One can adapt the argument used to show that, for a standard Brownian motion $W$, the laws of $W$ and $\sigma W$ on any interval $[0,t]$ with $t > 0$ and $\sigma^2\ne1$ are singular.
For every positive $u$, v$, let$E_u$E_v$ denote the space of $C^1$ real valued functions defined on $[0,t]$ such that the quadratic variation of their first derivative on $[0,t]$ exists and equals $u$. v$. Let$X=(X_s)_{0\le s\le t}$with$X_s=\displaystyle\int_0^sW_u\mathrm{d}u$. Then$[X\in E_{t}]$and$[\sigma X\in E_{\sigma^2t}]$are both almost sure events but$E_t$and$E_{\sigma^2t}$are disjoint hence the laws of$X$and$\sigma X$are singular. 2 added 15 characters in body One could try to can adapt the argument used to show that, for a standard Brownian motion$W$, the laws of$W$and$\sigma W$on any interval$[0,t]$with$t > 0$and$\sigma^2\ne1$are singular. For every positive$v$, u$, let $E_v$ E_u$denote the space of$C^1$real valued functions defined on$[0,t]$such that the quadratic variation of their first derivative on$[0,t]$exists and equals$v$. u$. Let $X$ denote the integral of $W$, indexed by X=(X_s)_{0\le s\le t}$with$[0,t]$. X_s=\displaystyle\int_0^sW_u\mathrm{d}u$. Then $[X\in E_{t}]$ and $[\sigma X\in E_{\sigma^2t}]$ are both almost sure events but $E_t$ and $E_{s}$ E_{\sigma^2t}$are disjoint for$t\ne s$hence the laws of$X$and$\sigma X$are singular. 1 One could try to adapt the argument used to show that, for a standard Brownian motion$W$, the laws of$W$and$\sigma W$on any interval$[0,t]$with$t > 0$and$\sigma^2\ne1$are singular. For every positive$v$, let$E_v$denote the space of$C^1$real valued functions defined on$[0,t]$such that the quadratic variation of their first derivative on$[0,t]$exists and equals$v$. Let$X$denote the integral of$W$, indexed by$[0,t]$. Then$[X\in E_{t}]$and$[\sigma X\in E_{\sigma^2t}]$are almost sure events but$E_t$and$E_{s}$are disjoint for$t\ne s$hence the laws of$X$and$\sigma X\$ are singular.