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Oct
2
comment Is random variable $X_i$ measurable on ${\mathcal F_{i+1}}$ or ${\mathcal F_{i-1}}$?
Peter: look at Shreve, vol. II, p. 66: "If $X$ is $\mathcal{G}$-measurable, then the information in $\mathcal{G}$ is sufficient to determine the value of $X$." In our case, since $X_i$ is $\mathcal{F}_{i+1}$-measurable, $\mathbb{E}[X_i | \mathcal{F}_{i+1}] = x_i$ is just a known constant. Think of $i$ as time -- the expectation of Google's share price yesterday, given a filtration up to today, is a deterministic number.
Oct
2
revised Is random variable $X_i$ measurable on ${\mathcal F_{i+1}}$ or ${\mathcal F_{i-1}}$?
added 47 characters in body
Oct
2
comment Is random variable $X_i$ measurable on ${\mathcal F_{i+1}}$ or ${\mathcal F_{i-1}}$?
Did: you wrote in the 1st comment that $\mathbb{E}[X_i | \mathcal{F}_{i+1}]$ is also a random variable, but I disagree. $\mathbb{E}[X_i | \mathcal{F}_{i+1}] = x_i$ is a known constant, since we know the value of $X_i$ from $\mathcal{F}_{i+1}$.
Oct
2
comment Find the conditional expectation $\mathbb{E}[X_2|\mathcal{F}_1]$
Did: my solution got around the issue you wrote about in yours that "$Y_n$ is independent of $\mathcal{F}_{n−1}$...Or not."
Oct
2
comment Find the conditional expectation $\mathbb{E}[X_2|\mathcal{F}_1]$
Did: I was trying to figure out what you meant by "except in the discrete case".
Oct
2
awarded  Commentator
Oct
2
comment Find the conditional expectation $\mathbb{E}[X_2|\mathcal{F}_1]$
Did: using my example above, I can still compute what $\mathbb{E}[X_2 | X_1 = 1]$ is, which evaluates to 13/8.
Oct
2
comment Find the conditional expectation $\mathbb{E}[X_2|\mathcal{F}_1]$
Did: referring to the original problem, you are saying that we cannot say what $\mathbb{E}[X_2 | X_1 = 1]$ is because the quantity is not well defined? I disagree. Given that $X_1 = 1$, $X_2$ is either 2 (with probability 3/4) or 0.5.
Oct
2
comment Find the conditional expectation $\mathbb{E}[X_2|\mathcal{F}_1]$
Did: is the original problem discrete? The $n$ in $X_n$ are integers.
Oct
2
revised Find the conditional expectation $\mathbb{E}[X_2|\mathcal{F}_1]$
edited body
Oct
2
revised Is random variable $X_i$ measurable on ${\mathcal F_{i+1}}$ or ${\mathcal F_{i-1}}$?
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Oct
2
comment Find the conditional expectation $\mathbb{E}[X_2|\mathcal{F}_1]$
Did: You are right about the problem with my original argument. I got rid of it in my answer.
Oct
2
revised Find the conditional expectation $\mathbb{E}[X_2|\mathcal{F}_1]$
deleted 54 characters in body
Oct
2
revised Is random variable $X_i$ measurable on ${\mathcal F_{i+1}}$ or ${\mathcal F_{i-1}}$?
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Oct
2
comment Is random variable $X_i$ measurable on ${\mathcal F_{i+1}}$ or ${\mathcal F_{i-1}}$?
Did: I disagree; let me write it more precisely -- is $\mathbb{E}[X_i | X_i = x_i] = x_i$ a random variable?
Oct
2
comment Is random variable $X_i$ measurable on ${\mathcal F_{i+1}}$ or ${\mathcal F_{i-1}}$?
Did: $\mathbb{E}[X_i | \mathcal{F}_{i+1}] = X_i$ is not a random variable, sinc $X_i$ is known. Is $\mathbb{E}[X_i | X_i] = X_i$ a random variable?
Oct
2
revised Is random variable $X_i$ measurable on ${\mathcal F_{i+1}}$ or ${\mathcal F_{i-1}}$?
added 3 characters in body
Oct
2
revised Is random variable $X_i$ measurable on ${\mathcal F_{i+1}}$ or ${\mathcal F_{i-1}}$?
Don't need to italicize the word "or" in the title
Oct
2
suggested suggested edit on Is random variable $X_i$ measurable on ${\mathcal F_{i+1}}$ or ${\mathcal F_{i-1}}$?
Oct
2
answered Is random variable $X_i$ measurable on ${\mathcal F_{i+1}}$ or ${\mathcal F_{i-1}}$?