On hedging European options in geometric fractional Brownian motion market model

Fractional stochastic differential equation with discontinuous diffusion

“…Towards this end, we observe that, for α < 1/2, (3.6) implies that Λ −1 is a convex function. Then, Remark 3.5 in [1] (see also Theorem 2.1 in [5]) yields…”

“…Consider 1) in case that this limit is well-defined, where we use the convention f r = 0 on [a, b] c . In this case D α a+ f is called the left-fractional derivative of f of order α.…”

Integral representation of random variables with respect to Gaussian processes

“…Later on Mishura et al [7] considered the same problem where standard Brownian motion W was replaced with fractional Brownian motion (fBm) B H with Hurst index H > 1 2 . In this case the authors considered generalised Lebesgue-Stieltjes integrals with respect to fBm which can be defined, thanks to results of Azmoodeh et al [1], for integrands of form f (B H u ) where f is a function of locally bounded variation. As an application of the results in [7], the authors considered financial implications of the results and gave a neg- It is interesting to note that while the stochastic integrals are defined in different ways, the results for standard Brownian motion and fBm are quite similar.…”

Integral representation with respect to fractional Brownian motion under a log-Hölder assumption

“…is well defined (see the Appendix). In particular, it is possible to take f with D α 1], and for such integrands, the definition agrees with the definition of the fractional integral given in [10]; see Remark on p. 340.…”