Solving Arbitrage Problem on the Financial Market Under the Mixed Fractional Brownian Motion With Hurst Parameter H ∈]1/2,3/4[

This study deals with the arbitrage problem on the financial market when the underlying asset follows a mixed fractional Brownian motion. We prove the existence and uniqueness theorem for the mixed geometric fractional Brownian motion equation. The semi-martingale approximation approach to mixed fractional Brownian motion is used to eliminate the arbitrage opportunities.

DISTRIBUTIONS OF QUADRATIC FUNCTIONALS OF THE FRACTIONAL BROWNIAN MOTION BASED ON A MARTINGALE APPROXIMATION

The present paper deals with the distributions related to the fractional Brownian motion (fBm). In particular, we try to compute the distributions of (ratios of) its quadratic functionals, not by simulations, but by numerically inverting the associated characteristic functions (c.f.s). Among them is the fractional unit root distribution. It turns out that the derivation of the c.f.s based on the standard approaches used for the ordinary Bm is inapplicable. Here the martingale approximation to the fBm suggested in the literature is used to compute an approximation to the distributions of such functionals. The associated c.f. is obtained via the Fredholm determinant. Comparison of the first two moments of the approximate with true distributions is made, and simulations are conducted to examine the performance of the approximation. We also find an interesting moment property of the approximate fractional unit root distribution, and a conjecture is given that the same property will hold for the true fractional unit root distribution.

LINEAR NONSTATIONARY MODELS—A REVIEW OF THE WORK OF PROFESSOR P.C.B. PHILLIPS

The work of Professor P.C.B. Phillips, even if it is focused on the area of linear nonstationary models, is enormous. So it is hard for me to explore the whole of his work in this paper. Therefore, I have decided to take up only a few results of his work. The topics chosen here are applications of the martingale approximation and the problem of choosing between stochastic and deterministic trends, which I discuss and, hopefully, extend.