I consider continuous time asset pricing models with stochastic differential utility incorporating decision makers' concern with ambiguity on true probability measure. In order to identify and estimate key parameters in the models, I use a novel econometric methodology developed recently by Park (2008) for the statistical inference on continuous time conditional mean models. The methodology only imposes the condition that the pricing error is a continuous martingale to achieve identification, and obtain consistent and asymptotically normal estimates of the unknown parameters. Under a representative agent setting, I empirically evaluate alternative preference specifications including a multiple-prior recursive utility. My empirical findings are summarized as follows: Relative risk aversion is estimated around 1.5-5.5 with ambiguity aversion and 6-14 without ambiguity aversion. Related, the estimated ambiguity aversion is both economically and statistically significant and including the ambiguity aversion clearly lowers relative risk aversion. The elasticity of intertemporal substitution (EIS) is higher than 1, around 1.3-22 with ambiguity aversion, and quite high without ambiguity aversion. The identification of EIS appears to be fairly weak, as observed by many previous authors, though other aspects of my empirical results seem quite robust. Next, I develop an approach to test for martingale in a continuous time framework. The approach yields various test statistics that are consistent against a wide class of nonmartingale semimartingales. A novel aspect of my approach is to use a time change defined by the inverse of the quadratic variation of a semimartingale, which is to be tested for the martingale hypothesis. With the time change, a continuous semimartingale reduces to Brownian motion if and only if it is a continuous martingale. This follows immediately from the celebrated theorem by Dambis, Dubins and Schwarz. For the test of martingale, I may therefore see if the given process becomes Brownian motion after the time change. I use several existing tests for multivariate normality to test whether the time changed process is indeed Brownian motion. I provide asymptotic theories for my test statistics, on the assumption that the sampling interval decreases, as well as the time horizon expands. The stationarity of the underlying process is not assumed, so that my results are applicable also to nonstationary processes. A Monte-Carlo study shows that our tests perform very well for a wide range of realistic alternatives and have superior power than other discrete time tests.