very simple modeling of stocks
Correct me if I’m wrong, but it seems the third part of the following problem serves as one of those ‘out of context’ problems that I mentioned earlier:
Let
be a probability space on which is defined a sequence of i.i.d. Gaussian random variables
with zero mean and unit variance. Consider the recursion
where
. This is a crude model for some nonnegative quantity that grows or shrinks randomly in every time step; for example, we could model the price of a stock this way, albeit in discrete time.
- Under which conditions on
and
do we have
in
?
- Show that if
, then
- Show that if there is no
such that
in any sense
- If we interpret
as the price of stock, then
if we invest one dollar in the stock at time 0. If
! Find such
. Would you consider investing in such a stock?
To prove that assertion, it’s enough to show that the sequence doesn’t converge in distribution. In class, 
in distribution iff 
for any bounded continuous 
. I spent forever trying to work with this definition, but I couldn’t get anything from it (can you?). But then I broke down and decided to use another definition: in distribution iff 
at every point of continuity of 
( and 
are the cdfs of 
and 
). In this case, the problem reduces to calculus.
Possibly relevant posts:
- Stochastic Lyapunov functions (5/6/2007)
- Solution to probability of matrix invertibility (5/18/2006)
- Expected maxima of subsequences (8/4/2008)