A note on parameter estimation for discretely sampled SPDEs

By Igor Cialenco, Yicong Huang. Stochastics and Dynamics, 2017

64 citations recorded by the source corpus.

We consider a parameter estimation problem for one-dimensional stochastic heat equations, when data is sampled discretely in time or spatial component. We prove that, the real valued parameter next to the Laplacian (the drift), and the constant parameter in front of the noise (the volatility) can be consistently estimated under somewhat surprisingly minimal information. Namely, it is enough to observe the solution at a fixed time and on a discrete spatial grid, or at a fixed space point and at discrete time instances of a finite interval, assuming that the mesh-size goes to zero. The proposed estimators have the same form and asymptotic properties regardless of the nature of the domain –bounded domain or whole space. The derivation of the estimators and the proofs of their asymptotic properties are based on computations of power variations of some relevant stochastic processes. We use elements of Malliavin calculus to establish the asymptotic normality properties in the case of bounded domain. We also discuss the joint estimation problem of the drift and volatility coefficient. We conclude with some numerical experiments that illustrate the obtained theoretical results.

Subfields: Computational Theory and Mathematics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/84c39addb4dded5b61cccfab5516cd1138e486a4

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Citations in the corpus

• Volatility estimation for stochastic PDEs using high-frequency observations (2017, background): “Literature The key insight in the pioneering work by Hübner et al. (1993) is that for a large class of parabolic differential equations the solution of the SPDE can be written as a Fourier series where each Fourier coefficient follows an ordinary stochastic differential equation of Ornstein-Uhlenbec”
• Parameter estimation for SPDEs based on discrete observations in time and space (2019, methodology): “[7] Cialenco, I. and Huang, Y. (2019).”