## HW4 MA5161. Due February 24

Exercise. Let $(\Omega, (\mathcal{F}_t)_{t \ge 0},\mathcal{F},\mathbb{P})$ be a filtered probability space that satisfies the usual conditions. We denote

$\mathcal{F}_{\infty}=\sigma \left( \mathcal{F}_t , t \ge 0 \right)$

and for $t\ge 0$, $\mathbb{P}_{/\mathcal{F}_t}$ is the restriction of $\mathbb{P}$ to $\mathcal{F}_t$. Let $\mathbb{Q}$ be a probability measure on $\mathcal{F}_{\infty}$ such that for every $t \ge 0$,

$\mathbb{Q}_{/\mathcal{F}_t} \ll \mathbb{P}_{/\mathcal{F}_t}.$

• Show that there exists a right continuous and left limited martingale $(D_t)_{t \ge 0}$ such that for every $t \ge 0$,$D_t=\frac{d\mathbb{Q}_{/\mathcal{F}_t}}{d\mathbb{P}_{/\mathcal{F}_t}},\text{ }\mathbb{P}-a.s.$
• Show that the following properties are equivalent:
1) $\mathbb{Q}_{/\mathcal{F}_\infty} \ll \mathbb{P}_{/\mathcal{F}_\infty}$;
2) The martingale $(D_t)_{t \ge 0}$ is uniformly integrable;
3) $(D_t)_{t \ge 0}$ converges in $L^1$;
4) $(D_t)_{t \ge 0}$ almost surely converges to an integrable and $\mathcal{F}_\infty$ measurable random variable $D$ such that $D_t =\mathbb{E}(D\mid \mathcal{F}_t), \quad t \ge 0.$

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