Despite my best effort, I think this write-up will have
* unclear, ambiguous points
, but first step is to write it down. This is first phase of thin->thick->thin.
Version 1: under RN measure , all traded asset  prices follows a GBM  with growth rate  similar to the riskfree money market account. The variance parameter of the GBM is unique to each asset.
Version 2: under RN measure , all traded asset  prices discounted to PV  by the riskfree money market account are martingales. In fact they are 0-drift GBM with individual volatilities.
Version 3: under RN measure , all traded asset  prices show an expected  return equal to the riskfree rate.
 many things are not really traded asset prices. See post on “so-called tradable”
 why we need to discount to present, and why “expected” return? because we are “predicting” the level of random walker /towards/ a target time later than the last revelation. The value before the revelation is “realized”, “frozen” and has no uncertainty, no volatility, no diffusion, and no bell-shaped distribution.
 no BM here. All models are GBM.
 see post on drift ^ grow rate