Article

On stochastic calculus related to financial assets without semimartingales

Bulletin Des Sciences Mathematiques - BULL SCI MATH 02/2011; 135(6). DOI: 10.1016/j.bulsci.2011.06.008
Source: arXiv

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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