stochastic Taylor expansion for stochastic partial differential equations (SPDEs) and apply this result to obtain cubature methods, i.e. high-order weak approximation schemes for SPDEs, in the spirit of Lyons and Victoir (Lyons & Victoir 2004 Proc. R. Soc. A 460, 169-198). We can prove a high-order weak convergence for welldefined classes of test functions if the process starts at sufficiently regular initial values. We can also derive analogous results in the presence of Lévy processes of finite type; here the results seem to be new, even in finite dimension. Several numerical examples are added.

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“…1Þ Filipović & Tappe (2006) in our finite activity case. The previous conditions are slightly more than standard for existence and uniqueness of mild solutions, i.e.…”

Section: Setting and Assumptionsmentioning

confidence: 71%

“…The main reference for equations of type (1.1) is the monograph of Da Prato & Zabczyk (1992). In the (general) Lévy case, we refer to Filipović & Tappe (2006) and Peszat & Zabczyk (2007), even though we do not need results of their strength in our case since all Lévy processes under consideration are of finite type.…”

Section: Introductionmentioning

confidence: 99%

Cubature on Wiener space in infinite dimension

Bayer

Teichmann

2008

Proc. R. Soc. A.

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“…0 < δ < y. Let f 1 : T x,y−δ −→ R + , where be a bounded function satisfying inequality 10) where g 1 : T x,y−δ −→ R + . Let f 2 : T x,y−δ −→ R + be a bounded function solving equation…”

Section: Remark 52mentioning

confidence: 99%

Forward rate models with linear volatilities

Barski

Zabczyk

2011

Finance Stoch

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Existence of solutions to the Heath-Jarrow-Morton equation of the bond market with linear volatility and general Lévy random factor is studied. Conditions for existence and non-existence of solutions in the class of bounded fields are presented. For the existence of solutions the Lévy process should necessarily be without the Gaussian part and without negative jumps. If this is the case then necessary and sufficient conditions for the existence are formulated either in terms of the behavior of the Lévy measure of the noise near the origin or the behavior of the Laplace exponent of the noise at infinity.

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“…For the proof we refer to [25], Chapter V.10, Theorem 57, in particular with respect to the conditioning on the jump part. The proof remains unchanged in the infinite dimensional setting, see [15] for the existence and uniqueness proof on separable Hilbert spaces.…”

Section: Decomposition Theorem For Jump-diffusions On Hilbert Spacesmentioning

confidence: 99%

Absolutely Continuous Laws of Jump-Diffusions in Finite and Infinite Dimensions with Applications to Mathematical Finance

Forster¹,

Lütkebohmert²,

Teichmann³

2009

SIAM J. Math. Anal.

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Abstract. In mathematical Finance calculating the Greeks by Malliavin weights has proved to be a numerically satisfactory procedure for finite-dimensional Itô-diffusions. The existence of Malliavin weights relies on absolute continuity of laws of the projected diffusion process and a sufficiently regular density. In this article we first prove results on absolute continuity for laws of projected jump-diffusion processes in finite and infinite dimensions, and a general result on the existence of Malliavin weights in finite dimension. In both cases we assume Hörmander conditions and hypotheses on the invertibility of the so-called linkage operators. The purpose of this article is to show that for the construction of numerical procedures for the calculation of the Greeks in fairly general jump-diffusion cases one can proceed as in a pure diffusion case. We also show how the given results apply to infinite dimensional questions in mathematical Finance. There we start from the Vasiček model, and add -by pertaining no arbitrage -a jump diffusion component. We prove that we can obtain in this case an interest rate model, where the law of any projection is absolutely continuous with respect to Lebesgue measure on R M .

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Existence of Lévy term structure models

Cited by 70 publications

References 50 publications

Cubature on Wiener space in infinite dimension

Cubature on Wiener space in infinite dimension

Forward rate models with linear volatilities

Absolutely Continuous Laws of Jump-Diffusions in Finite and Infinite Dimensions with Applications to Mathematical Finance

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