2015 Q&A Bank 2.12 vs 2.15 Question on use of the fact that if the underlying stochastic process is a standard Brownian motion, we can say: dWt = (1x dWt) + (0 x dt) In question 2.12, Xt is a function of a standard Brownian motion but in those solutions, the µ and σ are not replaced with 0's and 1's. Only in the last 2 where it deals with Bt directly do we do that. In question 2.15, Xt is a function of a Brownian motion so why is it done there? What's the difference between the two?
This is great question - thanks for asking it Q2.12 and Q2.15 are asking slightly different things about \(X_t\). Q2.12 is about finding the SDE of a function of \(X_t\) (general Brownian motion), which is in itself driven by \(B_t\) (standard Brownian motion). Q2.15 is about finding the SDE of the function \(X_t\) , which is in itself a function of \(B_t\) (standard Brownian motion). We could recast Q2.15 in the language of Q2.12 via: "Find the SDE of \(L_t=\alpha B_t^2+\beta\)". Then it would make sense to use that "\(1 \times \)" and "\(0 \times \)" trick to solve it (as is the case for Q2.12(e) and Q2.12(f)). Let me know if this doesn't make sense.
