Stochastic Processes – Problem 52: Determine whether the location of the maximum of on is unique

Let $(B_{t})_{t\geq0}$ be a one-dimensional standard Brownian motion. Determine whether the location of the maximum of $B_{t}+t$ on $[0,1]$ is unique.

Let $$X_t:=B_t+t,\qquad t\in[0,1],$$ where $(B_t)$ is standard Brownian motion. We show that the maximizer of $X_t$ on $[0,1]$ is almost surely unique.

Step 1. Define the event $$E:=\{\text{the argmax of }X_t\text{ on }[0,1]\text{ is unique}\}.$$ For standard Brownian motion without drift, it is classical that $$P\bigl(\text{argmax of }B_t\text{ on }[0,1]\text{ is unique}\bigr)=1.$$

Step 2. On the finite horizon $[0,1]$, the law of $X_t=B_t+t$ is absolutely continuous with respect to the law of $B_t$ (Cameron--Martin/Girsanov transform). In particular, for any measurable path event $A$, $$P_B(A)=0\iff P_X(A)=0.$$ Therefore zero-probability path events are preserved under adding deterministic drift $t$.

Step 3. Let $E^c$ be the event that the maximum is achieved at two or more distinct times. Under Brownian law, $P_B(E^c)=0$. By measure equivalence on $[0,1]$, this implies $$P_X(E^c)=0.$$ Hence $$P_X(E)=1.$$

Step 4 (equivalent geometric intuition). A tie of maxima would require a nontrivial flat-top phenomenon after subtracting a deterministic linear function. Brownian paths almost surely have no such plateau at their global maximum on a compact interval; the drift term does not create plateaus because it is smooth and strictly deterministic.

Therefore, the location of the maximum of $B_t+t$ on $[0,1]$ is almost surely unique.

Equivalent viewpoint (argmax law): the maximizer of Brownian motion with deterministic drift over a compact interval has a continuous distribution and therefore no atoms. In particular, $$P\bigl(\operatorname*{argmax}_{t\in[0,1]}(B_t+t)=u\bigr)=0\quad\text{for every }u\in[0,1],$$ which is consistent with almost-sure uniqueness of the maximizer.

Unique.