  In this work, we study the optimal discretization error of stochastic
integrals, in the context of the hedging error in a multidimensional It\^{o}
model when the discrete rebalancing dates are stopping times. We investigate
the convergence, in an almost sure sense, of the renormalized quadratic
variation of the hedging error, for which we exhibit an asymptotic lower bound
for a large class of stopping time strategies. Moreover, we make explicit a
strategy which asymptotically attains this lower bound a.s. Remarkably, the
results hold under great generality on the payoff and the model. Our analysis
relies on new results enabling us to control a.s. processes, stochastic
integrals and related increments.
