Suppose that a probability space $(\Omega, \Sigma, \mathbb{P})$ is given. Let $W=\{W_n\}_{n\in \mathbb{N}_0}$ be a sequence of $\mathbb{P}$-i.i.d real-valued random variables on $\Omega$. Furthermore, assume that $\mathbb{Q}$ be a new probablity measure on $\Sigma$, and $W$ is $\mathbb{Q}$-i.i.d and that $\mathbb{Q}_{W_1} \sim \mathbb{P}_{W_1}$. We denote by $\mathcal{F}^{W} = \{\mathcal{F}_n^W\}_{n\in \mathbb{N}}$ the natural filtration of $W$. Then, one can say that for every $n\in \mathbb{N}_0$ and for all $D\in \mathcal{F}_n^W = \sigma(W_1, W_2, .., W_n)$ \begin{equation} \mathbb{Q}(D) = \mathbb{E}^{\mathbb{P}}\Big[I_D \prod_{j=1}^{n}\frac{d\mathbb{Q}_{W_1}}{d\mathbb{P}_{W_1}}(W_j)\Big] \end{equation} or equivalently, \begin{equation} \frac{d\mathbb{Q}}{d\mathbb{P}}\Big|_{\mathcal{F}_n^W} = \prod_{j=1}^{n}\frac{d\mathbb{Q}_{W_1}}{d\mathbb{P}_{W_1}}(W_j) \end{equation} Now, my question is what if $W_j's$ is not independent. In this case, how does the Radon-Nikodym derivative $\frac{d\mathbb{Q}}{d\mathbb{P}}\Big|_{\mathcal{F}_n^W}$ look like?



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Browse other questions tagged or ask your own question.