Asymptotics in small time for the density of a stochastic differential equation driven by a stable Lévy process

Abstract

This work focuses on the asymptotic behavior of the density in small time of a stochastic differential equation driven by a truncated α-stable process with index α ∈ (0, 2). We assume that the process depends on a parameter β = (θ, σ)T and we study the sensitivity of the density with respect to this parameter. This extends the results of [5] which was restricted to the index α ∈ (1, 2) and considered only the sensitivity with respect to the drift coefficient. By using Malliavin calculus, we obtain the representation of the density and its derivative as an expectation and a conditional expectation. This permits to analyze the asymptotic behavior in small time of the density, using the time rescaling property of the stable process.MSC2010: 60G51; 60G52; 60H07; 60H20; 60H10; 60J75.

LAMN property for the drift and volatility parameters of a sde driven by a stable Lévy process

“…Based on the main ideas of [5] and using the results of [6], we extend in the present paper these results to α ∈ (0, 2) and prove that the LAMN property holds for the parameters (θ, σ) with rate r n = n du. The proof is mainly based on the L 2 -regularity property of the transition density (see Jeganathan [14]) and on Malliavin calculus (see for example Gobet [8] for the use of Malliavin calculus in the case of a diffusion process).…”

“…These assumptions are sufficient to ensure that (2.1) has an unique solution belonging to L p , ∀p ≥ 1, and that X β t admits a density, for t > 0 (see [24]). Moreover, it is proved in [6] that this density is differentiable with respect to β.…”

“…The introduction of the truncation function τ permits to address both issues and to avoid more technical proofs. In particular it permits to ensure that the process (n 1/α L t/n ) has no jump of size larger than 2n 1/α and consequently makes easier the control of the asymptotic behavior of the Malliavin weights (mainly studied in [6]). Moreover the exact stable behavior of the Lévy measure around zero (τ = 1) gives the equality between the rescaled process (n 1/α L t/n ) and the α-stable process (L α t ), and also the equality of the corresponding Poisson measures, on a set A n whose complementary has small probability (P(A c n ) ≤ C/n, see Lem.…”

“…This property is repeatedly used in our proofs (see for example the proof of Thm. 2.5) and is also essential to study the limit of the Malliavin weights in [6].…”

“…The proof is mainly based on the L 2 -regularity property of the transition density (see Jeganathan [14]) and on Malliavin calculus (see for example Gobet [8] for the use of Malliavin calculus in the case of a diffusion process). The L 2regularity property is established here by using the asymptotic behavior of the density of the process solution of (2.1) in small time as well as its derivative with respect to the parameter, given in [6] and based on the Malliavin calculus for jump processes developed by Bichteler et al [4]. It also requires a careful study of the asymptotic behavior of the information matrix based on one observation of the process, this is the subject of Section 3.…”