Let Z t be a bounded super martingale with continuous paths that is constant from some time t0 on. Show that for each t 0   We mentioned that one can prove the existence of M without using the...

Let Z_t
be a bounded super martingale with continuous paths that is constant from some time t0 on. Show that for each t₀

We mentioned that one can prove the existence of M without using the Doob–Meyer theorem. Here is how that argument starts. Let M be a bounded continuous martingale and for each n, define

Here [x] is the integer part of x. Prove that for each

One can then define

Chapter 10