This question is from Mathematics for Finance, a chapter on Ito's Formula. I could apply Ito's to calculate the required variable, but I could not follow through the second part, to find its expectation. Perhaps I don't understand the Brownian motion (most likely that I don't). Unfortunately I don't have the solution to this question.

PS is a multiplication factor to find the value of a deposit/sum of money after t periods, given that the deposit pays interest rate r, and assuming continous compounding.

Question.

The stochastic differential equation for the rate of inflation I is given by

Find the equation satisfied by the real interest rate R defined to be , where .

[Hint: Consider .]

Show that

.

Answer.

I will apply Ito's formula to find dR.

Let

therefore, we calculate df(I,t) from the equation of dI by applying Ito's formula:

by dividing this by , we get

Here where it gets tricky,

I know that E(dz)=0 as it is a Wiener process with mean=0

But what do I do with the remaining expectation on dt???

Somehow I need to show that and I can only see that if I have

BUT why T=1???

Is there an implicit assumption that values of are given on a per annum basis, therefore T=1?

thanks