# Girsanov's theorem and likelihood for random initial conditions

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.