Htykuut

0 $begingroup$ 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

2 $begingroup$ Direct limit of $mathbb {C^*}$ behaves well with quotients. The following solution is from mathoverflow: Direct limits do behave well with respect to quotients. Suppose $A$ is the direct limit of a sequence $(A_n)$ with connecting $*$ -homomorphisms $phi_n: A_n to A_{n+1}$ , and let $I$ be a closed ideal of $A$ . Then $I$ pulls back to an ideal $I_n$ of $A_n$ for each $n$ , and the connecting maps $phi_n$ are compatible with the quotients, i.e., they lift to connecting maps $tilde{phi}_n: A_n/I_n to A_{n+1}/I_{n+1}$ . Moreover, $A/I$ is then the direct limit of the sequence $(A_n/I_n)$ . This is easy because the maps $tilde{phi}_n$ have no kernel and hence are isometric, and the whole sequence isometrically embeds in $A/I$ . I understand the whole argument whatever is written