Volatility estimation for stochastic PDEs using high-frequency observations

Abstract

We study the parameter estimation for parabolic, linear, second-order, stochastic partial differential equations (SPDEs) observing a mild solution on a discrete grid in time and space. A high-frequency regime is considered where the mesh of the grid in the time variable goes to zero. Focusing on volatility estimation, we provide an explicit and easy to implement method of moments estimator based on squared increments. The estimator is consistent and admits a central limit theorem. This is established moreover for the joint estimation of the integrated volatility and parameters in the differential operator in a semi-parametric framework. Starting from a representation of the solution of the SPDE with Dirichlet boundary conditions as an infinite factor model and exploiting mixing-type properties of time series, the theory considerably differs from the statistics for semi-martingales literature. The performance of the method is illustrated in a simulation study.

Drift estimation for discretely sampled SPDEs

“…Moreover, if 4β < d, then consistency holds true, when N → ∞, when M, T are fixed. This, in particular implies that to estimate efficiently the drift parameter it is enough to observe the Fourier modes at one instant of time -a result that agrees with recent discoveries in [CH17,BT17] where the solution is sampled in physical domain. Under some additional technical assumptions on the growth rates of N, M and T , we also prove that the proposed estimator is also asymptotically normal, with the same rate of convergence √ T N Open problems and future work.…”

On generating fully discrete samples of the stochastic heat equation on an interval

“…based on both methods on the finite time horizon T = 1. The outcomes are compared with the following theoretical results: As shown in [1], the temporal quadratic variation satisfies for any finite M…”

“…In this article we consider a method for generating discrete samples X ti (y k ) on a regular grid ((t i , y k ), 0 ≤ i ≤ N, 0 ≤ k ≤ M ) ⊂ [0, T ] × [0, 1], where X is the weak solution to the stochastic partial differential equation (SPDE) 1], t ∈ [0, T ], X t (0) = X t (1) = 0, X 0 = ξ.…”

Estimation of the drift parameter for the fractional stochastic heat equation via power variation

“…with vanishing initial conditions, where (−∆) α 2 denotes the fractional Laplacian of order α ∈ (1,2], θ > 0 and W is a Gaussian noise which is white in time and white or correlated in space.…”

“…Firstly, we replace the standard Laplacian operator used in all the above references by a fractional Laplacian. On the other hand, we consider a simpler form, comparing to [2,17], of the differential operator. Secondly, we also consider a noise term which is correlated in space.…”