I have a couple of technical questions that, after searching the internet for hours, I have not been able to find an answer to. Mathematica's online instructions are not even addressing the issue at all for what I was able to find.
Question 1. Is TrasformedProcess assentially the same as using Ito-Doeblin formula? I suspect that it should be, because I tried the following experiment related to GeometricBrownianMotion and defined it in 3 different ways:
x0 =1;
\[Mu] = 1;
\[Sigma] = 2;
proc1 = GeometricBrownianMotionProcess[\[Mu], \[Sigma], x0];
proc2 = ItoProcess[\[DifferentialD]x[t] == \[Mu]*
x[t] \[DifferentialD]t + \[Sigma]*x[t] \[DifferentialD]w[t],
x[t], {x, x0}, t, w \[Distributed] WienerProcess[]];
proc3 = TransformedProcess[
Exp[\[Sigma]*x[t] + (\[Mu] - \[Sigma]^2/2)*t],
x \[Distributed] WienerProcess[0, 1], t];
I verified that the three processes have the same mean and variance. However, my question stems from the fact that, even using SeedRandom, they produce different paths. Similar, but different.
Using proc3 is the same as defining the geometric Brownian motion in proc1 and proc2, if Ito-Doeblin formula is used.
Question 2. When using TransformedProcess, or even just ItoProcess, is there a way to get a nice version of the formula, something that could be transported to Latex using TeXForm? For instance, say that I enter the following
proc = ItoProcess[{-x[t], Sqrt[1 + x[t]^2]}, x, t];
Any chance to get an output that looks like $$ dX_t = -X_t\, dt + \sqrt{1+X_t^2}\, dW_t~ $$
after using TeXForm or something that resembles it? It is not a big deal with ItoProcess. But with TransformedProcess, to me it is. Sure, if I just want to do simulations, I can go ahead and be ok. But any chance to see the SDE/diffusion for the transformed process instead of just seeing?
TransformedProcess[E^(\[FormalT] (\[Mu] - \[Sigma]^2/
2) + \[Sigma] \[FormalP][\[FormalT]]), \[FormalP] \[Distributed]
WienerProcess[0, 1], \[FormalT]]
Thank you.
