Math Problems Vault

Problem 9.1

Let $Z_{i}$ have a $N\left(0, \sigma_{i}^{2}\right)$ distribution, for $i = 1,2, \ldots , n$ . Prove that

\mathbb {E}\left[\max _{i}\left|Z_{i}\right|\right] \leq \sqrt{\mathbb {E}\left[\max _{i}\left|Z_{i}\right|^{2}\right]} \leq 2 \sigma \sqrt{1+\log n} \quad \text{ where } \sigma : = \max _{i} \sigma _{i}

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Remark.

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For Problem [1] we do not need to assume that the $\left\{ Z_{i}\right\}$ have a multivariate normal distribution. Nowhere in the argument do we need to know anything about the joint distribution. When the $Z_{i}$ are independent, or nearly so, the bound is quite good. For the extreme case of $n$ independent $N\left(0, \sigma^{2}\right)$ variables, the inequality from part (i) of Problem [2] shows that $\mathbb {E}\left[\max_{i \leq n}\left|Z_{i}\right|\right] \geq C_{0} \sigma \sqrt{1+\log n}$ for a universal constant $C_{0}$ . See Section 12.2 for a precise way-Sudakov's minorationof capturing the idea of approximate independence when the $\left\{ Z_{i}\right\}$ have a multivariate normal distribution. When the variables are highly dependent, the bound is not good: in the extreme cases where $Z_{i} \equiv Z_{1}$ , which has a $N\left(0, \sigma^{2}\right)$ distribution, $\mathbb {E}\left[\max_{i \leq n}\left|Z_{i}\right|\right] = \sigma \mathbb {E}\left[\left|N(0,1)\right|\right]$ , which does not increase with $n$ .

Problem 9.2

(i)(ii)(iii)

Let $Z_{1}, Z_{2}, \ldots$ , be independent random variables, each distributed $N(0,1)$ . Define $M_{n}: = \max_{i \leq n}\left|Z_{i}\right|$ and $\ell_{n}: = \sqrt{2 \log n}$ and $n(k): = 2^{k}$ . Use the tail bound from Appendix D,

(2 \pi )^{-1 / 2}\left(\frac{1}{x}-\frac{1}{x^{3}}\right) \exp \left(-\frac{x^{2}}{2}\right) \leq \mathbb {P}\left(Z_{1} > x\right) \leq \frac{1}{2} \exp \left(-\frac{x^{2}}{2}\right) \quad \text{ for } x > 0

to prove that $M_{n} / \sqrt{2 \log n} \rightarrow 1$ almost surely. Argue as follows.

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(i)

For each small $\epsilon > 0$ , show that there exist strictly positive constants $C$ and $\theta$ for which $\mathbb {P}\left(M_{n} \leq (1-\epsilon ) \ell_{n}\right) = \left(1-2 \mathbb {P}\left(Z_{1} > (1-\epsilon ) \ell_{n}\right)\right)^{n} \leq \exp \left(-C n^{\theta }\right)$ , for all $n$ large enough. Deduce that $\liminf_{n} M_{n} / \ell_{n} \geq 1$ almost surely.

(ii)

For each $\epsilon > 0$ , show that $\mathbb {P}\left(M_{n} \geq (1+\epsilon ) \ell_{n}\right) \leq n / n^{(1+\epsilon )^{2}}$ . Deduce that $\lim \sup_{k} M_{n(k)} / \ell_{n(k)} \leq 1$ almost surely.

(iii)

Use the inequality $M_{n} / \ell_{n} \leq M_{n(k+1)} / \ell_{n(k)}$ for $n(k) < n \leq n(k+1)$ , and the result from parts (i) and (ii), to conclude that $M_{n} / \ell_{n} \rightarrow 1$ almost surely.

Problem 9.3

Let $\left\{ \left(B_{t}, \mathcal{F}_{t}\right): 0 \leq t \leq 1\right\}$ be a centered Brownian motion whose sample paths need not be continuous. Show that there exists a centered Brownian motion $\left\{ B_{t}^{*}: 0 \leq t \leq 1\right\}$ with continuous sample paths, for which $\mathbb {P}\left(B_{t}^{*} \neq B_{t}\right) = 0$ for every $t$ . Argue as follows. Let $S_{n}: = \left\{ i / 2^{n}: i = 0,1, \ldots , 2^{n}\right\}$ and $S: = \cup_{n \in \mathbb {N}} S_{n}$ . For each $\omega$ define

D_{n}(\omega ) = \max _{0 \leq i < 2^{n}}\left|B\left(\frac{i+1}{2^{n}}, \omega \right)-B\left(\frac{i}{2^{n}}, \omega \right)\right|

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(i)

Use Problem [1] to prove that $\sum_{n} \mathbb {E}\left[D_{n}\right] < \infty$ . Deduce that there exists a negligible set $N$ such that $\epsilon_{n}(\omega ): = \sum_{i \geq n} D_{i}(\omega ) \rightarrow 0$ as $n \rightarrow \infty$ for $\omega \notin N$ .

(ii)

Let $s$ and $t$ be points in $S_{n}$ for which $\left|s-t\right| \leq 2^{-m}$ , where $n > m$ . For each $k$ , write $s_{k}$ for the largest value in $S_{k}$ for which $s_{k} \leq s$ , and define $t_{k}$ similarly. (Thus $s = s_{n}$ and $t = t_{n}$ .) Show that

\left|B(s)-B(t)\right| \leq \sum _{i = m}^{n-1}\left|B\left(s_{i+1}\right)-B\left(s_{i}\right)\right|+\left|B\left(s_{m}\right)-B\left(t_{m}\right)\right|+\sum _{i = m}^{n-1}\left|B\left(t_{i+1}\right)-B\left(t_{i}\right)\right| .

(iii)

Deduce that

\max \left\{ \left|B(s, \omega )-B(t, \omega )\right|: s, t \in S_{n} \text{ and }\left|s-t\right| \leq 2^{-m}\right\} \leq 2 \epsilon _{m}(\omega ),

and hence

\sup \left\{ \left|B(s, \omega )-B(t, \omega )\right|: s, t \in S \text{ and }\left|s-t\right| \leq 2^{-m}\right\} \leq 2 \epsilon _{m}(\omega ) .

(iv)

For each $t$ in $[0,1]$ and each $\omega \in N^{c}$ , define

B^{*}(t, \omega ) = \lim _{n \rightarrow \infty } \sup \left\{ B_{s}: s \in S \text{ and } t-2^{-n} \leq s \leq t\right\}

Define $B^{*}(t, \omega )$ to be zero if $\omega \in N$ . (Notice that $B_{t}^{*}$ is measurable with respect to the completion $\mathcal{F}_{t}^{*}$ .) Show that, for all $t, t^{\prime }$ in $[0,1]$ ,

\left|B^{*}(t, \omega )-B^{*}\left(t^{\prime }, \omega \right)\right| \leq 2 \epsilon _{m}(\omega ) \quad \text{ if }\left|t-t^{\prime }\right| < 2^{-m}

(v)

Show that $B^{*}(t, \omega ) = \lim_{n} B\left(t_{n}, \omega \right)$ almost surely, where $\left\{ t_{n}\right\}$ is the sequence defined in step (ii). For each $\epsilon > 0$ , deduce that $\mathbb {P}\left(\left|B_{t}^{*}-B_{t}\right| > \epsilon \right) \leq \mathbb {P}\left(\liminf \left\{ \left|B\left(t_{n}\right)-B_{t}\right| > \epsilon \right\} \right) \leq \underset {n}{\liminf } \mathbb {P}\left(\left|B\left(t_{n}\right)-B_{t}\right| > \epsilon \right) = 0$ . (Remember that $B(t)-B\left(t_{n}\right)$ has a $N\left(0, t-t_{n}\right)$ distribution.) Conclude that $B_{t}^{*} = B_{t}$ almost surely, for each $t$ .

Problem 9.4

Suppose $\left\{ B_{s}: s \in S\right\}$ is a Brownian motion, with $S$ the countable set of all dyadic rationals in $[0,1]$ , as in the previous Problem. Modify the argument from that Problem to construct a standard Brownian motion indexed by $[0,1]$ .

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Problem 9.5

Show that there exists a finite constant $C_{0}$ for which

\sum _{k = 0}^{\infty } 2^{k / 2}\left(\delta \wedge 2^{-k}\right) \sqrt{1+\log \left(2^{k}\right)} \leq C_{0} \sqrt{\delta +\delta \log (1 / \delta )} \quad \text{ for } 0 \leq \delta \leq 1

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Problem 9.6

Let $B$ be a standard Brownian motion indexed by [0,1]. Show that the process $X_{t} = B_{1}-B_{1-t}$ , for $0 \leq t \leq 1$ , is also a Brownian motion (with respect to own its natural filtration, not the filtration for $B$ ).

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Problem 9.7

(i)(ii)(iii)

Let $\left\{ \left(B_{t}, \mathcal{F}_{t}\right): 0 \leq t \leq 1\right\}$ be a Brownian motion with continuous sample paths. Define $\tau (\omega )$ as the smallest $t$ at which the sample path $B(\cdot , \omega )$ achieves its maximum value, $M(\omega )$ . (Note: $\tau$ is not a stopping time.) Follow these steps to show that the sets $\Omega_{t}: = \left\{ \tau = t\right\}$ for $0 \leq t \leq 1$ is a family suitable for the construction in Example $\langle 2 \rangle$ .

Example 2: given any uncountable partition $\left\{ \Omega_{t}: 0 \leq t \leq 1\right\}$ of $\Omega$ into $\mathcal{F}_{1}$ -measurable sets with $\mathbb {P}\left(\Omega_{t}\right) = 0$ for every $t$ (``a family suitable for the construction''), one can build a badly-behaved modification $B^{*}$ of $B$ , equal to $B$ almost surely at each fixed $t$ and satisfying the same finite-dimensional distributions, but with $\sup_{t}\left|B_{t}^{*}(\omega )\right|$ nonmeasurable, by reassigning $B^{*}(t, \omega )$ to an arbitrarily large nonmeasurable value on each $\Omega_{t}$ .

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(i)

Show that $\left\{ \tau \leq t\right\} = \left\{ \sup_{s \leq t} B_{s}(\omega ) = M(\omega )\right\}$ . Deduce that $\tau$ is $\mathcal{F}_{1}$ -measurable. Hint: Work with the process at rational times.

(ii)

For each $t$ in $[0,1)$ , show that $\left\{ \tau = t\right\} \subseteq \left\{ \sup_{t \leq s \leq 1}\left(B_{s}-B_{t}\right) \leq 0\right\}$ . Deduce via the result from Exercise $\langle 12 \rangle$ (the first time a standard Brownian motion hits a fixed positive level $a$ has density $a t^{-3 / 2} \exp \left(-a^{2} / 2 t\right) / \sqrt{2 \pi }$ with respect to Lebesgue measure on $\mathbb {R}^{+}$ , so it takes any particular value with probability zero) that $\mathbb {P}\left(\tau = t\right) = 0$ .

(iii)

Use Problem [6] to show that $\mathbb {P}\left(\tau = 1\right) = 0$ .

Problem 9.8

For all real $y$ , show that $e^{i y} = 1+i y+(i y)^{2} / 2!+\ldots +(i y)^{k} / k!+B_{k}(y)$ with $\left|B_{k}(y)\right| \leq \left|y\right|^{k+1} /(k+1)$ !.

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Problem 9.9

(i)(ii)

Let $X: = \mathbb {C}\left(\mathbb {R}^{+}\right)$ , equipped with its metric $d$ for uniform convergence on compacta.

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(i)

Show that $(X, d)$ is separable. Hint: Consider the countable collection of piecewise linear functions obtained by interpolating between a finite number of "vertices" with rational coordinates.

(ii)

Prove that the sigma-field $\mathcal{C}$ generated by the finite dimensional projections coincides with the Borel sigma-field $\mathcal{B}(X)$ . Hint: For one inclusion use continuity of the projections. For the other inclusion replace sup-norm distances by suprema over subsets of rationals.

Problem 9.10

(i)(ii)(iii)(iv)

Let $\left\{ B_{t}: t \in \mathbb {R}^{+}\right\}$ be a standard Brownian motion. For fixed constants $\alpha > 0$ and $\beta > 0$ show that $\mathbb {P}\left(\left\{ B_{t} \text{ eventually hits the line } \alpha +\beta t\right\} \right) = \exp (-2 \alpha \beta )$ , by following these steps. Write $\tau$ for the first hitting time on the linear barrier. For fixed real $\theta$ , let $X_{\theta }(t)$ denote the martingale $\exp \left(\theta B_{t}-1 / 2 \theta^{2} t\right)$ .

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(i)

For each fixed $t$ , show that $1 = \mathbb {E}\left[X_{\theta }(t \wedge \tau )\right]$ . Hint: You might consult Appendix E if you wish to be completely rigorous.

(ii)

If $\theta > 2 \beta$ , show that $0 \leq X_{\theta }(t \wedge \tau ) \leq \exp (\theta \alpha )$ .

(iii)

If $\theta > 2 \beta$ , show that $X_{\theta }(t \wedge \tau ) \rightarrow 0$ as $t \rightarrow \infty$ on the set $\left\{ \tau = \infty \right\}$ .

(iv)

Deduce that $1 = \mathbb {E}\left[\exp \left(\theta \alpha +\left(\theta \beta -1 / 2 \theta^{2}\right) \tau \right) \left\{ \tau < \infty \right\} \right]$ for each $\theta > 2 \beta$ . Then let $\theta$ decrease to $2 \beta$ to conclude the argument.

Problem 9.11

(i)(ii)

Let $\left\{ \left(B_{t}, \mathcal{F}_{t}\right): 0 \leq t \leq 1\right\}$ be a Brownian motion defined on a probability space $(\Omega , \mathcal{F}, \mathbb {P})$ . Let $\alpha$ be a positive real number.

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(i)

Show that the measure $\mathbb {Q}$ defined by $d \mathbb {Q} / d \mathbb {P}: = \exp \left(\alpha B_{1}-\frac{1}{2} \alpha^{2}\right)$ is a probability measure on $\mathcal{F}$ .

(ii)

Show that the process $X_{t}: = B_{t}-\alpha t$ for $0 \leq t \leq 1$ is a Brownian motion under $\mathbb {Q}$ . Hint: For fixed $s < t$ , real $\theta$ , and $F$ in $\mathcal{F}_{s}$ , show that

\begin{aligned} & \mathbb {Q} \exp \left(i \theta \left(X_{t}-X_{s}\right)+\frac{1}{2} \theta ^{2}(t-s)\right) F \\ & = \mathbb {E}\left[\exp \left(\alpha \left(B_{1}-B_{t}\right)-\frac{1}{2} \alpha ^{2}(1-t)\right)\right] \\ & \times \mathbb {E}\left[\exp \left((\alpha +i \theta )\left(B_{t}-B_{s}\right)-\frac{1}{2}(\alpha +i \theta )^{2}(t-s)\right)\right] \\ & \times \mathbb {E}\left[\exp \left(\alpha B_{s}-\frac{1}{2} \alpha ^{2} s\right) F\right] . \end{aligned}

Notice that right-hand side reduces to an expression that does not depend on $\theta$ , which identifies it as $\mathbb {Q} F$ .

Problem 9.12

Calculate the price of an option that allows you to purchase one unit of stock for a price $K$ at time 1 , if you wish.

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