Invariant density adaptive estimation for ergodic jump diffusion processes over anisotropic classes

in Journal of Statistical Planning and Inference (in press)

We consider the solution of a multivariate stochastic differential equation with Levy-type jumps and with unique invariant probability measure with density μ. We assume that a continuous record of ... [more ▼]

We consider the solution of a multivariate stochastic differential equation with Levy-type jumps and with unique invariant probability measure with density μ. We assume that a continuous record of observations is available. In the case without jumps, Reiss and Dalalyan [7] and Strauch [24] have found convergence rates of invariant density estimators, under respectively isotropic and anisotropic H ̈older smoothness constraints, which are considerably faster than those known from standard multivariate density estimation. We extend the previous works by obtaining, in presence of jumps, some estimators which have the same convergence rates they had in the case without jumps for d ≥ 2 and a rate which depends on the degree of the jumps in the one-dimensional setting. We propose moreover a data driven bandwidth selection procedure based on the Goldenshluger and Lepski method [11] which leads us to an adaptive non-parametric kernel estimator of the stationary density μ of the jump diffusion X. [less ▲]

Minimax rate of estimation for invariant densities associated to continuous stochastic differential equations over anisotropic Holder classes

E-print/Working paper (2021)

Rate of estimation for the stationary distribution of jump-processes over anisotropic Holder classes.

E-print/Working paper (2020)

We study the problem of the non-parametric estimation for the density π of the stationary distribution of the multivariate stochastic differential equation with jumps (Xt)0≤t≤T , when the dimension d is ... [more ▼]

We study the problem of the non-parametric estimation for the density π of the stationary distribution of the multivariate stochastic differential equation with jumps (Xt)0≤t≤T , when the dimension d is such that d ≥ 3. From the continuous observa- tion of the sampling path on [0, T ] we show that, under anisotropic H ̈older smoothness constraints, kernel based estimators can achieve fast convergence rates. In particu- lar, they are as fast as the ones found by Dalalyan and Reiss [9] for the estimation of the invariant density in the case without jumps under isotropic H ̈older smooth- ness constraints. Moreover, they are faster than the ones found by Strauch [29] for the invariant density estimation of continuous stochastic differential equations, under anisotropic H ̈older smoothness constraints. Furthermore, we obtain a minimax lower bound on the L2-risk for pointwise estimation, with the same rate up to a log(T) term. It implies that, on a class of diffusions whose invariant density belongs to the anisotropic Holder class we are considering, it is impossible to find an estimator with a rate of estimation faster than the one we propose. [less ▲]

Unbiased truncated quadratic variation for volatility estimation in jump diffusion processes

in Stochastic Processes and Their Applications (2020)

The problem of integrated volatility estimation for an Ito semimartingale is considered under discrete high-frequency observations in short time horizon. We provide an asymptotic expansion for the ... [more ▼]

The problem of integrated volatility estimation for an Ito semimartingale is considered under discrete high-frequency observations in short time horizon. We provide an asymptotic expansion for the integrated volatility that gives us, in detail, the contribution deriving from the jump part. The knowledge of such a contribution allows us to build an unbiased version of the truncated quadratic variation, in which the bias is visibly reduced. In earlier results to have the original truncated realized volatility well-performed the condition β> 1 /2 (2− α) on β (that is such that (1/ n)^β is the threshold of the truncated quadratic variation) and on the degree of jump activity α was needed (see Mancini, 2011; Jacod, 2008). In this paper we theoretically relax this condition and we show that our unbiased estimator achieves excellent numerical results for any couple (α, β). [less ▲]

Joint estimation for volatility and drift parameters of ergodic jump diffusion processes via contrast function

in Statistical Inference for Stochastic Processes (2020)

In this paper we consider an ergodic diffusion process with jumps whose drift coefficient depends on μ and volatility coefficient depends on σ, two unknown parameters. We suppose that the process is ... [more ▼]

In this paper we consider an ergodic diffusion process with jumps whose drift coefficient depends on μ and volatility coefficient depends on σ, two unknown parameters. We suppose that the process is discretely observed. We introduce an estimator of θ := (μ, σ), based on a contrast function, which is asymptotically gaussian without requiring any conditions on the rate at which the discretisation step goes to 0, assuming a finite jump activity. This extends earlier results where a condition on the step discretization was needed (see [15],[36]) or where only the estimation of the drift parameter was considered (see [2]). In general situations, our contrast function is not explicit and in practise one has to resort to some approximation. We propose explicit approximations of the contrast function, such that the estimation of θ is feasible under the condition that n∆n^k → 0 where k > 0 can be arbitrarily large. This extends the results obtained by Kessler [24] in the case of continuous processes. [less ▲]