Today we will look at Skorokhod embedding theorem. We consider a simple symmetric random walk where the probability for an event with value v is 1/2 and that against is 1/2. When we sample, we want the probability that the gambler's fortune will reach level b > x before reaching level a < x is given by (x-a)/(b-a). When we start the random walk from S0 = 0, the probability will reach level b > x before reaching level a < x equals -a / (b-a) and the probability that it will reach level a < x before b > x is b / (b-a).
These same probabilities can now be read as simple symmetric random walks with distribution as the range between a and b to reach b and a respectively.
We view the random walk this way. Given a 2 point probability distribution (p, 1-p) such that p is rational, we can always find integers a,b a < 0 < b such that pa + (1-p)b = 0.
This means we can simulate the distribution by generating a simple symmetric random walk starting from 0 and watching it till the first time it exits the interval [a,b] The probability it exits from the left point is p and the the probability it exits from the right point is 1-p. Having introduced the theorem, lets look at some more complicated probability distribution.
If the probabilities were to be many in a distribution, can we find a stopping time T such that the random walk has the given distribution ? This is where Skorokhod embedding theorem helps.
This theorem states that for a simple symmetric random walk and (pi, i belongs to Z) a given probability distribution on Z with zero mean which means satisfying the equation sum of i.pi = 0, then there exits a stopping time T where the probability for random walk to arrive at i = pi
These same probabilities can now be read as simple symmetric random walks with distribution as the range between a and b to reach b and a respectively.
We view the random walk this way. Given a 2 point probability distribution (p, 1-p) such that p is rational, we can always find integers a,b a < 0 < b such that pa + (1-p)b = 0.
This means we can simulate the distribution by generating a simple symmetric random walk starting from 0 and watching it till the first time it exits the interval [a,b] The probability it exits from the left point is p and the the probability it exits from the right point is 1-p. Having introduced the theorem, lets look at some more complicated probability distribution.
If the probabilities were to be many in a distribution, can we find a stopping time T such that the random walk has the given distribution ? This is where Skorokhod embedding theorem helps.
This theorem states that for a simple symmetric random walk and (pi, i belongs to Z) a given probability distribution on Z with zero mean which means satisfying the equation sum of i.pi = 0, then there exits a stopping time T where the probability for random walk to arrive at i = pi
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