I'm reading Shreve Stochastic Calculus II, theorem 5.4.9 (Second fundamental theorem of asset pricing),
This is the part that confuses me :
suppose there is only one risk-neutral measure. This means first of all that the filtration for the model is generated by the $d$-dimensional Brownian motion driving the assets. If that were not the case (i.e., if there were other sources of uncertainty in the model besides the driving Brownian motions), then we could assign arbitrary probabilities to those sources of uncertainty without changing the distributions of the driving Brownian motions and hence without changing the distributions of the assets. This would permit us to create multiple risk-neutral measures
This seems to be a proof by contradiction of the statement: only one risk-neutral mesure $\implies$ the driving Brownian motions are the only source of uncertainty.
I think the example of market model is the GBM: $$dS_i(t) = \alpha_i(t)S_i(t)dt+\sigma_i(t)S_i(t)dW_i(t) \quad i=1,...,d$$ Here the driving Brownian motion seems to refer to $W_i(t), i=1,...,d$, so I wonder what could be "other sources of uncertainty in the model besides the driving Brownian motions", could they be $\alpha_i(t) $ or $\sigma_i(t) $ or some $dB_i(t) $ for some random variable B added to the model?
Then the writer says we could "assign arbitrary probabilities to those sources of uncertainty", does it refer to the risk neutral probability measure we construct? But that measure only acts on the (preimage of ) random stock price, we cannot assign probabilities to any uncertainty...
I think "to create multiple risk-neutral measures", is related to the definition of risk neutral probability measure namely (i) It must be equivalent to the original probability measure and (ii) it must make the discounted stock price process a martingale
Anyone understand what the writer is saying?