On the sequential estimation of the trend parameter for a diffusion-type process with quadratic and non-quadratic loss functions

We consider the problem of sequential estimation of parameter $\theta$ corresponding to the process $\xi=\{\xi_t,\ t\ge 0\}$ with a stochastic differential
$$ d\xi_t=[\theta A_1(t,\xi)+A_0(t,\xi)]dt+B(t,\xi)dw_t,\qquad \xi_0=0. $$

Theorem. If the conditions 1)–4) of this paper are fulfilled, then the sequential estimation procedure $D_H=D(\tau_H,\delta_H)$, $0<H<\infty$, where $H$ is a given constant,
\begin{gather*} \tau_H(\xi)=\inf\biggl\{t:\int_0^tA_1^2(s,\xi)B^{-2}(s,\xi)\,ds=H\biggr\},\\ \delta_H(\xi)=H^{-1}\int_0^{\tau_H(\xi)}B^{-2}(t,\xi)A_1(t,\xi)[d\xi_t-A_0(t,\xi)dt], \end{gather*}
in the class of $\mathscr D_H$-unbiased sequential estimation procedures satisfying the conditions
\begin{gather*} \mathbf P\biggl\{\int_0^{\tau} A_1^2(t,\xi)B^{-2}(t,\xi)\,dt<\infty \biggr\}= \mathbf P\biggl\{\int_0^{\tau} A_1^2(t,w)B^{-2}(t,w)\,dt<\infty \biggr\}=1,\\ \mathbf E|\delta(\xi)|^{\alpha}<\infty,\qquad \mathbf E\int_0^\tau A_1^2(t,\xi)B^{-2}(t,\xi)\,dt\le H, \end{gather*}
is optimal in the following sense:
$$ \mathbf E|\delta_H(\xi)-\theta|^{\alpha}\le\mathbf E|\delta(\xi)-\theta|^{\alpha},\qquad \alpha\ge 1. $$

In the case of nonlinear relationship between the trend parameter and parameter $\theta$ the sequential estimation procedure $D_H=D(\tau_H,\delta_H)$ is asymptotically optimal when $H\to\infty$.