Hypothesis testing for diffusion processes with continuous observations: Direct computation of large deviation results for error probabilities

We propose statistical tests for deciding between two alternatives for diffusion processes observed continuously over a finite time interval. Our tests emphasize the large deviation aspects, or equivalently, the asymptotic behavior of probabilities of type I and type II errors and the rate at which these probabilities go to zero as the observation time increases. We obtain these rates using direct methods of calculation. We provide specific computational examples for diffusion processes commonly used in finance and show that the error probabilities for these cases go to zero exponentially fast. Applications in finance and economics are discussed.