Introduction to Stochastic Differential Equations (SDEs

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Stochastic differential equations

Stochastic differential
equations
HANPEN ROBOT

𝒁~𝑵 𝟎, 𝟏 ⇒ 𝑿 = 𝝈𝒁 + 𝝁~𝑵 𝝁, 𝝈 𝟐

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
⇓

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

Introduction to Stochastic Differential Equations (SDEs

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Introduction to Stochastic Differential Equations (SDEs

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!