Girsanov's theorem and likelihood for random initial conditions
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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 theo...