Approximation of a stochastic integral with respect to fractional Brownian motion by integrals with respect to absolutely continuous processes. (Ukrainian, English) Zbl 1118.60049

Androshchuk, T. O.

Teor. Jmovirn. Mat. Stat. 73, 17-26 (2005); translation in Theory Probab. Math. Stat., Vol 73, 19-29 (2006).

The author deals with the absolutely continuous process
\[ B_{t}^{H,\alpha}=\int_0^ts^{H-1/2}\,dY_s^{\alpha}=C_0\left(H-\frac{1}{2}\right) \int_0^t s^{H-1/2}\left[ \int_0^{\alpha s}(s-u)^{H-3/2} u^{1/2-H}\,dW_u \right]ds \]
converging in the mean square sense to a fractional Brownian motion
\[ B_{t}^{H}=\int_0^ts^{H-1/2}\,dY_s, \quad Y_t=C_0\int_0^t(t-s)^{H-1/2} s^{1/2-H}\,dW_s, \] with the Hurst parameter \(H\in(1/2,1)\). Sufficient conditions are proposed under which the integral with respect to the process \(B_{t}^{H,\alpha}\) converges as \(\alpha\to1\) to the corresponding integral with respect to the fractional Brownian motion \(B_{t}^{H}\).

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

60H05	Stochastic integrals
60G15	Gaussian processes

Keywords:

fractional Brownian motion; stochastic integral; convergence of integrals