MathIsimple – Simple, Friendly Math Tools & Learning

Core Definition

1 questions

Question 1: Brownian Motion Definition

Which of the following properties is NOT a defining characteristic of standard Brownian motion {B(t)...

Beginner

Not attempted

Core Properties

3 questions

Question 2: Increment Distribution

For standard Brownian motion {B(t); t ≥ 0}, the increment B(4) - B(1) follows which distribution?...

Beginner

Not attempted

Question 3: Covariance Function

For standard Brownian motion {B(t); t ≥ 0}, Cov[B(3), B(7)] equals:...

Intermediate

Not attempted

Question 4: Conditional Distribution

Let {B(t); t ≥ 0} be standard Brownian motion. Given B(5) = 2, the conditional distribution of B(2) ...

Intermediate

Not attempted

Advanced Properties

4 questions

Question 5: Reflection Principle

For standard Brownian motion {B(t); t ≥ 0}, the probability P{max_{0≤s≤t} B(s) ≥ a} equals:...

Advanced

Not attempted

Question 6: First Passage Time

For standard Brownian motion {B(t); t ≥ 0}, let T_a = inf{t > 0: B(t) = a}. The expected value E[T_a...

Advanced

Not attempted

Question 7: Quadratic Variation

For standard Brownian motion {B(t); t ≥ 0}, the quadratic variation [B, B](t) equals:...

Advanced

Not attempted

Question 10: Multidimensional Brownian Motion

For two-dimensional standard Brownian motion (B₁(t), B₂(t)), the correlation between B₁(t) and B₂(t)...

Advanced

Not attempted

Applications

2 questions

Question 8: Geometric Brownian Motion

Let S(t) = S₀e^{μt + σB(t)} be geometric Brownian motion. If S₀ = 100, μ = 0.05, σ = 0.2, and B(1) =...

Intermediate

Not attempted

Question 9: Ornstein-Uhlenbeck Process

The Ornstein-Uhlenbeck process dX(t) = -θX(t)dt + σdB(t) has which property?...

Advanced

Not attempted

Brownian Motion Practice

Ready to Start?