Asymptotic Inference for Jump Diffusions with State‐Dependent Intensity

Abstract

We establish the local asymptotic normality property for a class of ergodic parametric jump‐diffusion processes with state‐dependent intensity and known volatility function sampled at high frequency. We prove that the inference problem about the drift and jump parameters is adaptive with respect to parameters in the volatility function that can be consistently estimated.

Estimating functions for jump–diffusions

“…The model studied in [3] is more general than our model in one respect: the jump intensity is allowed to be state-dependent. In [32, Theorem 2.2], the model (4.1) is studied with only a one-dimensional drift parameter α in either of the cases (a) or (b) of Assumption 4.1.…”

“…Usually, the optimal rate of convergence and efficient asymptotic variance would be identified using results from the theory of local asymptotic normality. However, local asymptotic normality and, for infill asymptotics, local asymptotic mixed normality are ongoing areas of research for stochastic processes with jumps [3,5,27,28,31,32]. No results for general jump-diffusions have been established so far.…”

“…It can rather safely be conjectured that the optimal rate of convergence is √ n∆ n for drift and jump components of the parameter and √ n for diffusion components, and that the efficient asymptotic variance is as proposed in Section 4. These conjectures are motivated not only by local asymptotic normality results which cover particular submodels of (1.1) [3,27,31,32], but also by other asymptotic results [14,55,56].…”

Small-time expansions for state-dependent local jump–diffusion models with infinite jump activity

“…which is also clearly bounded since |∂ x F (x, w)/w| is bounded. For the third term, D 3 (x, w) w = (∂ x F (x, w)) 2 w F (x, w) α+1 ∂ r (ν(x, r)g(r) φε (r)r −α−1 ) r=F (x,w) ν(x, F (x, w))g(F (x, w)) φε (F (x, w)) .…”

“…(∂ x F (x, w)) 2 w F (x, w)(∂ 2 ν)(x, F (x, w)) ν(x, F (x, w)) , (∂ x F (x, w)) 2 wF (x, w) F (x, w)g (F (x, w)) g(F (x, w)) (∂ x F (x, w)) 2 w φ ε (F (x, w)) φε (F (x, w))…”