I need to calculate/upper bound the second moment of the variable

$t^{*} \triangleq \underset{t>\alpha}{argmax} \{W(t) - t^2\}$ where $W(t) \triangleq B(t) - B(t - \alpha), \alpha \in R^{+}$ and $B(t)$ is a standard Wiener process (Brownian motion).

Anyone has any useful idea on how to approach this problem?

I've been thinking of trying to bound the second moment by using the second moment of $\tilde{t}^{*} \triangleq \underset{t>0}{argmax} \{B(t) - t^2\}$, which intuitively I think should be higher than that of $t^{*}$ because the variance of the process $B(t)$ is growing with $t$ in contrast to that of $W(t)$ which is constant - but I couldn't formalize this claim.

Will appreciate any kind of help :)


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