Mixed fractional Brownian motion
Let \(B\) be the standard Brownian motion and \(B^H\) fractional Brownian motion with Hurst index \(H\in (0,1]\). If the Brownian motion \(B\) and the fractional Brownian motion \(B^H\) are independent and \(\alpha\in\mathbb{R} \setminus \{0\}\), define the mixed fractional Brownian motion \(M^{H,\alpha}\) by \(M^{H,\alpha} \doteq B+\alpha B^H\). If \(X\) is a stochastic process, denote by \(\mathbb{F}^X\) the filtration generated by \(X\). If \(f=\sum^{n-1}_{j=0} f_jI_{(t_j, t_{j+1}]}\) is a simple \(\mathbb{F}^X\)-predictable process on \((0,1]\): \(|f|\leq 1\), \(f_j\in F_{t_j}^X\) and \(0\leq t_0<t_1 <\cdots< t_n\leq 1\). If for every simple predictable process the random variables \(\sum^{n-1}_{j=0} f_j(X_{t_{j+1}}-X_{t_j})\) are bounded, the process \(X\) is a weak semimartingale. Two continuous processes \(X,Y\) are equivalent, if the measures \(\mathbb{P}_X\) and \(\mathbb{P}_Y\) are equivalent on \(C[0,1]\). The main results of this paper are: If \(H\in(0,1) \setminus \{\frac 12\}\), then the fractional Brownian motion \(B^H\) is not a weak semimartingale. If \(H\in(0,\frac 12) \cup(\frac 12,\frac 34]\), then the mixed fractional Brownian motion \(M^{H,\alpha}\) is not a weak semimartingale. If \(H\in (\frac 34,1]\), then the mixed fractional Brownian motion \(M^{H,\alpha}\) is equivalent to standard Brownian motion \(B\). The paper ends with some applications to pricing models in finance.
scientific article; zbMATH DE number 1786702