Nonparametric drift estimation for diffusions with jumps driven by a Hawkes process

Abstract

We consider a 1-dimensional diffusion process X with jumps. The particularity of this model relies in the jumps which are driven by a multidimensional Hawkes process denoted N. This article is dedicated to the study of a nonparametric estimator of the drift coefficient of this original process. We construct estimators based on discrete observations of the process X in a high frequency framework with a large horizon time and on the observations of the process N. The proposed nonparametric estimator is built from a least squares contrast procedure on subspace spanned by trigonometric basis vectors. We obtain adaptive results that are comparable with the one obtained in the nonparametric regression context. We finally conduct a simulation study in which we first focus on the implementation of the process and then on showing the good behavior of the estimator.

Appendix: Theoretical results

1.1 Gronwall’s lemma

Lemma 7.1

Let $\phi , \psi$ and y be three non negative continuous functions on [a, b], that satisfy

$$\begin{aligned} \forall t\in [a,b], \quad y(t)\le \phi (t)+\int _a^t\psi (s)y(s)ds. \end{aligned}$$

Then

$$\begin{aligned} \forall t\in [a,b], \quad y(t)\le \phi (t)+\int _a^t\phi (s)\psi (s)\exp \Big (\int _s^t\psi (u) du\Big )ds. \end{aligned}$$

1.2 Talagrand’s inequality

The following result follows from the Talagrand concentration inequality (see Klein and Rio 2005).

Theorem 7.2

Consider $n \in {\mathbb {N}}^*, {\mathcal {F}}$ a class at most countable of measurable functions, and $(X_i)_{i\in \{1,\ldots ,n\}}$ a family of real independent random variables. One defines, for all $f\in {\mathcal {F}}$,

$$\begin{aligned} \nu _n(f) =\frac{1}{n} \sum _{i=1}^{n} (f(X_i)-{\mathbb {E}}[f(X_i)]). \end{aligned}$$

Supposing there are three positive constants M, H and v such that $\underset{f\in {\mathcal {F}}}{\sup } \Vert f\Vert _{\infty } \le M$, ${\mathbb {E}}[\underset{f\in {\mathcal {F}}}{\sup } |\nu _n(f)| ] \le H$, and $\underset{f\in {\mathcal {F}}}{\sup } ({1}/{n}) \sum _{i=1}^{n} \mathrm {Var}(f(X_i)) \le v$, then for all $\alpha >0$,

$$\begin{aligned} {\mathbb {E}}\left[ \left( \underset{f\in {\mathcal {F}}}{\sup } |\nu _n(f)|^2-2(1+2\alpha )H^2 \right) _+ \right]\le & {} \frac{4}{b} \left( \frac{v}{n} \exp \left( -b \alpha \frac{n H^2}{v} \right) \right. \\&+ \left. \frac{49M^2}{b C^2(\alpha )n^2} \exp \left( -\frac{\sqrt{2}b C(\alpha )\sqrt{\alpha }}{7}\frac{nH}{M} \right) \right) \end{aligned}$$

with $C(\alpha )=(\sqrt{1+\alpha }-1) \wedge 1$, and $b=\frac{1}{6}$.