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Various approaches to stochastic processes exist, noting that key propertiessuch as measurability and continuity are not trivially satisfied. We introducea new theory for Gaussian processes using improper linear functionals. Using acollection of i.i.d. standard normal variables, we define Gaussian white noiseand discuss its properties. This is extended to general Gaussian processes onHilbert space, where the variance is allowed to be any suitable operator. Ourmain focus is $L^2$ spaces, and we discuss criteria for Gaussian processes tobe continuous in this setting. Finally, we outline a framework for statisticalinference using the presented theory with focus on the special case of$L^2[0,1]$. We introduce the Fredholm determinant into the functionallog-likelihood. We demonstrate that the naive functional log-likelihood is notconsistent with the multivariate likelihood. A correction term is introduced,and we prove an asymptotical result.

Stochastic modelling of Gaussian processes by improper linear functionals