A Market Driven By Weighted Fractional Brownian Motion And Related Analysis

In this article,we consider a market driven by the Weighted fractional Brownian motion(weighted-fBm)and discuss the European option pricing.As some related questions we study some stochastic analysis of weighted fracional Brownian.We first study some basic properties of weighted-fBm.The so-called weighted-fBm Ba,b with parameters a and b(a>-1,|b|<1,|b|<a+1)is a mean zero Gaussian process with the covariance E[Bta,bBsa,b]=1/2B(a+1,b+1)?0s^tua((t-u)b+(s-u)b+(s-u)b)du where B(·,·)is the beta function.Clearly,if a = 0,the process coincides with the standard fractional Brownian motion with Hurst parameter H =b+1/2 We show that some sample properties of weighted-fBm Ba,b such as local nondeterminism,the law of the iterated logarithm and intersection local time.Next,for-1<b<0,a<1-b we introduce the generalized quadratic covariation[f(Ba,b),Ba,b](w)defined by provided the limit exists in probability,where x(?)f(x)is a measurable function.Moreover,we give a generalized Ito formula as follows F(Ba,b)=F(0)+?0tf(Bsa,b)dBsa,b+1+a+b/2(1+b)(ta[f(Ba,b),B(a,b)]t(W)-a?0tsa-1[f(Ba,b),B(a,b)s(W)ds)for-1<6<0,a<1-b,where F ? C1(R)and F' = f satisfies a suitable condition,and under the same condition,we introduce the integral with respect to local time La,b(x,t)and show that the following generalized Bouleau-Yor identity:Finally,we consider the weighted fractional Black-Scholes formula,and obtained the European option pricing formula.