itos_lemma

itos_lemma

ito’s lemma is just the chain rule for stochastic integrals.

If X : dX_t = μ_tdt + σ_tdW_t, X₀ = x₀. With μ_t = μ(t, X_t) and σ_t = σ(t, X_t).

And Y : Y_t = f(t, X_t) where f is smooth and f : [0, T] × ℝ → ℝ.

Further easy way to remember: $dY_t = \partial_t f dt + \partial_x f d X_t + \frac{1}{2} \frac{\partial ^2f}{\partial x^2} (dX_t)^2$ where (dt)² = 0 dtdW_t = 0 and (dW_t)² = dt

ito’s lemma for multi-dimensions

Suppose dX_t = μdt + σ_t⁽¹⁾dW_t⁽¹⁾ + σ_t⁽²⁾dW_t⁽²⁾ and dY_t = αdt + β_t⁽¹⁾dW_t⁽¹⁾ + β_t⁽²⁾dW_t⁽²⁾, Where W⁽¹⁾ and W⁽²⁾ are independant Brownian motions.

Let Z_t = f(t, X_t, Y_t) for some smooth f.

Then $dZ_t= \partial_t f dt + \partial_x f d X_t + \partial_y f dY_t + \frac{1}{2} \partial_{xx} f(dX_t) ^2 + \frac{1}{2} \partial_{y y } f(d Y_t)^2 + \partial_{xy} f dX_t dY_t$.

Where (dt)² = dtdW_t⁽ⁱ⁾ = dW_t⁽¹⁾dW_t⁽²⁾ = 0 and (dW_t⁽ⁱ⁾)² = dt (think of this as a second order taylor expansion, especially dW_t⁽²⁾.

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