Distance between the fractional Brownian motion and the space of adapted Gaussian martingales

We consider the distance between the fractional Brownian motion defined on the interval [0, 1] and the space of Gaussian martingales adapted to the same filtration. As the distance between stochastic processes, we take the maximum over [0, 1] of mean-square deviances between the values of the processes. The aim is to calculate the function a in the Gaussian martingale representation integral(t)(0) a(s) dW(s) that minimizes this distance. So, we have the minimax problem that is solved by the methods of convex analysis. Since the minimizing function a cannot be either presented analytically or calculated explicitly, we perform discretization of the problem and evaluate the discretized version of the function a numerically.