Writing $A(t)=1+S_1S_2^{-1}$ as an Ito diffusion process.

Question

Let $W$ be a Wiener process/Brownian motian and let $$ \begin{align} \mathrm{d}S_1 &= 2S_1(t)dt +3S_1(t) dW\\ \mathrm{d}S_2 &= 4S_2(t)dt +5S_2(t) dW \end{align} $$

Now I'd like to write $A(t)=t+S_1 S_2^{-1}$ as an Ito diffusion process.

I'd appreciate any advice/help.

Answer 1

To be specific with your example, it will help to see that $$\begin{split} dS_2^{-1} &= -S_2^{-2}\left(dS_2\right) + \frac{1}{2}\cdot 2S_2^{-3} \left(dS_2\right)^2 \\ &= -\frac{dS_2}{S_2^2} + \frac{\left(dS_2\right)^2}{S_2^3} \\ &= -\frac{4S_2dt + 5S_2dW}{S_2^2} + \frac{25 S_2^2 dt}{S_2^3} \\ &= \frac{21dt - 5dW}{S_2} \end{split}$$ $$\begin{split} dA &= dt + d\left[S_1 S_2^{-1}\right] \\ &= dt + S_2^{-1} dS_1 + S_1 d S_2^{-1} + \frac{1}{2} dS_1 dS_2^{-1}\\ &= dt + \frac{S_1(2dt + 3dW)}{S_2} + S_1 \frac{21dt - 5dW}{S_2} + \frac{S_1(2dt + 3dW)(21dt - 5dW)}{2S_2} \\ &= dt + \frac{S_1}{S_2} \left(2dt + 3dW + 21dt - 5dW - \frac{15}{2} dt\right) \end{split}$$ and you can finish the arithmetic....