Let $W=(W_t)_{t\ge 0}$ be a standard Brownian motion starting from zero and $Z>0$ be an independent random variable. Fix $T>0$ and $C>0$. Denote by $\mathcal A$ the set of progressively measurable processes $\alpha=(\alpha_t)_{t\ge 0}$ such that

$$\alpha_t\ge 0 \quad \mbox{and}\quad \int_0^T\alpha_s ds \le C.$$

Define further for every $\alpha\in \mathcal A$

$$X^\alpha_t:=Z + W_t + \int_0^t \alpha_s ds, \quad \forall t\ge 0.$$

What is the optimiser for the optimisation problem

$$\sup_{\alpha\in\mathcal A} \mathbb P\big[\inf_{0\le t\le T}X^\alpha_t>0\big]?$$

My guess is to allocate the whole drift $C$ at time zero, i.e. $X^*_t:=Z+C+W_t$ (of course this strategy can be approximated by a suitable sequence $(\alpha^n)_{n\ge 1}\subset \mathcal A$), while I don't know how to prove it.

Any answer or comments are highly appreciated.