Densite en temps petit d'un processus de sauts

Léandre, Rémi

doi:10.1007/bfb0077628

Cited by 36 publications

(59 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(see Proposition III.6 in [14] and Corollary 1.1 in [20]). However, (5.2) may be hard to verify since, often, the transition density p t of a Lévy process are not explicitly given.…”

Section: Bounds For the Derivatives Of The Marginal Densitiesmentioning

confidence: 97%

“…The approach in [16] built on earlier results of Léandre [14]. In view of the previous bound, for values of t away from 0, the derivatives of p t are uniformly bounded, and condition (5.2) is then related to the behavior of p (k) t as t → 0.…”

Section: Bounds For the Derivatives Of The Marginal Densitiesmentioning

confidence: 98%

“…Section 28 in [21]), in which case, X t is known to admit a C ∞ -density p t (see e.g. [14,Theorem I.1] or [21,Proposition 28.3]). The characteristic exponent γ will be assumed to belong to C ∞ (R 0 ) and to satisfy either one of the following two bounds:…”

Section: Bounds For the Derivatives Of The Marginal Densitiesmentioning

confidence: 99%

See 2 more Smart Citations

Small-time expansions for the transition distributions of Lévy processes

Figueroa-López

Houdré

2009

Stochastic Processes and their Applications

View full text Add to dashboard Cite

Let X = (X t ) t≥0 be a Lévy process with absolutely continuous Lévy measure ν. Small-time expansions of arbitrary polynomial order in t are obtained for the tails P (X t ≥ y), y > 0, of the process, assuming smoothness conditions on the Lévy density away from the origin. By imposing additional regularity conditions on the transition density p t of X t , an explicit expression for the remainder of the approximation is also given. As a byproduct, polynomial expansions of order n in t are derived for the transition densities of the process. The conditions imposed on p t require that, away from the origin, its derivatives remain uniformly bounded as t → 0. Such conditions are then shown to be satisfied for symmetric stable Lévy processes as well as some tempered stable Lévy processes such as the CGMY one. The expansions seem to correct the asymptotics previously reported in the literature.

show abstract

“…(see Proposition III.6 in [14] and Corollary 1.1 in [20]). However, (5.2) may be hard to verify since, often, the transition density p t of a Lévy process are not explicitly given.…”

Section: Bounds For the Derivatives Of The Marginal Densitiesmentioning

confidence: 97%

Section: Bounds For the Derivatives Of The Marginal Densitiesmentioning

confidence: 98%

See 1 more Smart Citation

Small-time expansions for the transition distributions of Lévy processes

Figueroa-López

Houdré

2009

Stochastic Processes and their Applications

View full text Add to dashboard Cite

show abstract

“…Investigations on asymptotic behaviour of isotropic α-stable (α ∈ (0, 2)) convolution semigroups date back to 1923 and 1960, when Pólya [46] and Blumenthal and Getoor [4] obtained the first results in this direction. With respect to a further study of asymptotic behaviour of convolution semigroups in space and time we refer to [2,20,40,58,48,15,35,30] and references there. In recent papers [11,32,18] the case of unimodal and isotropic jump Lévy processes has been analyzed.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Spatial asymptotics at infinity for heat kernels of integro-differential operators

Kaleta

Sztonyk

2018

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

We study a spatial asymptotic behaviour at infinity of kernels p t (x) for convolution semigroups of nonlocal pseudo-differential operators. We give general and sharp sufficient conditions under which the limitsexist and can be effectively computed. Here ν is the corresponding Lévy density, T ⊂ (0, ∞) is a bounded time-set and E is a subset of the unit sphere in R d , d ≥ 1. Our results are local on the unit sphere. They apply to a wide class of convolution semigroups, including those corresponding to highly asymmetric (finite and infinite) Lévy measures. Key examples include fairly general families of stable, tempered stable, jump-diffusion and compound Poisson semigroups. A main emphasis is put on the semigroups with Lévy measures that are exponentially localized at infinity, for which our assumptions and results are strongly related to the existence of the multidimensional exponential moments. Here a key example is the evolution semigroup corresponding to the so-called quasi-relativistic Hamiltonian √ −∆ + m 2 − m, m > 0. As a byproduct, we also obtain sharp two-sided estimates of the kernels p t in generalized cones, away from the origin.

show abstract

“…(see e.g., Corollary 1 in Rüschendorf and Woerner (2002) as a special case for Lévy processes of Léandre (1987) and Picard (1997) for points that can be reached in one jump from 0). The regularity conditions referred to in the statement of the lemma are those of Theorem 1 in Rüschendorf and Woerner (2002).…”

Section: Appendix D: Proof Of Theoremmentioning

confidence: 99%

Disentangling Volatility from Jumps

Aı̈t-Sahalia¹

2003

View full text Add to dashboard Cite

Realistic models for financial asset prices used in portfolio choice, option pricing or risk management include both a continuous Brownian and a jump components. This paper studies our ability to distinguish one from the other. I find that, surprisingly, it is possible to perfectly disentangle Brownian noise from jumps. This is true even if, unlike the usual Poisson jumps, the jump process exhibits an infinite number of small jumps in any finite time interval, which ought to be harder to distinguish from Brownian noise, itself made up of many small moves.

show abstract

Densite en temps petit d'un processus de sauts

Cited by 36 publications

References 4 publications

Small-time expansions for the transition distributions of Lévy processes

Small-time expansions for the transition distributions of Lévy processes

Spatial asymptotics at infinity for heat kernels of integro-differential operators

Disentangling Volatility from Jumps

Contact Info

Product

Resources

About