In Mandelbrot(1968)'s paper, the fractional brownian motion, denoted by BH(t,ω)B_{H}(t,\omega),(t>0) is defined by BH(0,ω)=b0B_{H}(0,\omega)=b_{0} BH(t,ω)−BH(0,ω)=1Γ(H+12){∫−∞0[(t−s)H−1/2−(−s)H−1/2]dB(s,ω)+∫0t(t−s)H−1/2dB(s,ω)}B_{H}(t,\omega)-B_{H}(0,\omega)=\frac{1}{\Gamma(H+\frac{1}{2})}\{\int^{0}_{-\infty}[(t-s)^{H-1/2}-(-s)^{H-1/2}]dB(s,\omega)+\int^{t}_{0}(t-s)^{H-1/2}dB(s,\omega)\} I have difficulty understanding fractional brownian motion by self study.Is there an intuitive interpretation of this definition? Why time s can have negative value? Thanks!