Locally most powerful sequential tests for stochastic processes

For a continuous time stochastic process with distribution P[theta] depending on a one-dimensional parameter [theta] the problem of sequentially testing [theta] = 0 against [theta] > 0 is treated. We assume that the process of likelihood ratios has a certain representation which allows to obtain identities of the Wald type for stopping times. These identities are then used to derive a result on locally most powerful tests for which a problem of optimal stopping is solved.