Let $W$ be a Brownian motion and $\alpha, \beta$ be two progressively measurable processes taking values in $\mathbb R_+$ s.t. $\alpha_t\le \beta_t$ for all $t\ge 0$. Define respectively $X, Y$ by

$$X_t:=\int_0^t \alpha_s dW_s,\quad Y_t:=\int_0^t \beta_s dW_s,\quad \forall t\ge 0.$$

Can we find a probability space on which one has a Brownian motion $B$ and two stochastic processes $\tau, \sigma$ s.t. $0\le\tau_t\le \sigma_t$ for all $t\ge 0$ and

$$Law(X)=Law\big(B_{\tau}:=(B_{\tau_t})_{t\ge 0}\big),\quad Law(Y)=Law\big(B_{\sigma}:=(B_{\sigma_t})_{t\ge 0}\big)?$$

PS : Here we may assume any integrability conditions for $\alpha,\beta$ if needed.

Let $W$ be a Brownian motion and $\alpha$, $\beta$ be two progressively measurable processes taking values in $\mathbb R_+$ s.t. $\alpha_t\le \beta_t$ for all $t\ge 0$. Define respectively $X$, $Y$ by

$$X_t:=\int_0^t \alpha_s dW_s,\quad Y_t:=\int_0^t \beta_s dW_s,\quad \forall t\ge 0.$$

Can we find a probability space on which one has a Brownian motion $B$ and two stochastic processes $\tau, \sigma$ s.t. $0\le\tau_t\le \sigma_t$ for all $t\ge 0$ and

$$\DeclareMathOperator\Law{Law}\Law(X)=\Law\bigl(B_{\tau}:=(B_{\tau_t})_{t\ge 0}\bigr),\quad\Law(Y)=\Law\bigl(B_{\sigma}:=(B_{\sigma_t})_{t\ge 0}\bigr)?$$

PS : Here we may assume any integrability conditions for $\alpha$, $\beta$ if needed.