Math Problems Vault

Problem 17.1

Show that Proposition 17.2 and Corollary 17.3 remain valid for local martingales with continuous paths (cf. Definition 16.7).

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Problem 17.2

Extend Theorem 17.1 to local martingales with continuous paths.

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Problem 17.3

For $M, N \in \mathcal{M}_{T, \text{ loc }}^{c}$ we define the covariation by $\langle M, N\rangle_{t}: = \frac{1}{4}\left(\left\langle M+N\right\rangle_{t} - \left\langle M - N\right\rangle_{t}\right)$ . Show that

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(a)

$t \mapsto \langle M, N\rangle_{t}$ is well - defined, of bounded variation on compact $t$ - intervals (BV - process).

(b)

$\langle M, N\rangle$ is the unique BV - process such that $M N - \langle M, N\rangle \in \mathcal{M}_{T, \text{ loc }}^{c}$ .

(c)

$\langle M, N\rangle_{t} =$ ucp $- \lim_{\left|\Pi \right| \downarrow 0} \sum_{\Pi }\left(M_{t_{i} \wedge t} - M_{t_{i - 1} \wedge t}\right)\left(N_{t_{i} \wedge t} - N_{t_{i - 1} \wedge t}\right)$ .

(d)

$(M, N) \mapsto \langle M, N\rangle$ is bilinear.

(e)

$\left|\langle M, N\rangle \right| \leqslant \sqrt{\left\langle M\right\rangle } \sqrt{\left\langle N\right\rangle }$ .

Problem 17.4

(a)(b)(c)

Use Lemma 17.14 to show the following assertions:

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(a)

$\lim_{n} \mathbb {E}\left[\sup_{t \leqslant T}\left|X_{t}^{n} - X_{t}\right|^{2}\right] = 0 \Longrightarrow X^{n} \xrightarrow [n \rightarrow \infty ]{\text{ ucp }} X$ .

(b)

$X^{n} \xrightarrow [n \rightarrow \infty ]{\text{ ucp }} X, \quad Y^{n} \xrightarrow [n \rightarrow \infty ]{\text{ ucp }} Y \Longrightarrow X^{n}+Y^{n} \xrightarrow [n \rightarrow \infty ]{\text{ ucp }} X+Y$ .

(c)

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

Problem 17.5

(a)(b)(c)

Let $M \in \mathcal{M}_{T, \text{ loc }}^{c}$ . Show that the following maps are continuous, if we use ucpconvergence in both range and domain:

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(a)

$\mathcal{L}_{T, \text{ loc }}^{2}(M) \ni f \mapsto f^{2} \cdot \left\langle M\right\rangle$ .

(b)

$\mathcal{L}_{T, \text{ loc }}^{2}(M) \ni f \mapsto f \bullet M \in \mathcal{M}_{T, \text{ loc }}^{c}$ .

(c)

$\mathcal{M}_{T, \text{ loc }}^{c} \ni M \mapsto \left\langle M\right\rangle$ .

Problem 17.6

(a)(b)

Let $M \in \mathcal{M}_{T, \text{ loc }}^{c}$ and $f(t, \omega )$ be an adapted càdlàg (right continuous, finite left limits) process and $t \in [0, T]$ .

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(a)

Show that $f \in \mathcal{L}_{T}^{0}(M)$ , i.e. $\int_{0}^{T} f(t, \omega ) d\left\langle M\right\rangle_{t} < \infty$ a.s.

(b)

Show that for any sequence of partitions $\Pi$ of $[0, T]$ with $\left|\Pi \right| \downarrow 0$ we have $f^{\Pi } \bullet M \rightarrow f \bullet M$ in ucp - sense. Here

f^{\Pi }(t, \omega ): = \sum _{\Pi } f\left(t_{i - 1}, \omega \right) \; \mathbb {1}_{\left[t_{i - 1}, t_{i}\right)}(t) .

Problem 17.7

Let $M \in \mathcal{M}_{T, \text{ loc }}^{c}$ and $f_{n}$ be a sequence in $\mathcal{L}_{T}^{0}(M)$ . If $f_{n} \rightarrow f$ in ucp - sense, then $f \in \mathcal{L}_{T}^{0}(M)$ .

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Problem 17.8

Show that the process $f^{2} \bullet \left\langle M\right\rangle_{t}: = \int_{0}^{t}\left|f(s)\right|^{2} d\left\langle M\right\rangle_{s}$ appearing in Theorem 17.9.b) is adapted.

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