What's the expectation abd variable of the stochstic process giveb by

What's the expectation abd variable of the stochstic process giveb by

[math] dX_t = (X_t \, (1-X_t))^r \, dW_t [/math]

?

I'm mostly interested in r equals 1/2 or 1.

Other urls found in this thread:

en.wikipedia.org/wiki/Itô's_lemma#Mathematical_formulation_of_It.C3.B4.27s_lemma
en.wikipedia.org/wiki/Fokker–Planck_equation#One_dimension
en.wikipedia.org/wiki/Itô_diffusion
axiomsofchoice.org/kfv_._note
math.stackexchange.com/questions/598811/kolmogorov-backward-equation-for-itô-diffusion
twitter.com/SFWRedditVideos

I could add that it came up in modeling a distribution that can't take values outside of [0,1]. For r=1/2 it's a variation of the stochastic term in the "CIR model" and respecta the bounds I want

bump

senpaitachi

no takers at all?

I rewards with chocolate

explain briefly the theory behind this (only studied stuff like AR MA and ARMA kind of thing)

The ARMA model looks like a recurrence scheme where each new term is the previpus one, weighted by a phi and an independent/standalone fluctuation.
A stochastic differential equation scheme has x_i being x_{i-1} plus some function (set zero in the case I ask about) of that point, plus some function weigh multiplied by a fluctuation (dW)

I have 'a' equal zero and 'b' the quadratic function of X in the OP.
I wonder how the path of X will typically go. The function is not among the basic ones discussed in the books I now looked at.

Sorry for the crappy screenshots

ok tell me if this is bullshit for r=1:
separate variables
dXt/(Xt(1-Xt)) = dWt

when integarated, you get something like ln(Xt/(Xt-1)) = integral(dWt). Is this where you say it follows a normal distribution with std equal to sqrt(t) so N(0,t)

does that make any sense to you?

Such manipulations don't get you to an exact solution as (when thinking of a discretization) dW_t is not just a small positive quantity like in calculus, but instead a random value (normally distrubuted here, pic related).
I might use your rewriting in an algorithm to solve for X_t, but the stochastic Euler method sure is a more straight forward way to go and I did that (see third pic with the graph).

It's very unlikely that this equation has an exact silution, since I think to know that less complicated equations don't have one.
However, I'm only interested in the variance (or a similar meausre for the typical drift) of X_t anyway!