Locally most powerful sequential tests of a simple hypothesis vs one-sided alternatives

Let $X_1,X_2,...$ be a discrete-time stochastic process with a distribution $P_\theta$, $\theta\in\Theta$, where $\Theta$ is an open subset of the real line. We consider the problem of testing a simple hypothesis $H_0:$ $\theta=\theta_0$ versus a composite alternative $H_1:$ $\theta>\theta_0$, where $\theta_0\in\Theta$ is some fixed point. The main goal of this article is to characterize the structure of locally most powerful sequential tests in this problem. For any sequential test $(\psi,\phi)$ with a (randomized) stopping rule $\psi$ and a (randomized) decision rule $\phi$ let $\alpha(\psi,\phi)$ be the type I error probability, $\dot \beta_0(\psi,\phi)$ the derivative, at $\theta=\theta_0$, of the power function, and $\mathscr N(\psi)$ an average sample number of the test $(\psi,\phi)$. Then we are concerned with the problem of maximizing $\dot \beta_0(\psi,\phi)$ in the class of all sequential tests such that $\alpha(\psi,\phi)\leq \alpha\quad{and}\quad \mathscr N(\psi)\leq \mathscr N,$ where $\alpha\in[0,1]$ and $\mathscr N\geq 1$ are some restrictions. It is supposed that $\mathscr N(\psi)$ is calculated under some fixed (not necessarily coinciding with one of $P_\theta$) distribution of the process $X_1,X_2...$. The structure of optimal sequential tests is characterized.