  Using elementary arguments, we show how to derive $\mathbf{L}_p$-error bounds
for the approximation of frictionless wealth process in markets with
proportional transaction costs. For utilities with bounded risk aversion, these
estimates yield lower bounds for the frictional value function, which pave the
way for its asymptotic analysis using stability results for viscosity
solutions. Using tools from Malliavin calculus, we also derive simple
sufficient conditions for the regularity of frictionless optimal trading
strategies, the second main ingredient for the asymptotic analysis of small
transaction costs.
