Fractional Brownian Motion Integrals and Riemann-Sum Limits
Summary
The note asks whether an integral against fractional Brownian motion can be defined as a limit of ordinary Riemann sums when the Hurst parameter is above one half. It explains that several definitions of stochastic integration exist in this setting, and that they do not share all the same properties. One cited construction uses Riemann sums but may not have zero expectation, motivating alternatives such as a Wick-product integral and the Gaussian divergence, or Skorohod, integral.
It also describes a pathwise approach based on fractional derivatives, with conditions on the derivative order and the integrand’s function space. For sufficiently regular paths, specifically when the integrand is Hölder continuous with exponent greater than one minus the Hurst parameter, the integral can be interpreted as a Riemann–Stieltjes integral. The answer is a compact overview rather than a proof; the appropriate definition depends on the desired properties and assumptions about the integrand.
Key ideas
- A Riemann-sum limit is one possible definition of integration against fractional Brownian motion, but its properties require care.
- For Hurst parameters above one half, a cited Riemann-sum construction may not have zero mean.
- Wick-product and divergence-based definitions provide alternative notions of stochastic integration.
- A pathwise Riemann–Stieltjes interpretation is available under a Hölder regularity condition on the integrand.
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Full text
# approximating fBm stochastic integral
# approximating fBm stochastic integral
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?
$$\int_a^b f(t)dB_H(t) := \lim_{\Delta t_k \rightarrow0} \sum_k f(t_k)[(B_H(t_{k+1})-B_H(t_k)]$$
## Answer by user16651 (score 1, accepted)
https://quant.stackexchange.com/a/26327
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.