Fractional Brownian motion, random walks and binary market models
Using the kernel representation of the fractional Brownian motion \[ Z_{t}=\int_{0}^{t}z(t,s) dW_{s} \] with the deterministic kernel \[ z(t,s)= c_{H}(H-1/2)s^{1/2-H}\int_{s}^{t}u^{H-1/2}(u-s)^{H-3/2} du, \] where \(W_{t}\) is a standard Brownian motion, \(c_{H}\) is a normalizing constant, the author proves that the random walk \[ Z_{t}^{(n)}= \int_{0}^{t}z^{(n)}(t,s) dW^{(n)}_{s} \] converges weakly to the fractional Brownian motion \(Z_{t}\). Here \[ z^{(n)}(t,s)= n\int_{s-1/n}^{s}z([nt]/n,u) du, \quad W^{(n)}_{s}= n^{-1/2}\sum_{i=1}^{[nt]} \xi_{i}^{(n)}, \] \(\xi_{i}^{(n)}\) are i.i.d. random variables with \(E\xi_{i}^{(n)}=0\), \(D\xi_{i}^{(n)}=1\). On the base of this approximation a fractional binary market model is constructed and it is proved that this fractional binary market admits arbitrage opportunity.
scientific article; zbMATH DE number 1681755