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**Arbitrage-free models in markets with transaction costs.**
*(English)*
Zbl 1247.91173

Summary: In [Ann. Appl. Probab. 18, No. 2, 491–520 (2008; Zbl 1133.91422)], P. Guasoni, M. Rásonyi and W. Schachermayer studied financial markets which are subject to proportional transaction costs. The standard martingale framework of stochastic finance is not applicable in these markets, since the transaction costs force trading strategies to have bounded variation, while continuous-time martingale strategies have infinite transaction cost. The main question that arises out of [P. Guasoni et al., loc. cit.] is whether it is possible to give a convenient condition to guarantee that a trading strategy has no arbitrage. Such a condition was proposed and studied in [P. Guasoni, Ann. Appl. Probab. 12, No. 4, 1227–1246 (2002; Zbl 1016.60065); E. Bayraktar and H. Sayit, Quant. Finance 10, No. 10, 1109–1112 (2010; Zbl 1205.91178)], the so-called stickiness property, whereby an asset’s price is never certain to exit a ball within a predetermined finite time. In this paper, we define the multidimensional extension of the stickiness property, to handle arbitrage-free conditions for markets with multiple assets and proportional transaction costs. We show that this condition is sufficient for a multi-asset model to be free of arbitrage. We also show that d-dimensional fractional Brownian models are jointly sticky, and we establish a time-change result for joint stickiness.

### MSC:

91G10 | Portfolio theory |

91B25 | Asset pricing models (MSC2010) |

60G22 | Fractional processes, including fractional Brownian motion |