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Suppose I have the following stochastic integral:

$$\int_a^b f(t)dB_H(t)$$

with the term $dB_H(t)$ a fractional brownian motion with associated $H$ parameter.

Is it true that for $H \in (1/2,1)$, we have the following result?

$$\int_a^b f(t)dB_H(t) := \lim_{\Delta t_k \rightarrow0} \sum_k f(t_k)[(B_H(t_{k+1})-B_H(t_k)]$$

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  • $\begingroup$ Do you look for an approximation or isn't this the definition? If it were an Ito integral the limit would hold in L2 (quadratic) sense. $\endgroup$ – Ric May 2 '16 at 14:37
  • $\begingroup$ @Richard, Itô's integral $I_t = \int_a^b f(t) dM_t$ is generally defined for $M_t$ being a (semi-)martingale (and $f(t)$ a square-integrable, adapted process) (I should rather say that semi-martingales form the largest class of processes for which the Itô integral can be defined). Yet, the fractional Borwnian motion $B_H(t)$ is only martingale for $H=1/2$, hence for a Hurst coefficient $H \in (1/2,1)$ usual Itô stochastic calculus may not be applied. $\endgroup$ – Quantuple May 2 '16 at 15:23
  • $\begingroup$ Yes .. I know that only (!) for H=1/2 it is Ito. But the OP posted the question as question of approximation ... please tell me: how is it defined? Isn't it defined in a similar - but rigorous sense? $\endgroup$ – Ric May 2 '16 at 15:26
  • $\begingroup$ I'm afraid that we are at the same point then: I don't know exactly and I prefer not saying anything wrong. $\endgroup$ – Quantuple May 2 '16 at 16:04
  • $\begingroup$ Hi. Thanks for the replies. To answer Richard's question: I want to know if it is "correct" to approximate the integral in this fashion. (I know it's okay for H= 1/2. Perhaps, being more specific, I should say that I have already estimated H and that it is bigger than 1/2, but smaller than 1) $\endgroup$ – baluch_stan May 3 '16 at 13:01
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Your approach is correct. But unfortunately it is not applicable. In the case i.e. $\frac{1}{2}<H<1$, Dai and Heyde have defined a stochastic integral as limit of Riemann sums.Their approach does not satisfy the property $E[\int_{0}^{t}f(s)dB_{H}s]=0$ (Why?!!). For this reason, Duncan have introduced a new stochastic integral with zero mean which is the limit of Riemann sums defined by means of the Wick product. Nowadays, Since the fBm is a Gaussian process, one can apply the stochastic calculus of and introduce the stochastic integral as the divergence operator,that is, the adjoint of the derivative operator variations.Using the notions of fractional integral and derivative, Zhahle has introduced a path-wise stochastic integral. The integral of a process $f(s)$ on a time interval $[0, T]$ is defined as $$\int_{0}^{T}f(s)dB_{H}(s)=(-1)^{\beta}\int_{0}^{T} D_{0^{t}}^{\beta}D_{T-}^{1-\beta}B_{H}(s)f(s)ds+f(0+)B_{H}(T)$$ where $D^{\beta}$ denotes the fractional derivative of order $\beta$, and assuming $\beta>1−H$ and $f$ belongs to the space $I_{0+}^{\beta}(L^1([0,T])$.

I should mention when process $f$ has $\lambda$-Holder continuous paths with $\lambda>1 − H$, then this integral can be interpreted as a Riemann_Stieltjes integral.

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