Conditions of presence and absence of arbitrage for a model of \((B,S)\)-market defined by fractional Brownian motion (Q2755274)

The authors investigate properties of the \((B,S)\)-market with the random price process of a stock \((S_t,(F_t)_{t\geq 0},P)\): NEWLINE\[NEWLINES_t=\exp\{X_t\}:=\exp\left\{\int_0^t\nu(s) ds+ \int_0^t\mu(s) dB_s^H\right\}NEWLINE\]NEWLINE and with the random price process of bonds \((B_t,(F_t)_{t\geq 0},P)\): NEWLINE\[NEWLINEB_t=e^{rt}, \qquad r\geq 0,\;t\geq 0,NEWLINE\]NEWLINE where \(\nu(s)\) and \(\mu(s)\) are nonrandom measurable functions, \(B_s^H\) is a fractional Brownian motion with \(H\in(1/2,1)\). The problem of presence and absence of arbitrage for three types of \((B,S)\)-market is investigated. In the first case the \((B,S)\)-market is defined by the fractional stock. In this case the absence of the equivalent martingale measure is proved. An example is proposed where the arbitrage is possible. In the second case the \((B,S)\)-market is defined by the modified fractional stock [\textit{I. Norros, E. Valkeila} and \textit{J. Virtamo}, Bernoulli 5, No. 4, 571-587 (1999; Zbl 0955.60034)]. In this case the absence of arbitrage is proved. In the third case the \((B,S)\)-market is defined by a ``homogeneous'' kernel. In this case the absence of arbitrage is also proved.