show/hide this revision's text 3 edited body

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.
show/hide this revision's text 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.
show/hide this revision's text 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.