Math Problems Vault

Problem 3.1

Use the Lévy - Ciesielski representation $B(t) = \sum_{n = 0}^{\infty } G_{n} S_{n}(t), t \in [0,1]$ , to obtain a series representation for $X: = \int_{0}^{1} B(t) d t$ and find the distribution of $X$ .

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Problem 3.2

Let $\left(G_{n}\right)_{n \geqslant 0}$ be a sequence of iid Gaussian $\mathrm{N}(0,1)$ random variables, $\left(\phi_{n}\right)_{n \geqslant 0}$ be a complete ONS in $L^{2}([0,1], d t), B \in \mathscr {B}[0,1]$ be a measurable set and define $W_{N}(B): = \sum_{n = 0}^{N - 1} G_{n} \left\langle \; \mathbb {1}_{B}, \phi_{n} \right\rangle_{L^{2}}$ .

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(a)

Show that the limit $W(B): = \lim_{N \rightarrow \infty } W_{N}(B)$ exists in $L^{2}(\mathbb {P})$ .

(b)

Find $\mathbb {E} W(A) W(B)$ for sets $A, B \in \mathscr {B}[0,1]$ .

(c)

Show that $B \mapsto \mu (B): = \mathbb {E}\left[\left|W(B)\right|^{2}\right]$ is a measure. Is $B \mapsto W(B)$ a measure?

(d)

Let $\mathcal{S}$ denote all step functions of the form $f(t): = \sum_{j < \infty } c_{j} \; \mathbb {1}_{B_{j}}(t)$ where $c_{j} \in \mathbb {R}$ and $B_{j} \in \mathscr {B}[0,1]$ . Show that the operator $I(f): = \sum_{j} c_{j} W\left(B_{j}\right)$ is well - defined (i.e. independent of the particular representation of the step function).

(e)

Show that $\mathbb {E}\left[\left|I(f)\right|^{2}\right] = \int \left|f\right|^{2} d \mu$ and use this isometry between $L^{2}([0,1], \mu )$ and $L^{2}(\mathbb {P})$ to extend $I$ to all $f \in L^{2}([0,1], \mu )$ .

Remark. If we take $B = [0, t]$ , then $W_{t}: = W([0, t]), t \in [0,1]$ , is a Brownian motion. More on this topic can be found in [231, Chapters 9, 10].

Problem 3.3

(a)(b)

Let $\phi_{n} = H_{n}, n = 0,1,2, \ldots$ , be the Haar functions. The following steps show that the full sequence $W_{N}(t, \omega ): = \sum_{n = 0}^{N - 1} G_{n}(\omega ) S_{n}(t), N = 1,2, \ldots$ , converges uniformly for $\mathbb {P}$ - almost all $\omega \in \Omega$ .

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(a)

Let $\left(a_{n}\right)_{n \geqslant 0} \subset \mathbb {R}$ be a sequence satisfying $a_{n} = O\left(n^{\epsilon }\right)$ for some $\epsilon \in (0,1 / 2)$ . Show that $\sum_{n = 0}^{\infty } a_{n} S_{n}(t)$ converges absolutely and uniformly in $t \in [0,1]$ .

(b)

Let $\left(G_{n}\right)_{n \geqslant 0}$ be a sequence of iid $\mathrm{N}(0,1)$ random variables. Show that almost surely $\left|G_{n}\right| = O(\sqrt{\log n})$ .

Problem 3.4

Let $(S, d)$ be a complete metric space equipped with the $\sigma$ - algebra $\mathscr {B}(S)$ of its Borel sets. Assume that $\left(X_{n}\right)_{n \geqslant 1}$ is a sequence of $S$ - valued random variables such that

\lim _{n \rightarrow \infty } \sup _{m \geqslant n} \mathbb {E}\left[d\left(X_{n}, X_{m}\right)^{p}\right] = 0

for some $p \in [1, \infty )$ . Show that there is a subsequence $\left(n_{k}\right)_{k \geqslant 1}$ and a random variable $X$ such that $\lim_{k \rightarrow \infty } X_{n_{k}} = X$ almost surely.

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Remark.

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If the hypothesis is strengthened to $\lim_{n \rightarrow \infty } \mathbb {E}\left[\sup_{m \geqslant n} d(X_{n}, X_{m})^{p}\right] = 0$ , the argument simplifies considerably: this says $d_{n}: = \sup_{m \geqslant n} d(X_{n}, X_{m})^{p}$ converges to $0$ in $L^{p}(\mathbb {P}; \mathbb {R})$ , so some subsequence $(n_{k})_{k}$ has $\lim_{k \rightarrow \infty } \sup_{m \geqslant n_{k}} d(X_{n_{k}}, X_{m}) = 0$ almost surely, giving $d(X_{n_{k}}, X_{n_{l}}) \rightarrow 0$ as $k, l \rightarrow \infty$ ; completeness of $S$ then directly gives $X_{n_{k}} \rightarrow X$ .

Problem 3.5

Let $\left(X_{t}\right)_{t \geqslant 0}$ and $\left(Y_{t}\right)_{t \geqslant 0}$ be two stochastic processes which are modifications of each other. Show that they have the same finite dimensional distributions.

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Problem 3.6

Let $\left(X_{t}\right)_{t \in I}$ and $\left(Y_{t}\right)_{t \in I}$ be two processes with the same index set $I \subset [0, \infty )$ and state space. Show that

X, Y \text{ indistinguishable } \Longrightarrow X, Y \text{ modifications } \Longrightarrow X, Y \text{ equivalent. }

Assume that the processes are defined on the same probability space and $t \mapsto X_{t}$ and $t \mapsto Y_{t}$ are right continuous (or that $I$ is countable). Do the reverse implications hold, too?

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Problem 3.7

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a real - valued stochastic process with exclusively continuous sample paths. Assume that $\left(B_{q}\right)_{q \in \mathbb {Q} \cap [0, \infty )}$ satisfies (B0) - (B3). Show that $\left(B_{t}\right)_{t \geqslant 0}$ is a BM.

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Problem 3.8

Let $X^{\prime }, X^{\prime \prime }, Z$ be random vectors such that $X^{\prime } \; \perp \! \! \! \perp \; X^{\prime \prime }$ and $\left(X^{\prime }, X^{\prime \prime }\right) \; \perp \! \! \! \perp \; Z$ . Show that $X^{\prime }, X^{\prime \prime }, Z$ are independent.

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Problem 3.9

Give a direct proof of the formula (3.5) using the joint probability distribution $\left(W\left(t_{0}\right), W(t), W\left(t_{1}\right)\right)$ of the Brownian motion $W(t)$ .

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Problem 3.10

Show that there is a stochastic process $\left(W_{t}\right)_{t \in \mathbb {R}}$ satisfying (B0) - (B4).

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