  This paper does not suppose a priori that the evolution of the price of a
financial asset is a semimartingale. Since possible strategies of investors are
self-financing, previous prices are forced to be finite quadratic variation
processes. The non-arbitrage property is not excluded if the class ${\cal A}$
of admissible strategies is restricted. The classical notion of martingale is
replaced with the notion of ${\cal A}$-martingale. A calculus related to ${\cal
A}$-martingales with some examples is developed. Some applications to the
maximization of the utility of an insider are expanded.
