We consider a process given by the SDE , , with initial condition , where , , is a standard Wiener process, and are continuously differentiable functions. Assuming , , with some , we derive an explicit formula for the joint Laplace transform of and for all and for all . Our motivation is that the maximum likelihood estimator (MLE) of α can be expressed in terms of these random variables. As an application, we show that in case of , , where denotes the Fisher information for α contained in the observation , is a standard Wiener process and denotes equality in distribution. We also prove asymptotic normality of the MLE of α as for , . As an example, for all and , we study the process given by the SDE , , with initial condition . In case of , this process is known as an α-Wiener bridge, and in case of , this is the usual Wiener bridge.