Understanding Ito

[april2105]

Hi,
Q1 ii) in revision booklet 3 asks to find DX_t using Ito's lemma.
(April 2002 Q2)

I just have a few questions regarding the solution that will hopefully help me on my way.

1) in the tables the value you assign to a (or mu) is the term that's with the dt term and the b (or sigma) is the term with the dz. Here there is neither so how does it link up?

2) The notation confuses me. My do we go from df(B_t) to f(x) to dX_t?

3) the term that would correspond to the dellG/dellt bit on pg 46 of the tables isnt included, why is that? Do you need explicut t's in the model equation rather than just the "t" in B_t?

Thanks!

 

[Mike Lewry]

1) In the tables, a and b are the coefficients for your diffusion process x, which underlies G.

In this question, X is a function of B, so the underlying diffusion is just standard BM. so a=0 and b=1.

I find it easier to use Taylor directly as a function of whatever is given in the question. That way, I never need to worry about identifying a and b.

2) By writing X as f(B), we're just making it clear that X is a function of B.
Writing f(x) helps us focus on the function f while we're differentiating it. Writing f(B) here might be confusing as B is non-differentiable. That's not relevant though, as it's f we're differentiating. As long as your notation is clear, you don't need to follow this convention in exam answers.

3) Correct. There's no explicit t dependence so dellX/dellt=0