2.1 and $(\tilde{M}^{(1)})_{t \geq 0}$ is a non-negative $\tilde{P}$-martingale of mean $1$ with respect to the filtration $(\mathcal{G}_t)_{t \geq 0}$. The effect of $\tilde{M}^{(1)}$ if used as a change of measure martingale is to put some drift on $(\xi_t)_{t \geq 0}$.

For this particular paper we shall take $\tilde{M}^{(1)}$ to be either $$\tilde{M}^{(1)}_t = \mathrm{e}^{\lambda \xi_t - \frac{\lambda^2}{2}t} \text{ , } \qquad t \geq 0$$ or $$\begin{aligned}
\tilde{M}^{(1)}_t &= \mathrm{e}^{\lambda |\xi_t| - \lambda \tilde{L}_t - \frac{\lambda^2}{2}t}\\
&= \mathrm{e}^{\lambda \int_0^t sgn(\xi_s) \mathrm{d}\xi_s - \frac{\lambda^2}{2}t}
\text{ , } \qquad t \geq 0.\end{aligned}$$ The first choice is the classical Girsanov martingle which has the effect of adding constant drift $\lambda$ to $(\xi_t)_{t \geq 0}$. The second choice has the effect of adding instataneous drift $\lambda sgn(\cdot)$ to $(\xi_t)_{t \geq 0}$ so that if $\lambda < 0$ then this is drift of magnitude $|\lambda|$ towards the origin whereas if $\lambda > 0$ then this is drift of magnitude $\lambda$ away from the origin.

Note that we can decompose $(\tilde{M})_{t \geq 0}$ into a product of three martingales: $$\tilde{M}_t = \tilde{M}^{(1)}_t \tilde{M}^{(2)}_t \tilde{M}^{(3)}_t$$ where $$\label{m2}
\tilde{M}^{(2)}_t = m^{n_t'} \mathrm{e}^{- \hat{\beta}t} \times 
m_0^{n_t^0} \mathrm{e}^{- \hat{\beta}_0 \tilde{L}_t}$$ and $$\label{m3}
\tilde{M}^{(3)}_t = \prod_{n=1}^{n_t'} \frac{A_{n}'}{m} \times \prod_{n=1}^{n_t^0} \frac{A_{n}^0}{m_0}.$$ When 