# Differentiability of solutions of a stochastic differential equation

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.

Quantitative Finance Asked by user144410 on February 9, 2021

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.

Answered by Preston Lui on February 9, 2021

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