Problem 2

Question

Let $Z$ be normally distributed with mean zero and variance one under the measure $\mathbb{P} .$ What is the distribution of $\sqrt{t} Z ?$ Is the process $X_{t}=\sqrt{t} Z$ a Brownian motion?

Step-by-Step Solution

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Answer

$\sqrt{t} Z$ is normally distributed as $N(0, t)$, but $X_t=\sqrt{t} Z$ is not a Brownian motion.

1Step 1: Understanding the Standard Normal Distribution

First, we need to recall that a standard normal distribution, denoted as $Z$, has a mean ($\mu$) of 0 and a variance ($\sigma^2$) of 1. This distribution under the measure $\mathbb{P}$ is represented as $Z \sim N(0, 1)$.

2Step 2: Determine the Distribution of $\sqrt{t} Z$

We consider the random variable $\sqrt{t} Z$. Since $Z$ is a standard normal variable, the transformation $\sqrt{t} Z$ involves scaling $Z$ by $\sqrt{t}$. A scaled normal random variable $aZ$, where $a$ is constant, follows a normal distribution with mean $a\cdot 0 = 0$ and variance $a^2 \cdot 1 = a^2$. Therefore, $\sqrt{t} Z$ is normally distributed with mean 0 and variance $t$: \[\sqrt{t} Z \sim N(0, t)\].

3Step 3: Criterion for Brownian Motion

To determine whether $X_t=\sqrt{t} Z$ is a Brownian motion, we must check the conditions: (1) $X_0 = 0$, (2) $X_t$ must have independent increments, (3) increments $X_t - X_s$ for $t>s$ are normally distributed with mean 0 and variance $t-s$, and (4) $X_t$ must be continuous in $t$.

4Step 4: Evaluate if $X_t=\sqrt{t} Z$ is a Brownian Motion

For $X_t=\sqrt{t} Z$ to be a Brownian motion: - $X_0=\sqrt{0} Z = 0$, so condition 1 is satisfied.- $X_t$ does not have independent increments because the value is entirely determined by the same $Z$, failing condition 2 and 3.- Without independent increments, the distribution condition doesn't hold. Thus, $X_t$ is not continuous in the correct sense. Therefore, $X_t$ fails the criteria to be Brownian motion.

Key Concepts

Standard Normal DistributionIndependent IncrementsNormal DistributionVarianceMeasure

Standard Normal Distribution

A standard normal distribution is a fundamental concept in statistics, given its widespread application in various fields. It is represented by the notation \[ Z \sim N(0, 1) \]This indicates that the random variable $ Z $ follows a normal distribution characterized by two key parameters:

**Mean ($ \mu $):** The mean of a standard normal distribution is 0, indicating that the distribution is centered at zero. This makes it symmetric around the mean.

**Variance ($ \sigma^2 $):** The variance is 1. Variance measures the dispersion or spread of the distribution. A small variance like 1 implies that the values of $ Z $ are not too spread out from the mean.

In practical terms, it is used to standardize other normal distributions, allowing comparisons across different data sets. Each data point, in this way, can be understood through its relative position on the standard normal curve.

Independent Increments

The concept of independent increments is central to the theory of stochastic processes, which include Brownian motion. When we talk about independent increments in this context, we mean that the changes in the process are wholly unaffected by past values.
In mathematical terms, for a process $X_t$, increments over non-overlapping intervals are independent:

**Mathematical Expression:** For times $0 < s < t < u < v$, the increments $X_{t} - X_{s}$ and $X_{v} - X_{u}$ must be independent.

**Significance in Brownian Motion:** Brownian motion, or a standard Wiener process, crucially requires independent increments as one of its defining properties. This ensures unpredictability and consistent randomness throughout the process.

For the process $X_t = \sqrt{t} Z$, the lack of independent increments thereby disqualifies it from being a Brownian motion.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, symmetric about its mean $ \mu $. It is defined by two parameters:

**Mean ($ \mu $):** Acts as the central point of the distribution.

**Variance ($ \sigma^2 $):** Determines the spread of the distribution. A larger variance results in a wider curve.

This distribution is written as $X \sim N(\mu, \sigma^2)$. A key property of the normal distribution is that any linear transformation of a normal variable is also normally distributed. For example, if $Z$ is a standard normal variable and $\sqrt{t}$ is a constant, then $\sqrt{t}Z$ also follows a normal distribution but with adjusted parameters:
- **Mean:** Remains 0.- **Variance:** Becomes $t$, since $(\sqrt{t})^2 = t$.This transformation is integral in understanding the scaling impact on such variables.

Variance

Variance is a statistical measure that evaluates how spread out data points in a particular distribution are around the mean. In the context of a normal distribution, it is denoted by $ \sigma^2 $.
Key aspects of variance include:

**Formula:** Variance is calculated as the average of the squared differences from the mean. Mathematically, it is represented as \[\sigma^2 = \frac{1}{N}\sum_{i=1}^{N} (X_i - \mu)^2\]

**Importance:** It provides a quantitative measure of how much the values in a data set deviate from the mean. Larger variances mean more spread and variability, while smaller variances signify a tighter clustering of values around the mean.

In the random variable $\sqrt{t} Z$, the variance is adjusted by a factor of $t$ due to scaling, influencing the distribution's dispersion.

Measure

In probability theory, a measure refers to the mathematical foundation that allows us to assign a volume, area, or probability to different sets under consideration. It is a concept from measure theory, a branch of mathematics that studies ways to generalize notions of length, area, and volume.

**Measure in Probability Spaces:** In statistical contexts, measures help define probabilities on different sets of outcomes. The standard probability measure assigns probabilities to events in a sigma-algebra, a collection of subsets including the sample space.

**Application in Distributions:** The measure $\mathbb{P}$ helps us understand how a random variable like $Z$, under the standard normal distribution, behaves concerning probability. It underscores the transformative effects when we scale such a variable, for example, $\sqrt{t}Z$.

The concept of measure is integral in formalizing the probability and understanding how distributions can transform under scaling and other operations.

Problem 1

Problem 3

Other exercises in this chapter

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Problem 3

Suppose that $W_{t}$ and $\tilde{W}_{t}$ are independent Brownian motions under the measure $\mathbb{P}$ and let $\rho \in[-1,1]$ be a constant. Is the

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Problem 4

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Problem 5

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