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1d
comment Does Brownian Motion return to the origin infinitely soon?
You realize this sounds awfully phony, do you? (Unrelated: Please use @, not 2.)
1d
comment Girsanov theorem
No reaction to comments solving question asked, reposting duplicates... I am not overly optimistic about the quality of your interactions with the site if you persist in this approach.
1d
comment Evaluating $\lim_{x \to\infty }\left (x^{f(x)}-x \right )$
*Read $(f(x)-1)x\ln x\to c$ and $x^{f(x)}-x\to c$. (Unrelated: Please use @.)
1d
comment $e^{2\pi i x} = (e^{2\pi i})^x$: What happens if x is rational?
How would YOU define $z^{1/2}$ when $z$ is complex?
1d
comment Find the distribution of $Z = 1/X_1 + 1/X_2$
Example in action: math.stackexchange.com/a/30966
1d
comment Find the distribution of $Z = 1/X_1 + 1/X_2$
Did you try to apply the standard procedure? Where were you stuck?
1d
comment $e^{2\pi i x} = (e^{2\pi i})^x$: What happens if x is rational?
As has been emphasized several times on the site, there is no definition of $u^v$ for general complex numbers $(u,v)$. In particular, the identity $e^{2\pi ix}=(e^{2\pi i})^x$ is quite wrong when $x$ is not an integer.
1d
comment Markov chain simulation
Got something from the answers?
1d
comment Evaluating $\lim_{x \to\infty }\left (x^{f(x)}-x \right )$
As is easily shown using the approach in the deleted answer, if $(f(x)-1)\ln x\to c$ then $x^{f(x)}-x\to e^c-1$. Of course, if $(f(x)-1)\ln x\to+\infty$ then $x^{f(x)}-x\to+\infty$. And if $(f(x)-1)\ln x$ diverges because it oscillates then so does $x^{f(x)}-x$.
1d
comment The sum of the cubes of the reciprocal values of the roots of the equation $x^2+ax+1=0$ is?
$$\frac1{x^3}+\frac1{y^3}=\frac{(x+y)^3-3(xy)(x+y)}{(xy)^3}$$
1d
comment Possible to solve $A + P^{-1}AP = B$?
$$A=\sum_{n=0}^\infty(-1)^nP^{-n}BP^n$$
1d
comment Ito's lemma - mistake in text book?
Yes, provided $\langle X\rangle_t=t$. And one can recommend to make a distinction between the process $W$ such that $W_t=1+t+e^{X_t}$ and the function $G$ such that $W_t=G(t,X_t)$, here $G:(t,x)\mapsto1+t+e^x$. Itô's formula is most easily written down using the latter, as $$dW_t=\frac{\partial G}{\partial x}(t,X_t)dX_t+\frac{\partial G}{\partial t}(t,X_t)dt+\frac12\frac{\partial^2 G}{\partial x^2}(t,X_t)d\langle X\rangle_t.$$
2d
comment Derive probability mass function from probability-generating function
$$\sum_{x=0}^{\infty}3^{-x}\frac{z}{6}z^x=\sum_{x=0}^{\infty}\frac163^{-x}z^{x+1‌​}=\sum_{x=0}^{\infty}\frac123^{-x-1}z^{x+1}=\sum_{x=1}^{\infty}\frac123^{-x}z^{x}‌​$$
2d
comment Stochastic Differential Equation for Time Integral of Stochastic Process
The approach you recall works for diffusion processes, here $(Y_t)$ is not a diffusion, it is not even Markov. Note that $dY_t=e^{at+W_t}dt$ is not of the form $dY_t=\sigma(Y_t,t)dB_t+b(t,Y_t)dt$ because $e^{at+W_t}$ is not a function of $(t,Y_t)$, due to the factor $e^{W_t}$. The same reamarks would apply to the simpler setting of the process $(Z_t)$ solving $dZ_t=W_tdt$.
2d
comment $\mathbb E[X_i\mid X_1,…,X_n]=\mathbb E[X_i]$
@echo What is the factor between $(x-w)$ and $(x-y)$?
2d
comment Prove that for all prime numbers
@AlexM. Agreed, I was mentioning the fact only because an (odd) answer is posted.
2d
comment Stochastic Differential Equation for Time Integral of Stochastic Process
Indeed, $$dY=Xdt=e^{at+W}dt.$$
2d
comment Solving an integral to solve a statistical problem
The formula $\mathbb{P}( (Y_2| Y_1 \geq q_\alpha) \leq q_\alpha)$ is ill-formed since there is no such thing as a random variable $(Y_2| Y_1 \geq q_\alpha)$. Perhaps you mean to ask about $P(Y_2\leq q_\alpha| Y_1 \geq q_\alpha)$, please explain.
2d
comment Girsanov theorem
OP: Watch out, deliberate duplicates are frowned upon on this site.
2d
comment Prove that for all prime numbers
@AlexM. ...And the question as it is written now is obviously false.