On stochastic calculus related to financial assets without semimartingales

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 A of admissible strategies is restricted. The classical notion of martingale is replaced with the notion of A-martingale. A calculus related to 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.