Examples of a martingale with respect to two different probabilities:

Let \(W\) be a \(\mathbb{P}\)-BM, and set \(\left.d \mathbb{Q}\right|_{\mathcal{F}_{t}}=\left.L_{t} d \mathbb{P}\right|_{\mathcal{F}_{t}}\) where \(L_{t}=\exp \left(\lambda W_{t}-\frac{1}{2} \lambda^{2} t\right)\). Prove that the process \(X\), where

\[X_{t}=W_{t}-\int_{0}^{t} \frac{W_{s}}{s} d s\]

is a Brownian motion with respect to its natural filtration under both \(\mathbb{P}\) and \(\mathbb{Q}\).

(a) Under \(\mathbb{P}\), for any \(t,\left(X_{u}, u \leq t\right)\) is independent of \(W_{t}\) and is a Brownian motion.

(b) Replacing \(W_{u}\) by \(\left(W_{u}+\lambda u\right)\) in the definition of \(X\) does not change the value of \(X\).