# Why does risk-neutral price processes do not, in general, compose all arbitrage-free price processes?

I was reading reviewing my mathematical finance notes and I came across a remark I cant understand fully

Remark :Contrary to discrete time models, the risk-neutral price processes do not, in general, compose all arbitrage-free price processes in the sense of Proposition 4.3.2. given below Here $$\mathbb{Q}$$ is an ELMM(A measure equivalent to $$\mathbb{P}$$ such that all discounted price processes are local martingales)

The word 'compose' is really confusing me since Prop 4.3.2 claims that any risk neutral price process given the proposition above implies the existence of an ELMM(since the discounted contingent claim $$\frac{\pi_t}{X^0_t}$$ is a martingale by construction and hence a local martingale) which implies no arbitrage. How do I make sense of the remark? For completeness i am also attaching a picture of the remark below Edit: I think I understood. What the remark means is that there exists some arbitrage free processes (as a consequence of them being local martingales under $$\mathbb{Q}$$ which implies $$\mathbb{Q}$$ is ELMM) but not a risk neutral process of the particular form above described as a conditional expextation

• Try answering the question if you think you have a decent answer for it. Someone in the future can benefit! – Slade Mar 1 '19 at 4:14
• @Slade I will write a better answer later tonight when I have more time . Thank you for the advise – user3503589 Mar 1 '19 at 9:35
• I do agree with @Slade. It is a very good question to have an answer for in here. – Sanjay Mar 3 '19 at 22:25