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I was reading Oksendal's book which is Stochastic Differential Equations An Introduction with Applications. In Chapter 3, he introduces the concept of SDE by adding noise term to the coefficient. We can think noise term as a stochastic process such that:

$$\frac{dX_t}{dt}=b(t, X_t)+\sigma(t, X_t) W_t$$

where $W_t$ is any stochastic process (not white noise for now).

But for me, things are getting complicated here. He says;

Based on many situations, for example in engineering, one is led to assume that $W_t$ has, at least approximately, these properties:

(i) for $t_1 \neq t_2$, $W_{t_1}$ and $W_{t_2}$ are independent.

(ii) $W_t$ is stationary, i.e. the (joint) distribution of $\{W_{{t_1}+t},...,W_{{t_k}+t}\}$ does not depend on $t$.

(iii) ${\mathbb{E}}\left[W_t\right]=0$ for all $t$.

Here is my question: Why are we assuming that stochastic process $W_t$ has to satisfy these properties? Is this the only way to define SDEs or is there any other way? I also upload an image about related part.

enter image description here

Thanks in advance.

Yağız Berk Özdemir
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    The author says that these assumptions commonly hold in practice. When we make them certain calculations are simplified. You could certainly make other assumptions but you'd end up with a different theory. You seem to understand all of this, so what exactly is your question? – Brian Borchers Aug 31 '19 at 18:10
  • @BrianBorchers First of all, thanks for your answer. I am new to the topic of SDE. I really don't know why are we making these assumptions. Just to simplify our calculations? Or is there any other reason? Can't I use, for example, a stochastic process which has non-zero mean, is not stationary and independent? Is there any other way to define an SDE? Does using Wiener Process or making these assumptions is just one way to define them? Or is it the one and only way? Thanks again. – Yağız Berk Özdemir Aug 31 '19 at 18:29
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    Related question. It is possible to consider SDEs for a much larger class of processes than just processes with stationary and independent increments. – saz Aug 31 '19 at 19:54
  • @saz Thank you for your answer. I guess that we are using Wiener Process since it is easy to work with mathematically. Can I say that since any other process hasn't studied as much as Wiener Process, SDEs driven by Wiener Process are more general? – Yağız Berk Özdemir Aug 31 '19 at 21:35
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    @YağızBerkÖzdemir No, that's nor correct...SDEs driven by Wiener processes are not more general but less general. There is a deeper theory of SDEs which covers, as a particular case, SDE driven by Wiener processes. – saz Sep 01 '19 at 06:30

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