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Searching the web I've found some documents and among them I've stumbled across this one.

Brownian Motion by Peter Mörters (University of Bath).

It supposes as always that the brownian motion is continuous... But looking at Theorem 1.1 page 2 (attributed to Wiener 1923) saying that "Brownian motion exists". The steps of the proof are the following:

1. Construct a discreet version of the brownian motion on a diadic grid (I mean, by successive refinement, dividing every time interval in two equal parts at each iteration.
2. Consider the sequence of piecewise linear function obtained at each stage by linear interpolation
3. Prove almost sure uniform convergence

This seems totally satisfactory to me. Piecewise linear functions are continuous, uniform limit of continuous functions are continuous...

Anyone has an idea why on earth one has to suppose first that brownian motion is continuous ?

Am I missing something ?

[Edit : Ok, I now understand this is not a proof of continuity. I now understand that this is a proof of existence (non-contradiction), hence the name of the theorem. I doesn't prove that any stochastic process verifying the three axioms is indeed continuous. But still, such proof must exists.]

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Searching the web I've found some documents and among them I've stumbled across this one.

Brownian Motion by Peter Mörters (University of Bath).

It supposes as always that the brownian motion is continuous... But looking at Theorem 1.1 page 2 (attributed to Wiener 1923) saying that "Brownian motion exists". The steps of the proof are the following:

1. Construct a discreet version of the brownian motion on a diadic grid (I mean, by successive refinement, dividing every time interval in two equal parts at each iteration.
2. Consider the sequence of piecewise linear function obtained at each stage by linear interpolation
3. Prove almost sure uniform convergence

This seems totally satisfactory to me. Piecewise linear functions are continuous, uniform limit of continuous functions are continuous...

Anyone has an idea why on earth one has to suppose first that brownian motion is continuous ?

Am I missing something ?