We suppose $X$ solves our SDE $dX_{t}=-X_{t}dt+dW_{t}$ for $t\geq0$ with initial condition $X_{0}=0$ w.r.t to our measure $P$ on $(\Omega,\mathcal{F})$. $W_{t}$ is standard Wiener. This solution is given by \begin{align} X_{t}=e^{-t}X_{0}+\int_{0}^{t}e^{s-t}dW_{s}=\int_{0}^{t}e^{s-t}dW_{s} \end{align} Let $\mathcal{F}_{t}$ be the natural Filtration of our process $X_{t}$, i.e. $\mathcal{F}_{t}=\sigma(X_{s}:s\leq t)$. For $t>0$ we define a new prob. measures $\{Q_{\theta,t}:\theta\in\mathbb{R}\}$ on $\mathcal{F}_{t}$ by \begin{align} \frac{dQ_{\theta,t}}{dP^{t}}=exp[\theta X_{t}-\phi_{t}(\theta)]\,,\theta\in\mathbb{R} \end{align} where $P^{t}$ is the restriction of $P$ on the $\sigma$-field $\mathcal{F}_{t}$ and $\phi_{t}$ is the cumulant transform of $X_{t}$, i.e. $\phi_{t}(\theta):=\log(E[e^{\theta X_{t}}])$. So that we really define new prob. measures. $\phi_{t}(\theta)$ is given by \begin{align} \phi_{t}(\theta)=\frac{1}{4}\theta^{2}(1-e^{-2t}) \end{align} Question: Why doesn't there exist a class of probability measures $\{P_{\theta}:\theta\in\mathbb{R}\}$ such that $P_{\theta}^{t}=Q_{\theta,t}$? Where $P_{\theta}^{t}$ is again the restriction of $P_{\theta}$ to $\mathcal{F}_{t}$.
$\begingroup$
$\endgroup$
4
-
1$\begingroup$ I don't really see why would you care about all $\theta$ at the same time here. Does $P_\theta$ exist for all/any non-zero $\theta$? $\endgroup$– SBFCommented Apr 25, 2016 at 12:17
-
$\begingroup$ @Ilya Well of course this could be done by letting one $\theta$ fixed. Thats right. The set of $\theta$ here is the domain of the cumulant transform under $X_{t}$. It was directly transfered from a math-book. $\endgroup$– ziTCommented Apr 25, 2016 at 12:39
-
$\begingroup$ So what's the problem with existence of family, if its elements do exist? $\endgroup$– SBFCommented Apr 25, 2016 at 13:02
-
$\begingroup$ @Ilya Lets say we only watch one $\theta$. We can define prob. measures $Q_{\theta,t}$ on $\mathcal{F}_{t}$. The problem is to show, that there exists no prob. measure say $P_{\theta}$ on $\mathcal{F}_{\infty}$ such that, the measure is consistent in the sense that $P_{\theta}^{t}$ (restricted to $\mathcal{F}_{t}$ coincides with $Q_{\theta,t}$. $\endgroup$– ziTCommented Apr 25, 2016 at 13:24
Add a comment
|