On parameter estimation of stochastic delay differential equations with guaranteed accuracy by noisy observations

Uwe Küchler , Vjatscheslav A. Vasil’iev

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 27 pages

MSC 2000

34K50 Stochastic delay equations

60H10 Stochastic ordinary differential equations

62L10 Sequential analysis

62L12 Sequential estimation

Abstract
Let $(X(t), t\geq −1)$ and $(Y (t), t \geq 0)$ be stochastic processes satisfying \[ dX(t) = aX(t)dt + bX(t − 1)dt + dW(t) \] and \[dY (t) = X(t)dt + dV (t),\] respectively. Here $(W(t), t \geq 0)$ and $(V (t), t \geq 0)$ are independent standard Wiener processes and $\theta = (a, b)'$ is assumed to be an unknown parameter from some subset $\Theta$ of $R^2$. The aim here is to estimate the parameter $\theta$ based on continuous observation of $(Y (t), t \geq 0)$. Sequential estimation plans for $\theta$ with preassigned mean square accuracy $\epsilon$ are constructed using the so-called correlation method. The limit behaviour of the duration of the estimation procedure is studied if $\epsilon$ tends to zero.