Stochastic Differential Equations

[vertices]

If we have a DE of the following form:

\frac{dX}{dt}=b(t,X_t)+\sigma(t,X_t).W_t

and look for a stochastic process to represent the (second) noise term. Now my textbook tells me that the only process with 'continuous paths' is Brownian motion.

The noise term denotes random, indeterministic behaviour in the physical situation the DE is modelling.

Can someone please explain why is this is the case, and why it is significant?

 

[Dickfore]

What is N? Your parentheses, starting between b and t are not closed. And what is the dot between \sigma(t. X_{t}) and noise supposed to denote?

 

[vertices]

What is N? Your parentheses, starting between b and t are not closed. And what is the dot between \sigma(t. X_{t}) and noise supposed to denote?

I do apologise for my sloppiness. I've edited the post.

Now W_t has to satisfy the following conditions:

(i) t₁ does not = t₂ => W_t1 and W_t2 are independent.
(ii) {W_t} is stationary, i.e. the (joint) distribution of {W_t1+t,...,W_tk+t} does not depend on t.
(iii) E[W_t] = 0 for all t.

Now, my textbook says that "it turns out there does not exist any "reasonable" stochastic process satisfying (i) and (ii): Such a W_t cannot have continuous paths... The only such process with continuous paths is the Brownian motion B_t." And it simply gives an obscure reference "(See Knight (1981))"...

My question is firstly, why does the DE have to satisfy those three conditions? What is a reasonable stochastic process in this context, and why can't it have a continuous path? And why does Brownian motion in particular have a continuous path?

 

[vargasjc]

I have read that Brownian movement is used in certain stochastic processes, but I need further reading to be actually able to put some math into it. My research involves a stochastic process as well, but is an area of physics that so far I haven't been able to fully digest properly. Anyone who knows more should share for the enlightenment of all who are on the same situation, I say.