Consider a filtered probability space $(\Omega, (\mathcal F_n), \mathcal F, \mathbb P)$, where $\Omega$ is the set of sequences with value in some $E \subseteq \mathbb R^d$, and $\mathcal F$ is the canonical filtration generated by the coordinate maps $(X_n: \Omega \to \mathbb R^d)$.

Given a non-negative martingale $M$ with expectation $1$, there exists a probability measure $\mathbb Q \ll \mathbb P $ on $(\Omega, \mathcal F)$ such that: $$M_n = \left.\frac {\mathrm d \mathbb Q}{\mathrm d \mathbb P}\right|_{\mathcal F_n}.$$ This is a consequence of Kolmogorov's extension theorem. Conversely, for any $\mathbb Q \ll \mathbb P $, the expression above defines a martingale in the original probability space.

Now, let's look at martingale transforms. All martingale transforms of $X$ are martingales, but apart from the very specific case of the binomial model, the converse is false.

First question: is there a characterization of the measures $\mathbb Q \ll \mathbb P$ for which $\left(\left.\frac {\mathrm d \mathbb Q}{\mathrm d \mathbb P}\right|_{\mathcal F_n}\right)$ is a martingale transform of $X$?

Another interesting property of these Radon Nikodym representations is that, given some non-negative martingale tranform $(X \cdot H)$ with expectation $1$, the associated probability measure $\mathbb Q$ satifies: $$\mathbb E^{\mathbb Q} (\log (X \cdot H)) \ge \mathbb E^{\mathbb Q} (\log (X \cdot K)),$$ for all processes $K$ for which $(X \cdot K)$ is non-negative with expectation $1$. The proof is easy, just write both martingale transforms as Radon Nikodym derivatives and apply Jensen's inequality.

Second question: suppose I take $\mathbb Q \ll \mathbb P $. Consider the process $H$ that maximizes $\mathbb E^{\mathbb Q} (\log (X \cdot H))$. Are there known bounds linking $\left(\left.\frac {\mathrm d \mathbb Q}{\mathrm d \mathbb P}\right|_{\mathcal F_n}\right)$ and $(X \cdot H)_n$?

Consider a filtered probability space $(\Omega, (\mathcal F_n), \mathcal F, \mathbb P)$, where $\Omega$ is the set of sequences with value in some $E \subseteq \mathbb R^d$, and $\mathcal F$ is the canonical filtration generated by the coordinate maps $(X_n: \Omega \to \mathbb R^d)$. We will assume that $X$ is a martingale.

Given a non-negative martingale $M$ with expectation $1$, there exists a probability measure $\mathbb Q$ on $(\Omega, \mathcal F)$ such that: $$M_n = \left.\frac {\mathrm d \mathbb Q}{\mathrm d \mathbb P}\right|_{\mathcal F_n}.$$ This is a consequence of Kolmogorov's extension theorem. Conversely, for any $\mathbb Q \ll \mathbb P $, the expression above defines a martingale in the original probability space.

Now, let's look at martingale transforms. All martingale transforms of $X$ are martingales, but apart from the very specific case of the binomial model, the converse is false.

First question: is there a characterization of the measures $\mathbb Q \ll \mathbb P$ for which $\left(\left.\frac {\mathrm d \mathbb Q}{\mathrm d \mathbb P}\right|_{\mathcal F_n}\right)$ is a martingale transform of $X$?

Another interesting property of these Radon Nikodym representations is that, given some non-negative martingale tranform $(X \cdot H)$ with expectation $1$, the associated probability measure $\mathbb Q$ satifies: $$\mathbb E^{\mathbb Q} (\log (X \cdot H)) \ge \mathbb E^{\mathbb Q} (\log (X \cdot K)),$$ for all processes $K$ for which $(X \cdot K)$ is non-negative with expectation $1$. The proof is easy, just write both martingale transforms as Radon Nikodym derivatives and apply Jensen's inequality.

Second question: suppose I take $\mathbb Q \ll \mathbb P $. Consider the process $H$ that maximizes $\mathbb E^{\mathbb Q} (\log (X \cdot H))$. Are there known bounds linking $\left(\left.\frac {\mathrm d \mathbb Q}{\mathrm d \mathbb P}\right|_{\mathcal F_n}\right)$ and $(X \cdot H)_n$?