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.