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.



Maximise the probability that a drifted Brownian motion doesn't hit zero prior to $T$
The process you are describing is simply bigger than all the other: by the condition on $\alpha$
$$Z + W\_t + \int^t \alpha\_s ds \le Z + W\_t + C $$


