On the existence of smooth densities for jump processes

Picard, Jean

doi:10.1007/bf01191910

Cited by 161 publications

(145 citation statements)

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“…(1.1) with respect to the Lebesgue measure on R d , by means of the Malliavin calculus on the Poisson space initiated by Picard (cf. [9]). …”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In Sec. 3, we shall introduce the Malliavin calculus on the Poisson space initiated by Picard in [9] briefly, where the criterion on absolute continuity for the probability law of random variables is mentioned. See Theorem 3.1.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…We shall introduce the Malliavin calculus on the Poisson space initiated by Picard in [9]. Let U = [0, T ] × R m .…”

Section: Malliavin Calculus On the Poisson Spacementioning

confidence: 99%

“…Before giving the proof of Theorem 3.1, we shall introduce three propositions, which are cited from [9], and whose proofs are omitted.…”

Section: Malliavin Calculus On the Poisson Spacementioning

confidence: 99%

“…from Lemma 2.4 in [9], whereỸ (û) is the random function ofû obtained from Y (û) by symmetrization. Denote byZ(û) the random symmetrized function ofû obtained fromẐ(û) similarly.…”

Section: Proposition 32 (Cf [9] Lemma 14) For Any R-valued Randomentioning

confidence: 99%

See 4 more Smart Citations

Absolute Continuity for Solutions to Stochastic Functional Differential Equations With Jumps

Takeuchi

2007

Stoch. Dyn.

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We study absolute continuity for the probability law of solutions to stochastic functional differential equations driven by Lévy processes:where Xt is the path defined by Xt(s) = X(t+s) for s ∈ [−r, 0]. We apply methods from the Malliavin calculus on the Poisson space to show that non-degenerate conditions on A 1 , . . . , Am of the noise term guarantee the existence of densities.for θ ∈ R m and t ∈ [0, T ]. 153 Stoch. Dyn. 2007.07:153-185. Downloaded from www.worldscientific.com by NORTHWESTERN UNIVERSITY on 02/04/15. For personal use only. Stoch. Dyn. 2007.07:153-185. Downloaded from www.worldscientific.com by NORTHWESTERN UNIVERSITY on 02/04/15. For personal use only. Absolute Continuity for SFDEs with Jumps 155 Lebesgue densities by the Malliavin calculus on the Wiener space. Moreover, Bell and Mohammed prove in [1] that the polynomial-order degeneracy on the coefficients A i (1 ≤ i ≤ m) yields the smoothness of Lebesgue densities. Now we shall introduce the assumption on the Lévy measure µ(dz). Assumption 1.1. (i) There exists 0 < α < 2 such that lim inf ρ ↓ 0

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“…(1.1) with respect to the Lebesgue measure on R d , by means of the Malliavin calculus on the Poisson space initiated by Picard (cf. [9]). …”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…We shall introduce the Malliavin calculus on the Poisson space initiated by Picard in [9]. Let U = [0, T ] × R m .…”

Section: Malliavin Calculus On the Poisson Spacementioning

confidence: 99%

“…Before giving the proof of Theorem 3.1, we shall introduce three propositions, which are cited from [9], and whose proofs are omitted.…”

Section: Malliavin Calculus On the Poisson Spacementioning

confidence: 99%