Why Linear SDE Sample Paths Remain Nondifferentiable

Summary

The document asks whether solutions to a linear stochastic differential equation become differentiable when Brownian noise is filtered through the system’s dynamics. The question contrasts Brownian motion’s irregular paths with the intuition that a colored process might be smoother after filtering. It gives a linear equation with drift and Brownian forcing, along with a proposed solution representation, to frame the issue.

The answer argues informally that an Itô process is not differentiable, pointing to the variance of a Brownian increment divided by elapsed time as time approaches zero. This suggests that filtering in the stated dynamics does not remove the path irregularity induced by Brownian noise. However, the response is brief and explicitly non-rigorous; it does not carefully analyze the displayed solution or distinguish path regularity from properties such as continuity or mean-square differentiability. It is a conceptual prompt and partial explanation, rather than a full proof or a trading method.

Key ideas

• The question concerns whether linear dynamics smooth the sample paths of Brownian-driven processes.
• The response attributes nondifferentiability to the increasingly volatile behavior of Brownian increments over short intervals.
• A stochastic integral can produce a continuous process without making its sample paths differentiable.
• The supplied answer is informal and does not provide a rigorous proof for the displayed equation.

Tags

# Differentiability of solutions of a stochastic differential equation


# 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.

## Answer by Preston Lui (score 1)

https://quant.stackexchange.com/a/60182

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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