Given a filtered space $(\Omega, F,\mathcal{F}_{t})$ with rightcontinous filtration. We have a class of probability measures $P=\{P_{\theta}:\theta \in \Theta\}$ definied on the filtered space.

We assume there exists a $\sigma$-finite measure $\mu$ on $(\Omega,\mathcal{F})$, which locally dominates $P$:

\begin{align*} P_{\theta}^{t}<<\mu^{t} \end{align*}

where $P_{\theta}^{t},\mu^{t}$ is the restriction of our measures $P_{\theta},\mu$ to $\mathcal{F}_{t}$. Then we assume the Radon-Nikodym-Derivative of $P_{\theta}^{t}$ w.r.t $\mu^{t}$ are of the form \begin{align*} \frac{dP_{\theta}^{t}}{d\mu^{t}}=e^{<\gamma_{t}(\theta),B_{t}>+\phi_{t}(\theta)}\quad \theta \in \Theta,t\geq 0 \end{align*} Here $\gamma_{t}(\theta)$ is a $m$-dimensional real valued function of $\theta$ and $t$ which is RCLL (cadlag) function with respect to $t$.

For $\phi_{t}(\theta)$ we have the same, but just 1-dimensional. Both non-random.

$B_{t}$ is a $m$-dimensional real valued $\mathcal{F}_{t}$-adapted RCLL (cadlag) random-process.

This respresentation of the Radon-Nikodm-Derivative is not unique: for example take $\tilde{\gamma_{t}}=\frac{1}{2}\gamma_{t}$ and $\tilde{B_{t}}=2B_{t}$ as another form of representation.

Now it is stated: It is always possible to find a respresentation of the latter form, such that $B$ is a semimartingale. Why?

Additional stuff kept in mind: one can show, that $\dot{\gamma}_{t}(\theta)^{T}B_{t}-\dot{\phi}_{t}(\theta)$ is a $P_{\theta}$ square integrable martingale with cadlag paths. Thus its a semimartingale. Could this help me, for getting a semimartingale $\tilde{B}$?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.