Itô's Formula Theorem Theorem6.1Itô's formulaLet Xt=X0+∫0tμsds+∫0tσsdBsX_t = X_0 + \int_0^t \mu_s ds + \int_0^t \sigma_s dB_sXt=X0+∫0tμsds+∫0tσsdBs be an Itô process and f∈C1,2([0,T]×R)f \in C^{1,2}([0,T] \times \mathbb{R})f∈C1,2([0,T]×R). Then:f(t,Xt)=f(0,X0)+∫0t(∂f∂t+μs∂f∂x+12σs2∂2f∂x2)ds+∫0tσs∂f∂xdBs.f(t, X_t) = f(0, X_0) + \int_0^t \left(\frac{\partial f}{\partial t} + \mu_s \frac{\partial f}{\partial x} + \frac{1}{2}\sigma_s^2 \frac{\partial^2 f}{\partial x^2}\right) ds + \int_0^t \sigma_s \frac{\partial f}{\partial x} dB_s.f(t,Xt)=f(0,X0)+∫0t(∂t∂f+μs∂x∂f+21σs2∂x2∂2f)ds+∫0tσs∂x∂fdBs. The formula extends the chain rule to stochastic processes, with a second-order correction from the quadratic variation. Applications include derivative pricing, solving SDEs, and transforming processes.
Itô's Formula Theorem Theorem6.1Itô's formulaLet Xt=X0+∫0tμsds+∫0tσsdBsX_t = X_0 + \int_0^t \mu_s ds + \int_0^t \sigma_s dB_sXt=X0+∫0tμsds+∫0tσsdBs be an Itô process and f∈C1,2([0,T]×R)f \in C^{1,2}([0,T] \times \mathbb{R})f∈C1,2([0,T]×R). Then:f(t,Xt)=f(0,X0)+∫0t(∂f∂t+μs∂f∂x+12σs2∂2f∂x2)ds+∫0tσs∂f∂xdBs.f(t, X_t) = f(0, X_0) + \int_0^t \left(\frac{\partial f}{\partial t} + \mu_s \frac{\partial f}{\partial x} + \frac{1}{2}\sigma_s^2 \frac{\partial^2 f}{\partial x^2}\right) ds + \int_0^t \sigma_s \frac{\partial f}{\partial x} dB_s.f(t,Xt)=f(0,X0)+∫0t(∂t∂f+μs∂x∂f+21σs2∂x2∂2f)ds+∫0tσs∂x∂fdBs. The formula extends the chain rule to stochastic processes, with a second-order correction from the quadratic variation. Applications include derivative pricing, solving SDEs, and transforming processes.