On an estimator of the unknown drift coefficient parameter for an equation perturbed by Gaussian noise. (English. Russian original) Zbl 0746.60060

Bondarev, B. V.

Theory Probab. Math. Stat. 42, 1-12 (1991); translation from Teor. Veroyatn. Mat. Stat., Kiev 42, 3-13 (1990).

The author gives exponential inequalities for the probability of deviation estimator \[ \theta_ T = {\sum_{k=1}^ \infty \lambda_ k^{-1} \int_ 0^ T \varphi_ k(t) dx (t)\int_ 0^ T f\bigl( t,x(t)\bigr)\varphi_ k(t) dt\over \sum_{k=1}^ \infty \lambda_ k^{-1}\left( \int_ 0^ T f\bigl( t,x(t)\bigr) \varphi_ k(t) dt\right) ^ 2} \] for the unknown parameter \(\theta_ 0\) in the equation in \(\mathbb{R}\), \[ x'(t)=\theta_ 0 f\bigl( t,x(t)\bigr) + \xi'(t),\quad t>0;\qquad x(0)=\xi (0)=0, \] where \(f\) is a known nonrandom function, \(\theta_ 0\) is a parameter to be estimated from the realization \(\{ x(t): 0\leq t\leq T \}\), and \(\xi'\) is a Gaussian process with zero mean and known correlation function with eigenfunctions \(\{ \varphi_ k\}\) and eigenvalues \(\{ \lambda_ k\}\).

Reviewer: A.Ya.Dorogovtsev (Kiev)

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
62F25	Parametric tolerance and confidence regions
62M09	Non-Markovian processes: estimation

Keywords:

estimation of the drift coefficient; exponential inequalities; correlation function; eigenfunctions