We describe and implement statistical tests arguing for the presence of a Brownian component in the inventories and wealth processes of individual traders. Using intra-day data from the Toronto Stock Exchange, we provide empirical evidence of this claim. Both for regularly spaced time intervals, as well as for asynchronously observed data, the tests reveal with high significance the presence of a non-zero Brownian motion component. Furthermore, we extend the theoretical analysis of an existing optimal execution model to accommodate the presence of Ito inventory processes, and we compare empirically the optimal behavior of traders in such fitted models, to the actual behavior read off the data.