Densities for Stochastic Processes

Abstract

Let $\{x_\theta(t), \theta \varepsilon \Omega\}$ be a family of stochastic processes defined by their finite dimensional distributions; that is, $\{F_\theta\lbrack x(t_1), \cdots, x(t_n)\rbrack; \theta \varepsilon \Omega\}$ is given for all finite sets of time points $t_1, \cdots, t_n$. A general procedure for treating a statistical problem concerning this family has been to solve the problem for the finite dimensional families and then see what happens to the solution when limits are taken over suitably selected sets of time points. For example, if $\hat\theta\lbrack x(t_1), \cdots, x(t_n)\rbrack$ is an estimate of $\theta$ based on the finite dimensional family and it can be shown that the limit $$\hat\theta\lbrack x(t_1), \cdots, x(t_n)\rbrack \rightarrow \hat\theta\lbrack x(t)\rbrack$$ exists in some sense and is independent of the defining set $(t_1, t_2, \cdots)$, then this limit will usually provide an adequate estimate of $\theta$ for the process. Frequently the properties of the estimates $\hat\theta\lbrack x(t_1), \cdots, x(t_n)\rbrack$ can be extended to $\hat\theta\lbrack x(t)\rbrack$. An alternative approach to the problem is proposed by Grenander [1]. He introduces the likelihood ratio of two processes $P$ and $Q$ restricted to a finite number of time points, shows that it converges to a limit as the number of points goes to infinity and that this limit is the density of $P$ with respect to $Q$ if this density exists. He uses these results to derive numerous statistical results. The only criterion which he gives for the existence of the density is that the limit of the likelihood ratio be finite a.s. $P$. In applying this criterion he must always make use of some additional knowledge of the processes such as a.s. existence of certain integrals. In Section 2 these results are established very simply using the theory of martingales, and a criterion for the existence of the density is given which proves convenient in several applications. A condition also is given under which a density computed for a countable number of time points is valid for the continuous parameter process. Once the existence of the density is established standard statistical techniques can be applied directly. For example, sufficient statistics can often be found by inspection, or maximum likelihood methods can be used. Densities for a normal process with continuous covariance and unknown mean value function are derived in Section 3. Minimum variance unbiased estimates of regression coefficients are obtained. In [2] Cameron and Martin consider processes which are obtained from a Wiener process by linear transformations. They state conditions under which such a process is absolutely continuous with respect to the Wiener process, and they give a formula for computing the density. These results can be applied to the Ornstein Uhlenbeck process with covariance $\sigma^2e^{-\beta|s-t|}$ to obtain a family of densities for $2\beta\sigma^2 =$ constant. In Section 4 this family of densities is derived using the methods of Section 2. Using the same techniques it is shown that this family of Ornstein Uhlenbeck processes is mutually absolutely continuous. The maximum likelihood estimate of the correlation parameter $\beta$ is computed.