A linear mini-max estimator for the case of a quartic loss function

Let Y(t) be a stochastic process on [0,1] modeled as dY_t = θ(t)dt + dW(t), where W(t) denotes a standard Wiener process, and θ(t) is an unknown function assumed to belong to a given set O L ²[0,1]. We consider the problem of estimating the value L(θ), where L is a known continuous linear function defined on O, using linear estimators of the form (m,y) = m(t) dY(t)j m ε L [0,1]. We solve the problem for the case of a quartic loss function, and compare the solution of the quartic case with the solution for the case of a quadratic loss function.