The Ornstein-Uhlenbeck (O-U) model has been successfully applied to describe the response accuracy and response time in 2-alternative choice tasks. This paper analyses properties and performance of variants of the O-U model with absorbing and reflecting boundary conditions that limit the range of possible values of the integration variable. The paper focuses on choice tasks with pre-determined response time. The type of boundary and the growth/decay parameter of the O-U model jointly determine how the choice is influenced by the sensory input presented at different times throughout the trial. It is shown that the O-U models with two types of boundary hold close relationship and can achieve the same performance under certain parameter values. The value of the growth/decay parameter that maximizes the accuracy of the model has been identified. It is shown that when the boundaries are introduced, the O-U model may achieve higher accuracy than the diffusion model. This suggests that given the limited range of the firing rates of integrator neurons, the neural decision circuits could achieve higher accuracy employing leaky rather than linear integration in certain tasks. We also propose experiments that could distinguish between different models of choice in tasks with pre-determined response time.