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Oct
10
comment Why can I exchange the order of integration in a multiple Ito stochastic integral?
Mark: I think I know what the problem is. You simply cannot say $W_s = s^2$ since $W_s$ is Brownian motion.
Oct
10
comment Why can I exchange the order of integration in a multiple Ito stochastic integral?
Mark: $dW_s \sim \mathcal{N}(0, ds)$ is a random variable, while $ds$ is a deterministic quantity. For example, $\text{Var}(ds) = 0$.
Oct
10
comment Why can I exchange the order of integration in a multiple Ito stochastic integral?
How can $dW_s = 2s \, ds$? By definition $dW_s \equiv W_{s+ds} - W_s$.
Oct
3
comment Is random variable $X_i$ measurable on ${\mathcal F_{i+1}}$ or ${\mathcal F_{i-1}}$?
Thanks for pointing out my mistakes. I removed the erroneous part of the answer.
Oct
2
comment Find the conditional expectation $\mathbb{E}[X_2|\mathcal{F}_1]$
Did: the question says "$X_n=2*X_{n-1}$ with prob = $3/4$". So given $X_1$, the probability of $X_2 = 2 X_1$ is $3/4$.
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
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
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
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
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
comment Find the conditional expectation $\mathbb{E}[X_2|\mathcal{F}_1]$
Did: let me think about the argument some more, but do you agree that $\mathbb{E}[X_2 | \mathcal{F}_1] = \mathbb{E}[X_2 | X_1] = \frac{13}{8} X_1$?
Dec
11
comment Backward PDE for a mean-reverting stochastic process
I don't think you can write a PDE on $V$ since $V = V(x)$ only.
Dec
11
comment Backward PDE for a mean-reverting stochastic process
Thanks for down-voting my question -) Don't worry -- I will answer my own question in a few days.
Dec
11
comment Wiener Process $dB^2=dt$
I am aware of a fairly rigorous analysis that shows a term containing $dB_t^2$ in Ito's lemma, for example, converges to one that contains $dt$ almost surely, using the Borel-Cantelli lemma. See pages 4 to 6 in the lecture math.nyu.edu/faculty/goodman/teaching/StochCalc2012/notes/… .