Geometric versus arithmetic random walk the case of trended variables

The asymptotic behavior of the empirical means and variances for the geometric and arithmetic random walks are studied, when the underlying random walk is trended. Thus the effect of misspecifications can be described, and two tests are proposed. The first test uses a classical approach in model selection and is based on the comparison of estimated quasi likelihoods. The second one is obtained by estimating some nuisance parameters in a Neyman-Pearson test. Some bounds for the power functions are given, which suggest that the second test may be very powerful and better than the first one.