We will prove the following fact arising from finance.

- American call and European call with the same strike and maturity has the same price given that their underlying assets are the same non-dividend stock.

Of course, this fact shall not be generalized to put or some other scenarios.

We denote

- the stock price by random process ,
- and the strike by ,
- the maturity by ,
- interest rate by .

The mathematical counterpart of the claim, which we want to prove below, is that

We set . Because

- is convex;
- is a martingale;

we have

being a submartingale. This implies that

which in turn gives that

for any . The other direction of the inequality is obvious from operator.

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