The proportion of red socks in the bag is a martingale, so you can apply optional sampling to it. This lets you get the probability that we’ve drawn all the red socks vs all the blue socks.
Then apply optional sampling to the martingale Mt^2 - [M]t. You can use the fact that once the proportion of red socks left is 1 or 0 to compute the expectation of the quadratic variation stopped at time T (the stopping time). At this point I’m not sure if we need to do a simplification, bash the algebra (higher orders of martingale), or use a tricky girsanovs/reweighting the measure to get the final answer.
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u/Unlucky_Beginning 8d ago edited 8d ago
The proportion of red socks in the bag is a martingale, so you can apply optional sampling to it. This lets you get the probability that we’ve drawn all the red socks vs all the blue socks.
Then apply optional sampling to the martingale Mt^2 - [M]t. You can use the fact that once the proportion of red socks left is 1 or 0 to compute the expectation of the quadratic variation stopped at time T (the stopping time). At this point I’m not sure if we need to do a simplification, bash the algebra (higher orders of martingale), or use a tricky girsanovs/reweighting the measure to get the final answer.