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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