Bias of estimates of secondary parameters in linear-boundary sequential tests

The context is that of a sequential trial based on Brownian motion with
linear stopping boundaries, possibly truncated. Along with the monitoring
process, a secondary Gaussian process with constant mean is observed; the
mean is to be estimated once the monitoring process reaches a boundary.
We provide a formula for the conditional bias, conditioning on the final
position of the monitoring process; this formula can then be integrated
to obtain an overall bias. Special attention is given to evaluating bias -
mathematically and by Monte Carlo - of the Kaplan-Meier estimator of
one of the survival functions (and similarly for the Nelson-Aalen estimator of the corresponding cumulative hazard function) upon completion of
a survival-analysis-based.