• We are pleased to announce that the winner of our Feedback Prize Draw for the Winter 2024-25 session and winning £150 of gift vouchers is Zhao Liang Tay. Congratulations to Zhao Liang. If you fancy winning £150 worth of gift vouchers (from a major UK store) for the Summer 2025 exam sitting for just a few minutes of your time throughout the session, please see our website at https://www.acted.co.uk/further-info.html?pat=feedback#feedback-prize for more information on how you can make sure your name is included in the draw at the end of the session.
  • Please be advised that the SP1, SP5 and SP7 X1 deadline is the 14th July and not the 17th June as first stated. Please accept out apologies for any confusion caused.

Subject 103, April 2004, Question 1

lk1988

Keen member
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.
 
Back
Top