Predicting integrals of diffusion processes

Consider predicting the integral of a diffusion process Z in a bounded interval A، نويسنده , , based on the observations Z(t1n)، نويسنده , , …، نويسنده , , Z(tnn)، نويسنده , , where t1n، نويسنده , , …، نويسنده , , tnn is a dense triangular array of points (the step of discretization tends to zero as n increases) in the bounded interval. The best linear predictor is generally not asymptotically optimal. Instead، نويسنده , , we predict using the conditional expectation of the integral of the diffusion process، نويسنده , , the optimal predictor in terms of minimizing the mean squared error، نويسنده , , given the observed values of the process. We obtain that، نويسنده , , conditioning on the observed values، نويسنده , , the order of convergence in probability to zero of the mean squared prediction error is Op(n?2). We prove that the standardized conditional prediction error is approximately Gaussian with mean zero and unit variance، نويسنده , , even though the underlying diffusion is generally non-Gaussian. Because the optimal predictor is hard to calculate exactly for most diffusions، نويسنده , , we present an easily computed approximation that is asymptotically optimal. This approximation is a function of the diffusion coefficient.، نويسنده ,

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