Math Problems Vault

Problem 3.5.1

Let a filtration $\mathbb {F}=\left(\mathcal{F}_{n}, n \in \mathbb {N}\right)$ be given, and let $\tau_{1}, \tau_{2}, \ldots$ be Markov moments with respect to $\mathbb {F}$ . Prove that the random variables

\sum _{k=1}^{m} \tau _{k}, \quad \prod _{k=1}^{m} \tau _{k}, \quad \sup _{k} \tau _{k}, \quad \inf _{k} \tau _{k}

are also Markov moments with respect to $\mathbb {F}$ .

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Problem 3.5.2

Let a filtration $\mathbb {F}=\left(\mathcal{F}_{t}, t \geq 0\right)$ be given, and let $\tau_{1}, \tau_{2}, \ldots$ be Markov moments with respect to $\mathbb {F}$ . Prove that

\sum _{k=1}^{m} \tau _{k}, \quad \max _{k=1, \ldots , m} \tau _{k}, \min _{k=1, \ldots , m} \tau _{k}

are also Markov moments with respect to $\mathbb {F}$ .

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Problem 3.5.3

(a)(b)

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(a)

Let $\tau$ be a Markov moment with respect to a filtration $\mathbb {F}=\left(\mathcal{F}_{t}, t \in T\right)$ , where $T \subset \mathbb {R}_+$ . Prove that $\tau$ is an $\mathcal{F}_{\tau }$ -measurable random variable.

(b)

Let $\tau$ be a Markov moment with respect to a filtration $\mathbb {F}=\left(\mathcal{F}_{n}, n \in \mathbb {N}\right)$ , and let the random process $\left(X_{n}, n \in \mathbb {N}\right)$ be consistent with $\mathbb {F}$ . Prove that $X_{\tau }$ is $\mathcal{F}_{\tau }$ -measurable (where we set $X_{\tau } = +\infty$ if $\tau = +\infty$ ).

Problem 3.5.4

Let a filtration $\mathbb {F}=\left(\mathcal{F}_{t}, t \in T\right)$ , $T \subset \mathbb {R}$ , be given on $(\Omega , \mathcal{F}, \mathbb {P})$ . Prove that for Markov moments $\sigma , \tau$ with respect to $\mathbb {F}$ and for any $A \in \mathcal{F}_{\tau }$ ,

A \cap \left\{ \tau \leq \sigma \right\} \in \mathcal{F}_{\tau } \cap \mathcal{F}_{\sigma }.

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Remark.

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As a corollary, we obtain that if $\tau \leq \sigma$ , then $\mathcal{F}_{\tau } \subset \mathcal{F}_{\sigma }$ .

Problem 3.5.5

Let $\left(W_{t}, t \geq 0\right)$ be a Wiener process, and let $\tau_{x}=\inf \left\{ t \geq 0: W_{t}=x\right\}$ be the first time the level $x$ is reached. Compute the density of $\tau_{x}$ and find $\mathbb {E}\left[\tau_{x}\right]$ .

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Problem 3.5.6

Let $\left(W_t, t \geq 0\right)$ be a Wiener process. Find the density of the random variable

Y_{a}=\sup _{t \in [\tau _x, \tau _x+a]} W_{t}.

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Problem 3.5.7

Let $\left(W_{t}, t \geq 0\right)$ be a Wiener process. Prove that for any $0 \leq a<b \leq c<d$ , with probability 1 we have

\max _{s \in [a, b]} W_{s} \neq \max _{s \in [c, d]} W_{s}.

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Remark.

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Thus, the quantity $T_{[a,b]} = \operatorname *{arg\, max}_{s \in [a, b]} W_{s}$ is well defined.

Problem 3.5.8

Prove that the random variable $T=T_{[0,1]}\operatorname *{arg\, max}_{s \in [0, 1]} W_{s}$ has the arcsine distribution. Namely, prove that for $t \in (0,1)$ the distribution function of $T$ is:

F_T(t) = \mathbb {P}\left(T \leq t\right) = \frac{2}{\pi } \arcsin \sqrt{t}

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Problem 3.5.9

Let $\left(W_{t}, t \geqslant 0\right)$ be a Wiener process, and let $s < u < 0$ . Find

\mathbb {P}\left(W_t \text{ has no zeros on the interval }[s, u]\right).

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