Use Lemma 17.14 to show the following assertions:

a) \(\lim _{n} \mathbb{E}\left(\sup _{t \leqslant T}\left|X_{t}^{n}-X_{t}ight|^{2}ight)=0 \Longrightarrow X^{n} \xrightarrow[n ightarrow \infty]{\text { ucp }} X\).

b) \(X^{n} \xrightarrow[n ightarrow \infty]{\text { ucp }} X, \quad Y^{n} \xrightarrow[n ightarrow \infty]{\text { ucp }} Y \Longrightarrow X^{n}+Y^{n} \xrightarrow[n ightarrow \infty]{\text { ucp }} X+Y\).

c) \(\mathcal{E}_{T}\) is ucp-dense in \(\mathcal{L}_{T, \text { loc }}^{2}(M)\) for any \(M \in \mathcal{M}_{T, \text { loc }}^{c}\).

Data From Leema 17.14

Transcribed Image Text:

17.4 Lemma. (N,F), NN. is a continuous martingale, and E(N) < M-M (MEN-MEA) + 2N (MIN-MEN) =[(w 16x 16x*. (17.2) (17.3) (174) Proof. It is clear that tNs continuous. The martingale property follows from Theorem 15.11.a). For (17.2) we use Theorem 15.4: E [(N] = [(MM)(M (MT) The identity (17.3) follows from xE (M)] = xE[M - M]