I just bumped into the stochastic integral
$$
\int_\rho^1 (W_t - W_{t-\rho}) \,dW_t
$$
where $0 < \rho < 1$ is a constant and $W$ is a standard Wiener process. It would be nice if we have a closed-form representation of the cumulative distribution function, but for my work it suffices to show that **there is some simple transform to make its law $\rho$-invariant**, that is, to make its law not dependent on $\rho$.

Does it appear possible to any one? Thanks for any help in advance!

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**Additional Information**: I guessed so because when $\rho$ is large enough, $\int_\rho^1 (W_t - W_{t-\rho})\,dW_t \approx W_\rho (W_1 - W_\rho)$. I tried to multiply it by $\frac{1}{\sqrt{\rho(1-\rho)}}$, but in some simple Monte Carlo experiments the distribution of the stochastic integral doesn't seem to be $\rho$-invariant.