I would like to clarify a confusion I have.

It is well known that a Wiener process (Brownian motion) is nowhere differentiable. I have no difficulty in understanding that. But I am wondering about the solutions of stochastic differential equations (SDEs). For simplicity, suppose that we have a linear SDE $$ dX(t) = A\, dt + B\, dW(t) $$ with initial value $X(t_0) = X_0$, and where $W(t)$ is a standard Wiener process. Then the solution is given by $$ X(t) = \exp(A(t-t_0)) X_0 + \int_{t_0}^t \exp(A(t-\tau)) B \, dW(\tau). $$

I am wondering about the properties of this solution, which is a colored stochastic process. In particular, are the sample paths of $X(t)$ differentiable in $t$?

The source of my confusion is coming from the interpretation of white noise as a formal derivative of Wiener process, and that the nondifferentiability is due to the constant spectrum of white noise. But now $X$ is filtered noise, i.e. it is colored. So I would expect that the irregular behaviour of the noise is smoothened out by the dynamics of the linear system described by the drift part of the equation.


Rather non rigorously,

$\frac{W(t)}{t} \sim N(0,\frac{1}{t}) $

if $t \to 0$ , we can see the variance goes to infinity. Hence Ito process is not differentiable.


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