Quasi-likelihood estimation in a mixed fractional Black-Scholes model

Let $B^H=\{B_t^H, t \geq 0\}$ be a fractional Brownian motion with Hurst index $0<H<1$, and let $B=\{B_t, t \geq 0\}$ be an independent Brownian motion. In this study, we investigate the parameter estimation of a mixed fractional Black-Scholes model $ S_t^H=S_0^H+\mu\displaystyle \int_0^tS_s^H{\mathrm{d}}s+\sigma\displaystyle \int_{0}^{t}S_s^H{\mathrm{d}}(B_s+B_s^H)$,where $\sigma>0$, $\mu\in {\mathbb R}$ are two unknown parameters. Using quasi-likelihood estimation, when the system is observed at some discrete time instants $\{t_i=ih,i=0,1,2,\ldots,n\}$, we give estimations of the parameters $\mu$ and $\sigma$ provided $h=h(n)\to 0$, $nh\to \infty$ and $h^{1+\gamma}n\to 1$ for some $\gamma>0$, as $n\to \infty$. We present the asymptotic normality of the estimators based on the velocity of $nh^{1+\gamma}-1$ tending to zero as $n$ tends to infinity. Finally, we perform numerical calculus and simulations using factual data from the stock market to verify the effectiveness of the established estimators.