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Question 11 Let H be a real number such that 0 0} is called a fractional Brownian motion with Hurst parameter H if (i) X(0) = 0. (ii) X(t) is normally distributed with mean ( and variance (2H. (ini) {X(t) : t > 0} has stationary increments. (a) Prove that for a fractional Brownian motion {X (t) : t > 0}, the auto- covariance function is given by Kx (t, s) = E[X(t) X (s)] = (12H + $24 - It - $/24). (b) Prove that for H = 1/2, a fractional Brownian motion also has indepen dent increments and hence it is a Brownian motion, but for H # 1/2, a fractional Brownian motion does not have independent increments