Adaptive Estimation for Lévy Processes

Abstract

This chapter is concerned with nonparametric estimation of the Lévy density of a Lévy process. The sample path is observed at n equispaced instants with sampling interval Δ. We develop several nonparametric adaptive methods of estimation based on deconvolution, projection and kernel. The asymptotic framework is: n tends to infinity, Δ = Δ _n tends to 0 while n Δ _n tends to infinity (high frequency). Bounds for the $\mathbb{L}^{2}$-risk of estimators are given. Rates of convergence are discussed. Estimation of the drift and Gaussian component coefficients is studied. A specific method for estimating the jump density of compound Poisson processes is presented. Examples and simulation results illustrate the performance of estimators.

AMS Subject Classification 2000:

Primary: 62G05, 62M05

Secondary: 60G51

Appendix

The Talagrand Inequality The result below follows from the Talagrand concentration inequality given in [43] and arguments in [8] (see the proof of their Corollary 2 p. 354).

Lemma A.1 (Talagrand Inequality)

Let $Y _{1},\ldots,Y _{n}$ be independent random variables, let $\nu _{n,Y }(f) = (1/n)\sum _{i=1}^{n}[f(Y _{i}) - \mathbb{E}(f(Y _{i}))]$ and let $\mathcal{F}$ be a countable class of uniformly bounded measurable functions. Then for ε² > 0

$$ \displaystyle\begin{array}{rcl} & & \mathbb{E}\Big[\sup _{f\in \mathcal{F}}\vert \nu _{n,Y }(f)\vert ^{2} - 2(1 + 2\epsilon ^{2})H^{2}\Big]_{ +} {}\\ & & \qquad \leq \frac{4} {K_{1}}\left (\frac{v} {n}e^{-K_{1}\epsilon ^{2} \frac{nH^{2}} {v} } + \frac{98M^{2}} {K_{1}n^{2}C^{2}(\epsilon ^{2})}e^{-\frac{2K_{1}C(\epsilon ^{2})\epsilon } {7\sqrt{2}} \frac{nH} {M} }\right ), {}\\ \end{array} $$

with $C(\epsilon ^{2}) = \sqrt{1 +\epsilon ^{2}} - 1$, K₁ = 1∕6, and

$$\displaystyle{\sup _{f\in \mathcal{F}}\|f\|_{\infty }\leq M,\;\;\;\; \mathbb{E}\Big[\sup _{f\in \mathcal{F}}\vert \nu _{n,Y }(f)\vert \Big] \leq H,\;\sup _{f\in \mathcal{F}}\frac{1} {n}\sum _{k=1}^{n}\mathrm{Var}(f(Y _{ k})) \leq v.}$$

By standard density arguments, this result can be extended to the case where $\mathcal{F}$ is a unit ball of a linear normed space, after checking that $f\mapsto \nu _{n}(f)$ is continuous and $\mathcal{F}$ contains a countable dense family.

Lemma A.2 (The Rosenthal Inequality)

(see e.g. [33]) Let $(X_{i})_{1\leq i\leq n}$ be n independent centered random variables, such that $\mathbb{E}(\vert X_{i}\vert ^{p}) < +\infty$ for an integer p ≥ 1. Then there exists a constant C(p) such that

$$\displaystyle{ \mathbb{E}\left (\left \vert \sum _{i=1}^{n}X_{ i}\right \vert ^{p}\right ) \leq C(p)\left (\sum _{ i=1}^{n}\mathbb{E}(\vert X_{ i}\vert ^{p}) + \left (\sum _{ i=1}^{n}\mathbb{E}(X_{ i}^{2})\right )^{p/2}\right ). }$$

(A.1)

Lemma A.3 (The Young Inequality)

(see [35]) Let f be a function belonging to $\mathbb{L}^{p}(\mathbb{R})$ and g belonging to $\mathbb{L}^{q}(\mathbb{R})$ , let p,q,r be real numbers in [1,+∞] and such that

$$\displaystyle{\frac{1} {p} + \frac{1} {q} = \frac{1} {r} + 1.}$$

Then

$$\displaystyle{\|f \star g\|_{r} \leq \| f\|_{p}\;\|g\|_{q}.}$$

where f ⋆ g is the convolution product and $\|f\|_{p}^{p} =\int \vert f(x)\vert ^{p}\mathit{dx}$ . In particular, for p = 1, r = q = 2, we have $\|f \star g\|_{2} \leq \| f\|_{1}\;\|g\|_{2}$ .