Hi I understand the model answer in the revision books takes the SDE for d(log X) and then applies Ito's lemma. By substituting in the differential equation for dX then leads to the answer given in the question. But why do pick d(logX) to apply Ito's lemma? Why not some other function? Any help greatly appreciated! L
This question requires us to solve the SDE for geometric Brownian motion. This is described in Chapter 9, Section 1.3 of our Course Notes. As for "why logX(t)", carry on reading ... We start by "separating the variables", ie dividing through by X(t). This gives [1/X(t)]dX(t) on the left-hand side. Integrating 1/X(t) gives logX(t), which is the reason we apply Taylor (or Ito) to logX(t). We know that the first thing we do in Taylor is to differentiate the function and multiply it by dX(t), giving 1/X(t)dX(t), which is exactly what we want.
