On stochastic calculus related to financial assets without semimartingales

Bulletin des Sciences Mathématiques (Impact Factor: 1.19). 02/2011; 135(6). DOI: 10.1016/j.bulsci.2011.06.008
Source: arXiv


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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Available from: Francesco Russo, Feb 23, 2014
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    • "[43] and [44] Russo and Vallois initiated a theory of stochastic integration via regularization procedures. In later years this was further developed by them and several other authors (see [21] [42] [34] [48] [18] [6] [20], and also the lecture notes [46] and its references). The regularization procedure is connected to the celebrated forward and backward integrals which can be used to integrate with respect to more general processes than semimartingales. "
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    • "There are also other no-arbitrage results in related setups, see Bender et al. (2008), Bender (2011) and Coviello et al. (2011). The concept of wealth functionals is needed when defining allowed strategies. "
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