Non-parametric adaptive estimation of the drift for a jump diffusion process

Abstract

In this article, we consider a jump diffusion process (Xt) t≥0 observed at discrete times t = 0, ∆, . . . , n∆. The sampling interval ∆ tends to 0 and n∆ tends to infinity. We assume that (Xt) t≥0 is ergodic, strictly stationary and exponentially β-mixing. We use a penalized least-square approach to compute two adaptive estimators of the drift function b. We provide bounds for the risks of the two estimators.A 1. The functions b, σ and ξ are Lipschitz.A 2.1. The function σ is bounded from below and above:4. The Lévy measure ν satisfies:Under Assumption A1, the stochastic differential equation (1) admits a unique strong solution. According to Masuda (2007), under Assumptions A1 and A2, the process (X t ) admits a unique invariant probability ̟ and satisfies the ergodic theorem: for any measurable function g such that´|g(This distribution has moments of order 4. Moreover, Masuda (2007) also ensures that under these assumptions, the process (X t ) is exponentially β-mixing. Furthermore, if there exist two constants c and n 0 such that, for any x ∈ R, ξ 2 (x) ≥ c(1 + |x|) −n0 , then Ishikawa and Kunita (2006) ensure that a smooth transition density exists.A 3. 1. The stationary measure ̟ admits a density π which is bounded from below and above on the compact interval A:2. The process (X t ) t≥0 is stationary (η ∼ ̟(dx) = π(x)dx).The first part of this assumption is automatically satisfied if ξ = 0 (that is if (X t ) t≥0 is a diffusion process). The following proposition is very useful for the proofs. It is derived from Result 11. Proposition 1.Under Assumptions A1-A3, for any p ≥ 1, there exists a constant c(p) such that, if´R z 2p ν(dz) < ∞:

Adaptive nonparametric drift estimation of an integrated jump diffusion process

“…These assumptions are largely congruent to those in Schmisser (2014), who investigated the nonparametric estimation of b in the usual non-integrated setting.…”

“…In contrast to the kernel based approach, Comte et al (2009) use a model selection approach to construct adaptive nonparametric estimators of b and σ on a fixed compact interval in an integrated diffusion model without jumps. This work extends their approach for estimating ordinary univariate diffusions and was also pursued by Schmisser (2014) in the case of univariate jump diffusions. In view of these two papers, we will conduct an analogous approach for the case of estimating the drift in an integrated jump diffusion model.…”

“…Therefore: We follow straightly the proof of Theorem 2 in Schmisser (2014). To bound this term, we use a Bernstein inequality and, moreover, we need to apply a Markov inequality on the term exp(ν n (s)).…”

Bias Correction Estimation for a Continuous‐Time Asset Return Model with Jumps

“…Remark The BDG result is also denoted as Kunita's first inequality in Applebaum (). Schmisser () and Funke and Schmisser () stated its formulation for the detailed proof, one can also refer to them.…”

Nonparametric Gaussian inference for stable processes

“…Hanif (2012) extends this result to local linear smoothing, and Funke (2015) studies the case of infinite variation subject to moment conditions. Schmisser (2014) studies drift estimation for jump diffusions by means of a sieve regression, assuming finite fourth moments of the jump process. The case of α-stable innovations differs from these settings in that the driving Lévy process has infinite variation for α ≥ 1, and not even finite second moments.…”