# Likelihood ratio and pathwise sensitivity method for coupled SDEs

I have two coupled SDEs \begin{align*} dS_t=rS_tdt+V_tdW_t^{(1)},\\ dV_t=aV_tdt+b(V_t)dW_t^{(2)},\\ \end{align*} where $$W_t^{(1)}$$ and $$W_t^{(2)}$$ are independent Brownian motions, initial input data are $$S_0$$ and $$V_0$$, $$a(\cdot)$$ and $$b(\cdot)$$ are sufficiently well-behaved, and I use an Euler-Maruyama discretisation with $$N$$ timesteps. How exactly should one calculate the derivative of a payoff function $$\mathbb{E}f(S)$$ with respect to $$S_0$$ in this case?

In particular, I am confused as to how to apply the chain rule in this case due to the dependence of $$S_t$$ on $$V_t$$? For example, say with the likelihood ratio method, would my integral form of the probability density be formed by $$2N$$ integrals ($$N$$ with respect to the $$S$$ and $$N$$ with respect to $$V$$)? (Similarly for the pathwise sensitivity approach, I am confused as to how construct the chain rule.)

Any help is greatly appreciated!