Advanced Process

8-12 Hours

Brownian Motion

The Wiener process: continuous-time random motion with fundamental applications

Brownian Motion Calculator

Calculate covariance, hitting probabilities, and general Brownian motion properties

Calculation Type

Time s

Time t

Definition & Properties - Standard Brownian Motion

A standard Brownian motion (Wiener process) $\{B(t), t \geq 0\}$ is a stochastic process with:

$B(0) = 0$
Independent increments
For $s < t$ , $B(t) - B(s) \sim N(0, t-s)$
Continuous sample paths (almost surely)

Key Properties:

$E[B(t)] = 0$
$\text{Var}(B(t)) = t$
$\text{Cov}(B(s), B(t)) = \min(s, t)$

Reflection Principle

For $a > 0$ , the probability that Brownian motion reaches level $a$ before time $t$ is:

P\left(\max_{0 \leq s \leq t} B(s) \geq a\right) = 2P(B(t) \geq a) = 2(1 - \Phi(a/\sqrt{t}))

This is used to calculate hitting time probabilities: $P(T_a \leq t) = 2P(B(t) \geq a)$ .

Geometric Brownian Motion

Geometric Brownian motion models stock prices:

S(t) = S_0 \exp\left(\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma B(t)\right)

where $\mu$ is the drift, $\sigma$ is the volatility, and $B(t)$ is standard Brownian motion.

Example 1: Covariance

Problem:

Find $\text{Cov}(B(2), B(5))$ for standard Brownian motion.

Solution:

$\text{Cov}(B(2), B(5)) = \min(2, 5) = 2$ .

Example 2: Hitting Time

Problem:

Find $P(T_2 \leq 4)$ where $T_2$ is the hitting time of level 2.

Solution:

Using reflection principle: $P(T_2 \leq 4) = 2P(B(4) \geq 2) = 2(1 - \Phi(2/\sqrt{4})) = 2(1 - \Phi(1)) \approx 0.318$ .

Example 3: General Brownian Motion

Problem:

For $X(t) = \mu t + \sigma B(t)$ with $\mu = 1$ and $\sigma = 2$ , find the distribution of $X(3) - X(1)$ .

Solution:

$X(3) - X(1) = \mu(3-1) + \sigma(B(3) - B(1)) = 2 + 2(B(3) - B(1))$ .

Since $B(3) - B(1) \sim N(0, 2)$ , we have $X(3) - X(1) \sim N(2, 8)$ .

Practice Quiz

Questions

Correct

Accuracy

What is the covariance of standard Brownian motion

\text{Cov}(B(s), B(t))

for

s \leq t

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What is the distribution of

B(t)

for standard Brownian motion?

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What is the reflection principle for Brownian motion?

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What is the hitting time

T_a

for Brownian motion?

Not attempted

What is geometric Brownian motion?

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What is the self-similarity property of Brownian motion?

Not attempted

What is a Brownian bridge?

Not attempted

What is the variance of

B(t)

for standard Brownian motion?

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What property does Brownian motion have regarding differentiability?

Not attempted

What is the distribution of increments

B(t) - B(s)

for

s < t

Not attempted

Frequently Asked Questions

Why is Brownian motion called the Wiener process?

Norbert Wiener provided the first mathematical construction of Brownian motion in 1923, proving its existence rigorously. While Robert Brown observed the phenomenon in 1827 and Einstein explained it physically in 1905, Wiener's mathematical formulation is why it's also called the Wiener process.

What does sample path continuity mean?

Sample path continuity means that for every outcome ω, the function t → B(t,ω) is continuous (no jumps). This is remarkable because even though the process has continuous paths, those paths are nowhere differentiable—they're infinitely rough at all scales.

How does the reflection principle work?

The reflection principle says: if Brownian motion hits level 'a' before time t, and we reflect the path about 'a' afterwards, we get the same distribution. Mathematically, P(max B(s) ≥ a) = 2P(B(t) ≥ a). This symmetry is powerful for calculating hitting time and maximum distributions.

Why is Brownian motion self-similar?

For any c>0, {B(ct)/√c} has the same distribution as {B(t)}. This means zooming in or out looks statistically the same—a fractal property. It's why Brownian motion appears equally rough at all time scales. The exponent H=1/2 is called the Hurst parameter.

What's the difference between Brownian motion and geometric Brownian motion?

Standard Brownian motion B(t) can be negative (symmetric around 0). Geometric Brownian motion S(t)=S₀e^{μt+σB(t)} is always positive, making it suitable for stock prices. The 'geometric' means percentage changes (not absolute) are normally distributed.