Why does $$d(X_tY_t)=X_tdY_t+Y_tdX_t+dX_tdY_t$$ hold for Ito process? $X_t$ and $Y_t$ are both the Ito process.
2026-04-01 22:49:20.1775083760
Why does $d(X_tY_t)=X_tdY_t+Y_tdX_t+dX_tdY_t$ hold for Ito process?
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This is known as the product rule relating to the Ito integral and can be proved if you can prove that $$X_t^2=X_0^2+2\int^t_0X_sdX_s+[X]_t$$ and using the clever fact that the two formulas $$(X_t+Y_t)^2=(X_0+Y_0)^2+2\int^t_0(X_s+Y_s)d(X_s+Y_s)+[X+Y]_t$$ and $$(X_t-Y_t)^2=(X_0-Y_0)^2+2\int^t_0(X_s-Y_s)d(X_s-Y_s)+[X-Y]_t$$ can be subtracted to give the product rule formula. Subtract the first equation from the second and divide by four to get your identity!
The first identity can be proved using the definition of quadratic variation.
Also note that $$d(X_tY_t)=X_tdY_t+Y_tdX_t+dX_tdY_t$$ is shorthand for $$X_tY_t=X_0Y_0+\int^t_0X_sdY_s+\int^t_0Y_sdX_s+[X,Y]_t$$