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Applying Itô’s Product Rule to Correlated Brownian Motions

Original question

Can you explain how to apply Itô's product rule when two stochastic processes are driven by correlated Brownian motions?
For continuous Itô processes, d(XtYt)=Xt dYt+Yt dXt+d[X,Y]td(X_tY_t)=X_t\,dY_t+Y_t\,dX_t+d[X,Y]_t. If the Brownian coefficients are btb_t and dtd_t and the Brownian drivers have instantaneous correlation ρ\rho, then d[X,Y]t=ρbtdt dtd[X,Y]_t=\rho b_t d_t\,dt. The sign and size of this extra drift depend on both exposures and the correlation.

Compute the cross-variation before differentiating the product

Let W1W^1 and BB be independent Brownian motions, and write W2=ρW1+1−ρ2BW^2=\rho W^1+\sqrt{1-\rho^2}B for constant ρ\rho in [−1,1][-1,1]. Bilinearity of quadratic covariation gives

[W1,W2]t=ρ[W1,W1]t+1−ρ2[W1,B]t=ρt.[W^1,W^2]_t=\rho[W^1,W^1]_t+\sqrt{1-\rho^2}[W^1,B]_t=\rho t.
Now let dXt=at dt+bt dWt1dX_t=a_t\,dt+b_t\,dW^1_t and dYt=ct dt+dt dWt2dY_t=c_t\,dt+d_t\,dW^2_t. The drift terms have finite variation, so the product rule becomes
d(XtYt)=(Xtct+Ytat+ρbtdt) dt+Xtdt dWt2+Ytbt dWt1.d(X_tY_t)= (X_tc_t+Y_ta_t+\rho b_td_t)\,dt +X_td_t\,dW^2_t+Y_tb_t\,dW^1_t.
You get the same result by applying the multivariate Itô formula to f(x,y)=xyf(x,y)=xy. Its mixed derivative is 11. The two off-diagonal terms in the Hessian sum cancel the prefactor of one half, leaving the entire covariation term.

You can see the sign of the extra drift from a single step. The product gains contain (ΔX)(ΔY)(\Delta X)(\Delta Y) in addition to the two first-order terms. The leading noise pieces are bXΔW1b_X\Delta W^1 and bYΔW2b_Y\Delta W^2. For an interval of length hh, the expected product of the Brownian shocks is ρh\rho h. As the grid is refined, this produces the covariation term ρbXbY dt\rho b_Xb_Y\,dt. Zero correlation removes it even though the two first-order stochastic terms are still present.

For an original exercise, take Xt=3+2Wt1X_t=3+2W_t^1, Yt=4+3Wt2Y_t=4+3W_t^2 and ρ=−0.4\rho=-0.4. The product has drift (−0.4)(2)(3)=−2.4(-0.4)(2)(3)=-2.4 per year even though each factor is a driftless martingale. The stochastic integrals have mean zero on finite intervals, giving
E[X1.5Y1.5]=12−2.4(1.5)=8.4.\mathbb E[X_{1.5}Y_{1.5}]=12-2.4(1.5)=8.4.
Check this by expanding the terminal product and using E[W1.51W1.52]=ρ(1.5)=−0.6\mathbb E[W_{1.5}^1W_{1.5}^2]=\rho(1.5)=-0.6. Multiplying the two separate means would give 1212 because it omits the covariance. If the drivers were independent, the covariation term would vanish and the expected product would stay 1212 for this example. With a common driver, set ρ=1\rho=1 and the entire coefficient product would enter the drift.
The argument extends to models in which one process drives the other's diffusion coefficient. For example, if the loading of XX is bt=Ytb_t=\sqrt{Y_t} and the loading of YY is dt=ηYtd_t=\eta\sqrt{Y_t}, with Yt≥0Y_t\ge0, their local covariation is ρηYt dt\rho\eta Y_t\,dt. The current state affects the correction even though ρ\rho is constant. This is the type of cross term that appears when applying Itô's lemma to a price with stochastic variance.

References and next steps