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1979)). Moreover, there is no problem in extending this construction to a two-sided Brownian motion: As the two sides are not at all correlated, starting from 0, one can symmetrically evolve both sides of the process and receives a process over the whole time line. 1), it is obvious that fractional Brownian motion also relates to an inﬁnite past. Unfortunately, it is not promising to carry over the above idea of a two-sided symmetric approximation if one wants to model fractional Brownian motion by a binomial tree.
5) where t Mt = and c1 = 2HΓ 0 1 1 c1 s 2 −H (t − s) 2 −H dBsH , 3 −H Γ 2 H+ 1 2 −1 . The process Mt is a martingale with independent increments, zero mean and variance function where EMt2 = c22 t2−2H , cH √ . c2 = 2H 2 − 2H It is called the fundamental martingale. 5) of the Radon–Nikodym derivative by dP a 1 = exp −aMt − a2 c22 t2−2H dP 2 . From this representation it is easy to see that for the case H = 12 we obtain the well-known change of measure formula. The generalization of this drift removal theorem from the simple fractional Brownian motion to fractional integrals can also be done (see Bender (2003b), p.
Instead, we will concern ourselves with the more intuitive Riemann sum approaches. With respect to the latter, the discrete time considerations in the next chapter will deliver deeper insight. Chapter 3 Fractional Binomial Trees Binomial trees are discrete approximations of stochastic processes where at every discrete point in time the process has two possibilities: it either moves upwards or descends to a certain extent. Each alternative occurs with a certain probability adding up to 1. Consequently, two factors determine the characteristics of the resulting discrete process: The probability distributions of the single steps as well as the extent of the two possible shifts at each step.