Math Problems Vault

Problem 16.1

Let $T < \infty$ . Do we have $\mathcal{L}_{T}^{0} = \mathcal{L}_{T, \text{ loc }}^{2}$ ?

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Problem 16.2

Let $\tau$ be a stopping time. Show that $\; \mathbb {1}_{[0, \tau )}$ is $\mathscr {P}$ measurable.

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Problem 16.3

If $\sigma_{n}$ is localizing for a local martingale, so is $\tau_{n}: = \sigma_{n} \wedge n$ .

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Remark.

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Consequence: this yields the maximal inequality $\mathbb {E}\left[\sup_{s\leqslant T}\left|M(s\wedge \tau_{n})\right|^{2}\right] \overset {\text{Doob}}{\leqslant } 4\mathbb {E}\left[\left|M(\tau_{n})\right|^{2}\right] \leqslant 4\mathbb {E}\left[\left|M(n)\right|^{2}\right]$ .

Problem 16.4

The factor $\; \mathbb {1}_{\left\{ \sigma_{n} > 0\right\} }$ is used in the definition of a local martingale, in order to avoid integrability requirements for $M_{0}$ . More precisely, if $\left(\sigma_{n}\right)_{n \geqslant 1}$ is a localizing sequence, then we have

$M_{0} \in L^{1}(\mathbb {P}) \& \left(M_{t}\right)_{t \geqslant 0}$ is a local martingale $\Longleftrightarrow \left(M_{\sigma_{n} \wedge t}\right)_{t \geqslant 0}$ is a martingale.

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Problem 16.5

Let $\left(I_{t}, \mathscr {F}_{t}\right)_{t \geqslant 0}$ be a continuous adapted process with values in $[0, \infty )$ and a.s. increasing sample paths. Set for $u \geqslant 0$

\sigma _{u}(\omega ): = \inf \left\{ t \geqslant 0: I_{t}(\omega ) > u\right\} \quad \text{ and } \quad \tau _{u}(\omega ): = \inf \left\{ t \geqslant 0: I_{t}(\omega ) \geqslant u\right\}

Show that

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(a)

$\sigma_{u} \geqslant t \Longleftrightarrow I_{t} \leqslant u$ and $\tau_{u} > t \Longleftrightarrow I_{t} < u$ .

(b)

$\tau_{u}$ is an $\mathscr {F}_{t}$ stopping time and $\sigma_{u}$ is an $\mathscr {F}_{t+}$ stopping time.

(c)

$u \mapsto \sigma_{u}$ is right continuous and $u \mapsto \tau_{u}$ is left continuous.

(d)

$\tau_{u} = \sigma_{u - } = \lim_{\epsilon \downarrow 0} \sigma_{u - \epsilon }$ .

(e)

$u \mapsto \tau_{u}, u \mapsto \sigma_{u}$ are continuous, if $t \mapsto I_{t}$ is strictly increasing.