This document introduces stochastic differential equations by showing how a random variable sequence X can be represented as the sum of random noise Z and a drift term μ over small time intervals Δt. As the time interval approaches 0, the change in X is represented by the stochastic differential equation dX = σdZ + μdt, where σ scales the random noise and μ is the drift rate.

1. Stochastic differential
equations
HANPEN ROBOT

2. 𝒁~𝑵 𝟎, 𝟏 ⇒ 𝑿 = 𝝈𝒁 + 𝝁~𝑵 𝝁, 𝝈 𝟐

3. Let {𝑋𝑡} and {𝑍𝑡} be random variable sequences.
𝒁 𝒕+𝚫𝐭 − 𝒁 𝒕~𝑵 𝟎, 𝚫𝒕
𝑿 𝒕+𝚫𝐭 − 𝑿 𝒕 = 𝝈 𝒁 𝒕+𝚫𝐭 − 𝒁 𝒕 + 𝝁𝚫𝒕 ~ 𝑵 𝝁𝚫𝒕, 𝝈 𝟐
𝚫𝒕
⇓

4. 𝑿 𝒕+𝚫𝐭 − 𝑿 𝒕 = 𝝈 𝒁 𝒕+𝚫𝐭 − 𝒁 𝒕 + 𝝁𝚫𝐭
⇓ (Δ𝑡 → 0)
∴ 𝒅𝑿 𝒕 = 𝝈 𝒅𝒁 𝒕 + 𝝁 𝒅𝒕
Very Simple!