On stochastic calculus related to financial assets without semimartingales

Abstract : 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.
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Rosanna Coviello, Cristina Di Girolami, Francesco Russo. On stochastic calculus related to financial assets without semimartingales. Bulletin des Sciences Mathématiques, Elsevier, 2011, 135, pp.733-774. ⟨10.1016/j.bulsci.2011.06.008⟩. ⟨inria-00564756⟩

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