# William S. Wong

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 Oct2 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. Oct2 revised Is random variable $X_i$ measurable on ${\mathcal F_{i+1}}$ or ${\mathcal F_{i-1}}$? added 47 characters in body Oct2 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 $\mathcal{F}_{i+1}$. Oct2 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." Oct2 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". Oct2 awarded Commentator Oct2 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. Oct2 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. Oct2 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. Oct2 revised Find the conditional expectation $\mathbb{E}[X_2|\mathcal{F}_1]$ edited body Oct2 revised Is random variable $X_i$ measurable on ${\mathcal F_{i+1}}$ or ${\mathcal F_{i-1}}$? added 25 characters in body Oct2 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. Oct2 revised Find the conditional expectation $\mathbb{E}[X_2|\mathcal{F}_1]$ deleted 54 characters in body Oct2 revised Is random variable $X_i$ measurable on ${\mathcal F_{i+1}}$ or ${\mathcal F_{i-1}}$? added 32 characters in body Oct2 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? Oct2 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? Oct2 revised Is random variable $X_i$ measurable on ${\mathcal F_{i+1}}$ or ${\mathcal F_{i-1}}$? added 3 characters in body Oct2 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 Oct2 suggested suggested edit on Is random variable $X_i$ measurable on ${\mathcal F_{i+1}}$ or ${\mathcal F_{i-1}}$? Oct2 answered Is random variable $X_i$ measurable on ${\mathcal F_{i+1}}$ or ${\mathcal F_{i-1}}$?