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 ▲]

Spectral characterization of the quadratic variation of mixed Brownian–fractional Brownian motion

in Statistical Inference for Stochastic Processes (2013), 16(2), 97-112

Dzhaparidze and Spreij (Stoch Process Appl, 54:165–174, 1994) showed that the quadratic variation of a semimartingale can be approximated using a randomized periodogram. We show that the same ... [more ▼]

Dzhaparidze and Spreij (Stoch Process Appl, 54:165–174, 1994) showed that the quadratic variation of a semimartingale can be approximated using a randomized periodogram. We show that the same approximation is valid for a special class of continuous stochastic processes. This class contains both semimartingales and non-semimartingales. The motivation comes partially from the recent work by Bender et al. (Finance Stoch, 12:441–468, 2008), where it is shown that the quadratic variation of the log-returns determines the hedging strategy. [less ▲]