Math Problems Vault

Problem 3.4.1

(a)(b)(c)(d)

Let $\left(W_{t}, t \geqslant 0\right)$ be a Wiener process. Prove that the following processes are also Wiener processes:

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(a)

the Wiener process reflected about the abscissa axis: $X_{t}:=-W_{t}$ ;

(b)

the rescaled Wiener process: $X_{t}:=\sqrt{c} W_{t / c}, c>0$ ;

(c)

the Wiener process started at the point $t_0 > 0$ : $X_{t}:=W_{t+t_0}-W_{t_0}$ , where $t_0 > 0$ is an arbitrary fixed moment in time;

(d)

the Wiener process reflected at the point $t_0 > 0$ :

X_t := \begin{cases} W_t, & t < t_0 \\ 2W_{t_0} - W_t, & t \geqslant t_0 \end{cases},

where $t_0 > 0$ is an arbitrary fixed moment in time.

Problem 3.4.2

(a)(b)(c)

Let $\left(B_{t}, t \in [0,1]\right)$ be a Gaussian process with zero mean function and covariance function $r(s, t)=\min (t_1, t_2)-t_1 t_2$ .

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(a)

Prove that such a process exists;

(b)

Prove that the process

\tilde{W}_s := (s+1)B_{\frac{s}{s+1}}, \; s \geq 0

is a Wiener process.

(c)

Is $B_t$ a process with independent increments?

Remark.

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The process $(B_t)$ is called a Brownian bridge, since its largest variance is attained at the point $\frac{1}{2}$ , and at the endpoints of the interval $[0,1]$ it is a.s. equal to zero.

Problem 3.4.3

Let $W_{t}^{1}, \ldots , W_{t}^{d}$ be independent Wiener processes. Prove that with probability 1 the process $W_{t}=\left(W_{t}^{1}, \ldots , W_{t}^{d}\right)$ (a multidimensional Wiener process) will leave the ball of an arbitrary fixed radius $r$ centered at the origin of the space $\mathbb {R}^{d}$ .

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Problem 3.4.4

Prove that there exists a Gaussian process $X=\left(X_{t}, t \in \mathbb {R}_{+}^{d}\right)$ with zero mean function and covariance function

r(s, t)=\prod _{k=1}^{d} \min (s_k, t_k),

where $s=\left(s_{1}, \ldots , s_{d}\right) \in \mathbb {R}_{+}^{d}, t=\left(t_{1}, \ldots , t_{d}\right) \in \mathbb {R}_{+}^{d}$ .

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Problem 3.4.5

Let $\left(W_{t}, t \geqslant 0\right)$ be a Wiener process. Prove that with probability 1 its trajectory has unbounded variation on an arbitrary interval $[a, b] \subset \mathbb {R}_{+}$ , i.e., that

\sup _{T} \sum _{i=1}^{n}\left|W_{t_{i+1}}-W_{t_{i}}\right|=+\infty \text{ a.s., }

where $T=\left\{ a=t_{0}<\ldots <t_{n}=b\right\}$ is a partition of the interval $[a, b]$ .

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Problem 3.4.6

Let $\left(W_{t}, t \geq 0\right)$ be a Wiener process. For each $n \in \mathbb {N}$ consider partitions of the interval $[a ; b]$ with points

a=t_{n ; 0}<t_{n ; 1}<\ldots <t_{n ; N(n)}=b, \quad \text{where } N(n) \in \mathbb {N}

and

\Delta T(n):=\max _{k=0 ; \ldots ; N(n)}\left(t_{n ; k+1}-t_{n ; k}\right) \xrightarrow [n \to +\infty ]{} 0

Set

U_{n}:=\sum _{k=0}^{N(n)-1}\left(W_{t_{n ; k+1}}-W_{t_{n ; k}}\right)^{2}

Find the mean-square (i.e., $L^2$ ) limit of the quantities $U_{n}$ as $n \rightarrow +\infty$ .

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Problem 3.4.7

Let the sequence of positive numbers $\left\{ t_{n}, n \in \mathbb {N}\right\}$ be such that $\sum_{n=1}^{\infty } t_{n}^{-1 / 2}<\infty$ . Prove that then $\left|W_{t_{n}}\right| \rightarrow +\infty$ a.s. as $n$ increases. Here $\left(W_{t}, t \geq 0\right)$ is a Wiener process.

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Problem 3.4.8

Let $\left(X_{t}, t \geq 0\right)$ be a Gaussian process with stationary independent increments. Prove that there exist constants $a, b \in \mathbb {R}$ such that the process

Y_{t}= \frac{X_{t}-a t}{b}

is a Wiener process.

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Problem 3.4.9

Let $\left\{ \xi_n, n \in \mathbb {Z}_+\right\}$ be independent $N(0,1)$ random variables. Prove that the process

X_{t} = \frac{\xi _{0} t}{\sqrt{\pi }} + \sqrt{\frac{2}{\pi }} \sum _{k=1}^{\infty } \frac{\sin (kt)}{k} \xi _{k}

is a Wiener process on the interval $[0, \pi ]$ (the series is understood as the limit of its partial sums in $L^2$ ).

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