Math Problems Vault

Problem 32.1

Let $w(t)$ be a Wiener process. Is the process $\eta (t)$ Markovian, if

\eta (t) = \begin{cases} w(t), & \text{if } \max \limits _{0 \leq s \leq t} w(s) < a, \\ a, & \text{otherwise.} \end{cases}

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Remark.

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A stochastically continuous homogeneous process with independent increments $\xi (t)$ is called a Wiener process if $\xi (0) = 0$ and $\xi (1)$ has a normal distribution. A Wiener process $\xi (t)$ is called standard if $\xi (1)$ has the standard normal distribution. Throughout the rest of this section, continuous modifications of the Wiener process are considered.

Problem 32.2

Let $w(t)$ be a Wiener process. Is the process $\eta (t)$ Markovian, if

\eta (t) = \begin{cases} w(t), & \text{if } w(t) \leq a, \\ 2a - w(t), & \text{otherwise.} \end{cases}

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Problem 32.3

Let $w(t)$ be a Wiener process. Is the process $\eta (t) = w(t) - \left[w(t)\right]$ Markovian, where $\left[x\right]$ is the integer part of $x$ ?

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Problem 32.4

Find the covariance function of the Wiener process.

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Problem 32.5

Prove that the Wiener process is not differentiable in probability.

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Problem 32.6

Let $w(t)$ be a standard Wiener process. Find the covariance function of the random process $w^{(0)}(t) = w(t) - tw(1)$ , considered on the time interval $t \in [0, 1]$ .

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Problem 32.7

Let $w(t)$ be a standard Wiener process. Find the covariance function of the process $e^{-bt}w(ae^{2bt})$ , where $a$ and $b$ are real numbers (this process is called the Ornstein--Uhlenbeck process).

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Problem 32.8

Let $w(t)$ be a standard Wiener process. Find the joint distribution of $w(t)$ and $\displaystyle \int_{0}^{t} w(s)\, ds$ , $t \geq 0$ .

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Problem 32.9

Let $w(t)$ be a standard Wiener process. Prove that the random process $w^{(1)}(t) = tw(1/t)$ is also a standard Wiener process.

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Problem 32.10

Let $w(t)$ be a Wiener process with zero drift. Prove that the process

\eta (t) = \begin{cases} w(t), & t \leq T, \\ 2w(T) - w(t), & t > T, \end{cases}

is also a Wiener process.

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Problem 32.11

Let $w(t)$ be a three-dimensional Wiener process starting at zero. Prove that with probability 1

\lim _{t \to \infty } \left|w(t)\right| = \infty .

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Problem 32.12

(a)(b)

Let $w(t)$ be a standard Wiener process. Find the distributions of the following stochastic integrals:

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(a)

$\displaystyle \int_{0}^{t} s^{2}\, dw(s)$ ;

(b)

$\displaystyle \int_{0}^{t} e^{s}\, dw(s)$ .

Problem 32.13

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

Let $w(t)$ be a standard Wiener process. Determine which of the following processes are Wiener processes:

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(a)

$-w(t)$ ;

(b)

$w(2t) - w(t)$ ;

(c)

$\sqrt{t}w(1)$ ;

(d)

$cw(t/c^{2})$ .

Problem 32.14

Let $w_{1}(t)$ and $w_{2}(t)$ be independent standard Wiener processes. For which $c_{1}$ and $c_{2}$ is the random process $c_{1}w_{1}(t) + c_{2}w_{2}(t)$ a standard Wiener process?

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Problem 32.15

Let $w(t)$ be a standard Wiener process. Find the covariance function of the random process $w^{(0)}(t) = (1-t)w(t/(1-t))$ , $t \in [0, 1)$ , $w^{(0)}(1) = 0$ .

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Problem 32.16

Prove that the quadratic variation of the standard Wiener process $w(t)$ is finite, i.e., that

\sum _{j=0}^{n-1} \Bigl(w\Bigl(\frac{(j+1)t}{n}\Bigr) - w\Bigl(\frac{jt}{n}\Bigr)\Bigr)^{2}

converges in mean square to $t$ as $n \to \infty$ .

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Problem 32.17

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

Let $w(t)$ be a standard Wiener process. Find the mean-square limit as $n \to \infty$ of:

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(a)

$\displaystyle \sum_{j=0}^{n-1} w\Bigl(\frac{jt}{n}\Bigr)\Bigl(w\Bigl(\frac{(j+1)t}{n}\Bigr) - w\Bigl(\frac{jt}{n}\Bigr)\Bigr)$ ;

(b)

$\displaystyle \sum_{j=0}^{n-1} w\Bigl(\frac{(j+1)t}{n}\Bigr)\Bigl(w\Bigl(\frac{(j+1)t}{n}\Bigr) - w\Bigl(\frac{jt}{n}\Bigr)\Bigr)$ ;

(c)

$\displaystyle \sum_{j=0}^{n-1} \frac{1}{2}\Bigl(w\Bigl(\frac{jt}{n}\Bigr) + w\Bigl(\frac{(j+1)t}{n}\Bigr)\Bigr)\Bigl(w\Bigl(\frac{(j+1)t}{n}\Bigr) - w\Bigl(\frac{jt}{n}\Bigr)\Bigr)$ ;

(d)

$\displaystyle \sum_{j=0}^{n-1} w\Bigl(\frac{(j+1/2)t}{n}\Bigr)\Bigl(w\Bigl(\frac{(j+1)t}{n}\Bigr) - w\Bigl(\frac{jt}{n}\Bigr)\Bigr)$ .