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## Sunday, February 15, 2015

### optional stopping theorem, my take

label - stoch

Background -- There's no way to beat a fair game. Your winning always has an expected value of 0, because winning is a martingale, i.e. expected future value for a future time is equal to the last revealed value.

Now, what if there's a stopping time i.e, a strategy to win and end the game? Is the winning at that time still a martingale? If it's not, then we found a way to beat a fair game.

For a Simple Random Walk (coin flip) with upper/lower bounds, answer is intuitively yes, it's a martingale.

For a simple random walk with only an upper stopping bound (say \$1), answer is -- At the stopping time, the winning is the target level of \$1, so the expected winning is also \$1, which is Not the starting value of \$0, so not a martingale! Not limited to the martingale betting strategy. So have we found a way to beat the martingale? Well, no.

"There's no way to beat a martingale in __Finite__ time"

You can beat the martingale but it may take forever. Even worse (a stronger statement), the expected time to beat the martingale and walk away with \$1 is infinity.

The OST has various conditions and assumptions. The Martingale Betting Strategy violates all of them.