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Consider as an example the Stochastic Differential Equation $$ \text{d}Y(t) = -\kappa Y(t) \text{d}t + \sigma \text{d} B(t), \qquad t \geq 0 $$

where $B(t)$ is a standard Brownian motion, $\kappa$ and $\sigma$ are positive constants. If $Y(0) = 0$, Girsanov's theorem tells that the distribution of the process $Y(t)$ is absolutely continuous w.r.t. that of a standard Brownian. If $\sigma$ is known, this theorem defines a likelihood function for an (ideal) continuously observed path $t \in [0, \, T]$ with $T >0$. A maximum likelihood estimator for $\kappa$ can be consequently be derived as described by Phillips and Yu.

Now assume that $Y(t)$ is stationary, which implies that $Y(0)$ is centred normal with variance $\sigma^2/(2 \kappa)$. Can we still then use Girsanov's theorem to define a likelihood function? If yes, what relations exist between this infill or continuous record likelihood and the simply defined likelihood function arising from partial observations $Y(t_i)$ when the instants $t_i$ tend to fill the fixed interval $[0,\,T]$?

More generally how can we cope with random initial conditions for a continuously observed process having a state-space representation? As a major difference with the example above, the initial state will no longer be observed.

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