TALK by Elena YUDOVINA
### November 15, 2012

### A toy limit order book

#### Elena Yudovina (University of Michigan)

I consider a Markov process inspired by a toy model of a
limit order book. "Bid" and "ask" orders arrive in time; the prices
are iid uniform on [0,1]. (I'll discuss some extensions.) When a match
is possible (bid > ask), the highest bid and lowest ask leave the
system. This process turns out to have surprising dynamics, with three
limiting behaviours occurring with probability one. At low prices (< 0.21...), bids eventually never leave; at high prices (>0.78...), asks eventually never leave; and in between, the system "ought to" be positive
recurrent. I will show how we can derive explicitly the limiting
distribution of certain marginals for the middle prices; this makes it
possible to extract the numerical values above from a 0-1 Law result.

http://www-personal.umich.edu/~yudovina/