Conditional CDF ST9-10 pp41

[Porter Ricks]

The formula for the CDF of Xt|Xt-1 in the notes ST9-10 pp41 conditions on Xt-1 = xt-1.

Assignment 3 solutions X3.11 p 16 the same formula conditions on Xt-1 <= xt-1.

So which is it?

 

[dee22]

The formula for the CDF of Xt|Xt-1 in the notes ST9-10 pp41 conditions on Xt-1 = xt-1.

Assignment 3 solutions X3.11 p 16 the same formula conditions on Xt-1 <= xt-1.

So which is it?

I presume the "<=" means less than or equal to. My apologies if I've misunderstood, and feel free to disregard the below in that case.

I've lost my X3 solutions so I can't say for sure if what's there is correct I'm afraid, but it doesn't seem to make any sense to condition on the "less than or equal to" condition as the density function conditions on "equals to" and this shouldn't change moving between cdf and pdf as they are just different descriptions of the same distribution.

i.e. f_X (x|A) = d/dx ( P(X< x | A) )

(Plus the derivation largely depends on the conditioning fixing the volatility so you can express the pdf of X in terms of the pdf of Z, which isn't the case if you condition on <=.)

Although the notation used in this section of the notes and in QRM are a little unclear, and I don't trust the formulae on p41 of the notes at the bottom either (the square both sides step seems wrong in case you care!) so I'm sort of hoping this doesn't come up.

Now back to memorising lists..... or possibly bed.