# Stochastic Differential Equations

1. Aug 15, 2010

### 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?

Last edited: Aug 15, 2010
2. Aug 15, 2010

### 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?

3. Aug 15, 2010

### vertices

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

Now Wt has to satisfy the following conditions:

(i) t1 does not = t2 => Wt1 and Wt2 are independent.
(ii) {Wt} is stationary, i.e. the (joint) distribution of {Wt1+t,...,Wtk+t} does not depend on t.
(iii) E[Wt] = 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 Wt cannot have continuous paths... The only such process with continuous paths is the Brownian motion Bt." 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?

4. Sep 2, 2010

### 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.