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The Property of Mixed Fractional Brownian Motion and the Application in Finance

Author: YuZheng
Tutor: ZuoLiTan
School: Donghua University
Course: Applied Mathematics
Keywords: Fractional Brownian motion Mixed fractional Brownian motion LOCAL TIME It (o ^ ) Formula INTERSECTION LOCAL Collision local time of European options
CLC: F830
Type: Master's thesis
Year: 2009
Downloads: 217
Quote: 1
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Abstract


Assuming the B H < / sup> = { the B t H }, t ≥ 0} Hurst exponent H ∈ (0,1) the fractional Brownian motion , B H is a center Gauss process makes E [ the B t H ] = 0 , t ≥ 0 and covariance in this article some of our research and mixed fractional Brownian motion . So-called mixed fractional Brownian motion is a linear combination of independent fractional Brownian motion ( see literature [ 15,72 ]) , here we only consider a Hurst exponent H ∈ (0,1) fractional Brownian motion B H < / sup> a linear combination with an independent Brownian motion B : wherein a, b ??∈ R. Points: First of all, the conditions ( ? ) ≤ H lt ; 1 next we consider empowering local mixed fractional Brownian motion M H (a, b) ( ? ) Where f is a certain function. Secondary Association worse by the use of this integral f (M H ) and M H [f (M H ), M H ] as portrayed : and thus absolutely continuous function we have established the following generalized It (?) formula where the integral is a Wick -It ( ? ) type stochastic integral promotion of the fractional Brownian movement of It (?) formula . In fact, the same results as fractional Brownian motion is not established , due to the fractional Brownian motion of the secondary Association variation equal to zero. Secondly , we study the the selfing local d ≥ 2 - the the peacekeeping mixed fractional Brownian motion M H (a , b ) as well as two independent 1 - dimensional mixture fraction of Brownian Motion M H < sub> 1 (a 1 , b 1 ) with M - H 2 < / sup > we prove (a 2 , b 2 ) collision local two random variables the L T < / sub> and l T < / sub> L 2 exist , and they are smooth ( in the case of Meyer-Watanabe ) . Finally, for the ( ? ) Lt; H lt; 1 we study an application with mixed fractional Brownian motion , consider a financial market driven by mixed fractional Brownian motion where the integral ( ? ) Is a Wick -It ( ? ) type stochastic integral , we get a mixed fraction of the risk-neutral pricing formula .

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CLC: > Economic > Fiscal, monetary > Finance, banking > Finance, banking theory
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