  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
$\mathcal{A}$ of admissible strategies is restricted. The classical notion of
martingale is replaced with the notion of $\mathcal{A}$-martingale. A calculus
related to $\mathcal{A}$-martingales with some examples is developed. Some
applications to no-arbitrage, viability, hedging and the maximization of the
utility of an insider are expanded. We finally revisit some no arbitrage
conditions of Bender-Sottinen-Valkeila type.
