# Maximise the probability that a drifted Brownian motion doesn't hit zero prior to $T$

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.

The process you are describing is simply bigger than all the other: by the condition on $$\alpha$$ $$Z + W_t + \int^t \alpha_s ds \le Z + W_t + C$$