Degradation data are an important source of products' reliability information. Though stochastic degradation models have been widely used for fitting degradation data, there is a lack of efficient and accurate methods to get their confidence intervals, especially in small sample cases. In this paper, based on the Wiener process, a doubly accelerated degradation test model is proposed, in which both the drift and diffusion parameters are affected by the stress level. The point estimates of model parameters are derived by constructing a regression model. Furthermore, based on the point estimates of model parameters, the interval estimation procedures are developed for the proposed model by constructing generalized pivotal quantities. First, the generalized confidence intervals of model parameters are developed. Second, based on the generalized pivotal quantities of model parameters, using the substitution method the generalized confidence intervals for some interesting quantities, such as the degradation rate ${\mu }_0$, the diffusion parameter ${\sigma }_0^2$, the reliability function $R(t_0)$ and the mean lifetime $E(T)$, are obtained. In addition, the generalized prediction intervals for degradation amount $X_0(t)$ and remaining useful life at the normal use stress level are also developed. Extensive simulations are conducted to investigate the performances of the proposed generalized confidence intervals in terms of coverage percentage and average interval length. Finally, a real data set is given to illustrate the proposed model.