We’re interested in the end in functions where we want to estimate some parameters of features in based on observations in . The application in the end will be multivariate financial data of high dimension but before we consider this relatively complex problem we consider and a function that is one-dimensional, and we consider (**) which has the nice property (see this) . Now even powers of have the expectation which can be used to evaluate (*) using the Ito formula with and then using the explicit expressions for the Hermite polynomials. We can thus tabulate the expectations (*) as functions of . These explicit functions of can then be used to evaluate the coefficients of (**) using observations of some observed univariate time series, thereby giving us parameters for the function which can be considered the functional. This is a solution to the problem of determining of some twice-differentiable function of in the one-dimensional case.

Now for the multidimensional case let $\Phi:\mathbf{R}^d\rightarrow\mathbf{R}^N$ be . When then the above approach is sufficient for each component .

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