Transformation formulas for fractional Brownian motion

We derive a Molchan–Golosov-type integral transform which changes fractional Brownian motion of arbitrary Hurst index KK into fractional Brownian motion of index HH. Integration is carried out over [0,t][0,t], t>0t>0. The formula is derived in the time domain. Based on this transform, we construct a prelimit which converges in L2(P)L2(P)-sense to an analogous, already known Mandelbrot–Van Ness-type integral transform, where integration is over (−∞,t](−∞,t], t>0t>0.