# Infill likelihood for a continuously observed continuous-time process

Consider a continuous-time stochastic process $$y(t)$$ having the following linear (Gaussian) state-space representation for $$t \geq 0$$

$$\left\{ \begin{array}{c c l} \text{d}{\boldsymbol{\alpha}}(t) &=& \mathbf{A} \boldsymbol{\alpha}(t)\, \text{d}t + \text{d}\boldsymbol{\zeta}(t)\\ y(t) &=& \alpha_1(t) \end{array} \right.$$

where $$\boldsymbol{\zeta}(t)$$ is $$p$$-dimensional Wiener process process and $$\mathbf{A}$$ is a $$p \times p$$ transition matrix. Assume that the process $$\boldsymbol{\alpha}(t)$$ is stationary, so the initial state $$\boldsymbol{\alpha}(0)$$ and $$y(0)$$ are assumed to be drawn from their respective stationary distribution. Assume as well that the state-space representation is observable. Examples of this framework include Continuous-time Auto-Regressive (CAR) processes such as the famous Ornstein–Uhlenbeck.

Consider the process on a bounded time interval, say $$t \in [0,\, 1]$$. A well-known context is that of partial observations: the process $$y(t)$$ is observed at times $$t_i$$ with $$t_1 < \dots < t_n$$. The likelihood is easily defined and it can be computed in $$O(n)$$ operations using Kalman Filtering. Now

1. How can we define the infill likelihood a.k.a. the continous record likelihood, corresponding to the (ideal) case where a complete continuous path $$y(t)$$ would be observed on a time interval $$[0, \, 1]$$?

2. What relation would then exist between the infill likelihood and the limit of the likelihood for the partial observations when the observation times $$t_k$$ fill the interval $$[0, \, 1]$$? The filling condition can be expressed as: $$\max_k \{t_{k} - t_{k-1}\}$$ tends to zero with $$t_0:=0$$ and $$t_{n+1}:=1$$.

Concerning 1. I know that an infill log-likelihood can be obtained by using Girsanov's theorem, as explained by Brockwell, Davis and Yang or by Phillips and Yu. However, this likelihood seems to be conditional on the initial value $$y(0)$$, while $$y(0)$$ is assumed here to be random with its distribution depending on the model parameters. As a side-effect, the defined infill-likelihood does not preserve the time-reversibility which arises from the assumptions here. Intuition suggests that a contribution accounting for the initial state should be found in the log-likelihood.

• The term "infill likelihood" is used by some authors in an rather informal acceptation to evocate the likelihood of an infinity of observations such as $\{y(t):\, 0 \leq t \leq 1\}$. Since nothing exists like a Lebesgue measure in an infinite dimensional space, the set of observations does not have a "density". The likelihood is defined by using the Radon-Nikodym derivative w.r.t. a Gaussian measure which does not depend on the parameters e.g. the distribution of a standard Wiener process. – Yves Nov 28 '18 at 14:11
• When applying Girsanov's theorem as P&Y do on p. 5-6, it is assumed that the distribution of the process $y(t)$ is absolutely continuous w.r.t. that of the standard Wiener $B(t)$. But this implies that the probability of $\{y(0) \neq 0\}$ is zero because the distribution of $y(0)$ is absolutely continuous w.r.t. that of $B(0)$ which is Dirac. For the Ornstein-Ulhenbeck example, no use is made of the stationary distributionw which is centred normal with s.d. $\sigma_0 / \sqrt{2 \kappa}$ (on top of p. 6 $\kappa$ seems to be rather $-\kappa$ with $\kappa > 0$). – Yves Dec 4 '18 at 13:54