# Symbol “.” in the derive of Quanto Adjustment

I am reading "Analysis, Geometry and Modeling in Finance". In section 2.10.2 which derives the quanto adjustment, it states that (in page 46) by definition the process $$S_t^{d/f}S_t^f$$ is the foreign asset valued in the domestic currency and therefore should be driven under $$\mathbb{P}^d$$ by $$\frac{dS_t^{d/f}S_t^f}{S_t^{d/f}S_t^f} = r_ddt+\sigma_S.dW^d_t+\sigma_{d/f}.dW_t^d$$

My question is:

1) What is the "$$.$$" in $$\sigma_S.dW^d_t$$ and $$\sigma_{d/f}.dW_t^d$$? I cannot find it in "Symbol Description" of the book.

If it is just multiplication, why there is no correlation in the the multi-dimension Ito's lemma? (theorem 2.2 in page 21) It say $$dW^i_t.dW^j_t=\delta_i^jdt$$

2) Why can we formulate the $$S_t^{d/f}S_t^f$$ like this? Is it an assumption?

• It’s a dot product, and there is no correlation since it’s standard Brownian motion – starovoitovs Jun 19 '19 at 11:36
• Answering the second question, that is simply "the foreign asset valued in domestic currency" in formula. – Vitomir Jun 19 '19 at 15:02