Part a: Consider a Wiener process, $W_t$ and denote by ${\mathscr{F}_t}_{(t \geq 0)}$ the natural filtration generated by W. Let $\mathbb{R}_{+} = \{x : x \geq 0\}$ and $\mathscr{B}$ be a sigma algebra of Borel subsets of $\mathbb{R}_{+}$. Consider now an augmented filtration, ${\tilde{\mathscr{F}_t}} :=\mathscr{F}_t \vee \mathscr{B}$. Let $\xi \geq 0$ be a random variable. Consider a process $X_t = W_t + \xi$. Prove that $(X_t,{\tilde{\mathscr{F}_t}})$ is a martingale.
If we assume that John Dawkins assumption about the meaning of this augmented filtration is true, then this part of the proof is trivial.
Part b: Let $\tau$ be the first hitting time of $X_t$ to the linear boundary $b(t) = 2t$. Find the distribution of $\xi$ such that $$P(X_{\tau} < t) = 1 - (1 + \lambda t + (\lambda t)^2/2) + e^{-\lambda t}$$
My remarks: I need help understanding what ${\tilde{\mathscr{F}_t}} :=\mathscr{F}_t \vee \mathscr{B}$ even means. I usually denote the $\vee$ symbol as the maximum. It does not make sense to me to take the maximum of two filtrations. Can someone please clarify this? How do I start part b? I am looking for a hint.
Guess: $\xi$ is independent of $(W_t)_{t\ge 0}$, and $\tilde{\mathscr F}_t$ should be $\sigma(\xi)\vee\mathscr F_t$