  An explicit martingale representation for random variables described as a
functional of a Levy process will be given. The Clark-Ocone theorem shows that
integrands appeared in a martingale representation are given by conditional
expectations of Malliavin derivatives. Our goal is to extend it to random
variables which are not Malliavin differentiable. To this end, we make use of
Ito's formula, instead of Malliavin calculus. As an application to mathematical
finance, we shall give an explicit representation of locally risk-minimizing
strategy of digital options for exponential Levy models. Since the payoff of
digital options is described by an indicator function, we also discuss the
Malliavin differentiability of indicator functions with respect to Levy
processes.
