Compensator that makes t times a Wiener integral a martingale

Question

For a standard Brownian motion $W$ , let $Y_t=\int_0^t s\,dW_s$ and $Z_t=tY_t-a_t$ , where $a_t=k\int_0^t Y_s\,ds$ is chosen so that $Z$ is a martingale. Determine $k$ .

Short answer

$k=1$ . By the product rule, $d(tY_t)=Y_t\,dt+t\,dY_t=Y_t\,dt+t^2\,dW_t$ . The $Y_t\,dt$ drift is exactly cancelled by $da_t=kY_t\,dt$ when $k=1$ .

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Lessons behind this question

Itô's Formula
The Optional Stopping Theorem
Discrete-Time Martingales

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