I am trying to understand the connection between white noise and the Wiener process in the context of SDEs. At the beginning one starts with a differential equation including white noise $\xi_t$, e.g.,
$\frac{dX_t}{dt}=a(X_t,t)+b(X_t,t)\xi_t$.
and some initial value $X_0$. This is equivalent to the integral equation
$X_t=X_0+\int_{0}^{t}a(X_s,s)ds+\int_{0}^{t}b(X_s,s)\xi_sds$.
And now comes the step I do not understand. The second integral is equal to the Ito integral
$\int_{0}^{t}b(X_s,s)dW_s$
where $\{W_t\}_{t\geq0}$ is a Wiener process.
Why is that? How can one explain this in a mathematical rigorous way? Many textbooks just argue that the white noise is some kind of derivative of the Wiener process, i.e., $"\frac{dW}{dt}=\xi_t"$ but do not go into more detail. Why can we "replace $\xi_tdt$ by $dW_t$"?
For reference, here is the definition of a white noise process I am working with:
A white noise process is defined to be a generalized wide-sense stationary Gaussian process $Z_t$ with mean zero and covariance function $E[Z_sZ_t]=\delta_0(t-s)$. Here $\delta_0$ is the Dirac Delta function at 0.
Best Answer
As I remember, a stationary process is a white noise if
One should remember the property of a white noise. If the covariance function equals to $${K_\xi}_t (\tau)=2\pi c \delta(\tau)$$ then $\xi_t$ is a white noise. In order to prove it one should use the definition given above and remember the property of the Dirac delta function: $$\int_{-\infty}^{+\infty}\ \delta(s) e^{-i\nu s }ds =1. $$ Now let's consider the derivative of the Wiener process $W_t$. It's covariance function equals to ${K_W}_t(t_1,t_2)=\sigma ^2 min(t_1,t_2)$. It's easy to see that the usual derivative of the Wiener process doesn't exist.
$$\frac{\partial {K_W}_t}{\partial t_1}=\left\lbrace \begin{matrix} \sigma^2,\, t_1<t_2\\ 0, \,\,t1>t_2 &\\ \end{matrix}\right. $$ But the distribution exists $$\frac{\partial^2 {K_W}_t}{\partial t_1 \partial t_2}= \sigma^2 \delta (t_2-t_1). $$ It follows from $E[W_t]=0$, that $E\left[\frac{d W_t}{dt}=0\right]$. The derivative of the Wiener process is also a wide-sense stationary process. We conclude that the derivative of the Wiener process is a white noise having intensity equals to $c=\frac{\sigma^2}{2\pi}$.