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By Thao N. X.

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Extra resources for A Basis Analog of the H -function of Several Variables

Example text

Recall the finite-interval representation by Norros and Valkeila (1999) that underlies this approximation: It is evident that the conditional expectation of the process based on information up to time t can also be represented as a finite-interval of classical Brownian motion. We just cut off the independent future part. More formally, we obtain T E BTH |Ft = E 0 z(T, s) dBs |Ft t = T z(T, s) dBs + E 0 z(T, s) dBs |Ft t t = 0 z(T, s) dBs . 2 Binomial Approximation of the Conditional Moments of FBM 41 that is, we use the same coefficients as for BTH but only summing up to time t.

So, despite the improvement of an unconditionally unbiased process, the game becomes unfair as soon as information about the past can be exploited. Consequently, one should also expect arbitrage within the Wick-product-based setup. Indeed, taking for example a look at Fig. 9, one recognizes both for the ordinary product and the Wick-product-based process one conspicuous peculiarity: Concerning the nodes before last, one can see that—starting from the top node—both descending branches show upward direction.

So, in order to evolve BtH , it is always necessary to factor the whole history—also beyond null—into the calculation, hence an independent symmetric modeling is inadequate. Dasgupta (1998) suggests a way out that is able to incorporate the influence of an infinite past. 2). Dasgupta (1998) proves that this process weakly converges to fractional Brownian motion as n gets larger. However, there is one crucial drawback within this approximation procedure. The effort of calculating appropriate n approximations is extremely high, as we obtain 22 +n+1 terminal values within the nth step.

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A Basis Analog of the H -function of Several Variables by Thao N. X.


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