On stochastic calculus related to financial assets without semimartingales

Rosanna Coviello, Cristina Di Girolami and Francesco Russo

july, 2011

Publication type:

Paper in peer-reviewed journals

Journal:

Bulletin Sciences Mathématiques, vol. 135, pp. 733774

DOI:

10.1016/j.bulsci.2011.06.008

HAL:

inria-00564756

arXiv:

1102.2050

Keywords :

$A$-martingale; weak $k$-order Brownian motion; no-semimartingale; utility maximization; insider; no-arbitrage; viability; hedging

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

BibTeX:

@article{Cov-DiG-Rus-2011,
    author={Rosanna Coviello and Cristina Di Girolami and Francesco Russo },
    title={On stochastic calculus related to financial assets without 
           semimartingales },
    doi={10.1016/j.bulsci.2011.06.008 },
    journal={Bulletin Sciences Mathématiques },
    year={2011 },
    month={7},
    volume={135 },
    pages={733774},
}