@mathdude,
 bearish's iid argument is rigorous as long as you interpret "returns" as [$]x_t = \log S_t/S_{t-1}[$], and the variance of [$]x_t[$] exists.

Frank Bruni -- Liberal Censorship -- Bill Maher show

plus, another good clip from his show, on the same topic

Jordan Peterson -- Bill Maher show

Speaking of free speech and porn: 
Why Every Member of Congress Gets a Monthly Porn Delivery

Thanks, Alan.  Mulling it over. :D What comes immediately to mind is ppauper, actually, and the photo he liked to post whenever writing about AOC, e.g.,: ppauper:  the "democratic socialists" are fighting amongst themselves. Sandy Ocasio wants to tax the rich at 70% https://cdn.cnn.com/cnnnext/dam/...

There's no accounting for taste. I'd like to hear trackstar's vote: AOC or:

My campaign is promoting her to honorary super-model. If elected, we'll throw in a Medal of Freedom: The Presidential Medal of Freedom is awarded by the president of the United States "for especially meritorious contribution to (1) the security or national interests of the United States, or (2) worl...

Unrelated, but if they're going to subpoena John Bolton, they have to subpoena Hope Hicks. Why? Oh come on :D ... https://i.4pcdn.org/pol/1519870931505.png       I have some news that will make both of you happy.   Hope Hicks expected to return to White House - CNN Feb 13 "Long seen as a stabilizin...

Yes, what you are describing is a jump-diffusion. There is an Ito's lemma, a generator, an evolution PIDE, etc, etc. There are many books discussing such processes. Cont and Tankov is good. So is "Option Valuation under Stochastic Volatility II"  

https://en.wikipedia.org/wiki/Burgers%27_equation It would be interesting if Alan and Paul can produce explicit solution for (3A)/(3B) [$]u(x,t) = (ax + b)/(at + 1)[$] for the initial condition [$]u(x,0) = ax + b.[$] I said earlier I don't know much about shocks. But, reading that Wikipedia link, i...

I demand a recount! Anyway, it's not even 6am -- don't you people sleep?

The named function just happens to be the answer! Exam question 1: Find the solution of ... Answer: Define [$]\mbox{Lewis}_1[$] as the answer to question 1. Trivially and wlog the answer to question 1 is [$]\mbox{Lewis}_1[$]. Exam question 2: Etc. !!! Well, I'm glad the thread title is called "Sill...

I think it's just recognizing a "named function". So, take n=6  In Mathematica, what I posted evaluates to [$]1764 - 720 \, \gamma \approx 1348.4[$], using Euler's constant [$]\gamma[$].   Since [$]1348.4 \not= 6[$] that answer's Daniel's question. If it was truly circular, I don't think one could g...

Via gamma fn, looks messy!

That works. 
Since [$]\frac{d}{dz} \log \Gamma[z] = \psi(z)[$], the Digamma function (A&S, 6.3.1), then 

[$] \frac{dn!}{dn} = \psi(n+1) \, n![$], where of course [$]n![$] is interpreted everywhere as [$]\Gamma(n+1)[$]. 

That's very good and clear, Alan. I get the same solution as your integral approach. BTW do you use some kind of Leibniz' rule for this? I might have missed something along the way. Thanks. No Leibniz rule needed for what I wrote; you solve [$]\frac{dX}{dt} = b(X)[$] simply by writing it as [$]\fra...

Let me switch to (mostly) book notation for a moment. My main goal in Appendix 1.2 is to make the standard PDE-SDE probability connection for parabolic PDE's on the real line (so no bc). So, for each PDE treated, one finds the (formal) probabilistic solution: run an SDE and take an expectation. Effe...