Bernstein-von Mises type theorem for a class of Hilbert space valued stochastic differential equations. (English) Zbl 0990.60061

Prakasa Rao, B. L. S.

Theory Probab. Math. Stat. 63, 151-158 (2001) and Teor. Jmovirn. Mat. Stat. 63, 138-144 (2000).

The author proves a Bernstein-von Mises type theorem for a class of Hilbert space valued stochastic differential equations \(dZ(t)=\theta AZ(t)+dW(t),\;Z(0)=z_0,\;t\geq 0\), where \(\{Z(t), t\geq 0\}\) is an \(H\)-valued stochastic process, \(H\) is a real separable Hilbert space; \(A\) is the infinitesimal generator of a strongly continuous semigroup acting on \(H\); \(W(t)\) is an \(H\)-valued Wiener process with nuclear covariance operator; the parameter \(\theta\) is unknown and real-valued. Also, it is proved that the Bayes estimator of \(\theta\) is strongly consistent, asymptotically normal and asymptotically efficient under some regularity conditions.

Reviewer: A.D.Borisenko (Kyïv)

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
62F15	Bayesian inference

Keywords:

infinite dimensional stochastic differential equations; Bernstein-von Mises theorem; asymptotic properties; Bayes estimation