Statistical analysis of tempo-spatial stochastic integral processes
Final Report Abstract
3 Zusammenfassung Chong [1] started the project by considering the problem of estimating stochastic volatility for a class of second-order parabolic stochastic PDEs. Assuming that the solution is observed at a high temporal frequency, he uses limit theorems for multipower variations and related functionals to construct consistent nonparametric estimators and asymptotic confidence bounds for the integrated volatility process. As a byproduct of our analysis, he also obtains feasible estimators for the regularity of the spatial covariance function of the noise. In [5], Chong extends this model by considering the stochastic heat equation driven by a multiplicative Gaussian noise that is white in time and spatially homogeneous in space. Assuming that the spatial correlation function is given by a Riesz kernel of order α ∈ (0, 1), he proves a central limit theorem for power variations and other related functionals of the solution. To our surprise, there is no asymptotic bias despite the low regularity of the noise coefficient in the multiplicative case. We trace this back to cancellation effects between error terms in second-order limit theorems for power variations. a Chong and Pham [4] propose a novel class of tempo-spatial OrnsteinˆUhlenbeck processes as solue tions to L´vy-driven Volterra equations with additive noise and multiplicative drift. After formulating conditions for existence and uniqueness of solutions, they derive an explicit solution formula and discuss distributional properties such as stationarity, second-order structure, and short versus long memory. Furthermore, they analyze in detail path properties of the solution process. In particular, they introduce a a different notions of c`dl`g paths in space and time and establish conditions for the existence of versions with these regularity properties. The theoretical results are accompanied by illustrative examples. e In [3] we estimate model parameters of L´vy-driven causal continuous-time autoregressive moving average random fields by fitting the empirical variogram to the theoretical counterpart using a weighted least squares (WLS) approach. Subsequent to deriving asymptotic results for the variogram estimator, we show strong consistency and asymptotic normality of the parameter estimator. Furthermore, we conduct a simulation study to assess the quality of the WLS estimator for finite samples. For the simulation, we utilize numerical approximation schemes based on truncation and discretization of stochastic integrals and we analyze the associated simulation errors in detail. Finally, we apply our results to real data of the cosmic microwave background. ˙ Given a sequence Lǫ of L´vy noises, Chong and Delerue [2] derive necessary and sufficient conditions in e ˙ terms of their variances σ 2 (ǫ) such that the solution to the stochastic heat equation with noise σ 2( ǫ)−1 Lǫ converges in law to the solution of the same equation with Gaussian noise. These results apply to both equations with additive and multiplicative noise and hence lift the findings of [9] and [26] for finitee dimensional L´vy processes to the infinite-dimensional setting without making distributional assumptions on the solutions such as infinite divisibility. One important ingredient of the proof is to characterize the solution to the limit equation by a sequence of martingale problems. To this end, it is crucial to view a a the solution processes both as random fields and as c`dl`g processes with values in a Sobolev space of negative real order. ˙ For a sequence Lǫ of L´vy noises with variance σ 2 (ǫ), Delerue [7] proves the Gaussian approximation e ˙ of the solution uǫ to the stochastic wave equation driven by σ 2 (ǫ)−1 Lǫ and thus extends the result of [2] to the class of hyperbolic stochastic PDEs. That is, he finds a necessary and sufficient condition in terms of σ 2 (ǫ) for uǫ to converge in law to the solution to the same equation with Gaussian noise. Furthermore, uǫ is shown to have a space-time version with a c`dl`g property determined by the wave kernel, and its a a derivative ∂t uǫ a c`dl`g version when viewed as a distribution-valued process. These two path properties a a are essential to the proof of the normal approximation as the limit is characterized by martingale problems that necessitate both elements. Our results apply to additive as well as to multiplicative noises. In [6] the authors consider the problem of estimating volatility based on high-frequency data when o the observed price process is a continuous Itˆ semimartingale contaminated by microstructure noise. Assuming that the noise process is compatible across different sampling frequencies, they argue that it typically has a similar local behavior to fractional Brownian motion. For the resulting class of processes, called mixed semimartingales, they derive consistent estimators and asymptotic confidence intervals for the roughness parameter of the noise and the integrated price and noise volatilities, in all cases where these quantities are identifiable. This model can explain key features of recent stock price data, most notably divergence rates in volatility signature plots that vary considerably over time and between assets.
Publications
- Volterra-type Ornstein-Uhlenbeck processes in space and time. Stochastic Process. Appl., 128(9):3082–3117, 2018
V.-S. Pham and C. Chong
(See online at https://doi.org/10.1016/j.spa.2017.10.012) - Normal approximation of the solution to the stochastic wave equation with L´vy noise. 2019
T. Delerue
(See online at https://doi.org/10.48550/arXiv.1911.01811) - High-frequency analysis of parabolic stochastic PDEs with multiplicative noise. 2020
C. Chong
(See online at https://doi.org/10.48550/arXiv.1908.04145) - High-frequency analysis of parabolic stochastic PDEs. Ann. Statist., 48(2):1143–1167, 2020
C. Chong
(See online at https://doi.org/10.1214/19-AOS1841) - Normal approximation of the solution to the stochastic heat equation with Lévy noise. Stoch. Partial Differ. Equ. Anal. Comput., 8(2):362–401, 2020
C. Chong and T. Delerue
(See online at https://doi.org/10.1007/s40072-019-00148-4) - Estimation of causal CARMA random fields. Scand. J. Stat., 48(1):132–163, 2021
C. Klüppelberg and V.-S. Pham
(See online at https://doi.org/10.1111/sjos.12444) - Mixed semimartingales: Volatility estimation in the presence of fractional noise. 2021
C. Chong, T. Delerue, and G. Li
(See online at https://doi.org/10.48550/arXiv.2106.16149)
