Stochastic Differential Equations

Itô vs Stratonovich · Geometric Brownian Motion · Fokker-Planck
Mean X(T)—
Std X(T)—
Itô correction—
E[X(T)] theory—
Itô: dX = μX dt + σX dW
Solution: X(t) = X₀ exp((μ−σ²/2)t + σW(t))
Stratonovich adds σ²/2 to drift
Itô SDE: dX = f(X,t)dt + g(X,t)dW. The Itô integral ∫g dW uses left-endpoint evaluation — non-anticipating, martingale property preserved.
Itô's lemma: df = (∂f/∂t + f∂f/∂x + ½g²∂²f/∂x²)dt + g∂f/∂x dW. The extra ½g²f″ term has no classical analog.
Stratonovich uses midpoint rule: ∫g∘dW. Obeys classical chain rule but is not a martingale. Stratonovich drift = Itô drift + ½g(∂g/∂x).
Fokker-Planck governs the probability density ∂ₜp = −∂ₓ(fp) + ½∂ₓₓ(g²p). GBM has log-normal stationary distribution.