A utility based approach to information for stochastic differential equations

A Bayesian perspective is taken to quantify the amount of information learned from observing a stochastic process, Xt, on the interval [0, T] which satisfies the stochastic differential equation, dXt = S([theta], t, Xt)dt+[sigma](t, Xt)dBt. Information is defined as a change in expected utility when the experimenter is faced with the decision problem of reporting beliefs about the parameter of interest [theta]. For locally asymptotic mixed normal families we establish an asymptotic relationship between the Shannon information of the posterior and Fisher's information of the process. In particular we compute this measure for the linear case (S([theta], t, Xt) = [theta]S(t, Xt)), Brownian motion with drift, the Ornstein-Uhlenbeck process and the Bessel process.