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 [E. Clément and A. Gloter, Local asymptotic mixed normality property for discretely observed stochastic dierential equations driven by stable Lévy processes. Stochastic Process. Appl. 125 (2015) 2316–2352.] 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.
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Contributor : Emmanuelle Clément <>
Submitted on : Friday, April 20, 2018 - 11:27:50 AM
Last modification on : Tuesday, May 14, 2019 - 12:46:01 PM

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Emmanuelle Clément, Arnaud Gloter, Huong Nguyen. Asymptotics in small time for the density of a stochastic differential equation driven by a stable lévy process. ESAIM: Probability and Statistics, EDP Sciences, 2018, 22, pp.58-95. ⟨10.1051/ps/2018009⟩. ⟨hal-01772290⟩

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