The Joint Law of the Times of the Maxima of Correlated Brownian Motions

Let $W_1$ and $W_2$ be standard Brownian motions on $[0, 1]$ with correlation $\rho$, and $\theta_1$, $\theta_2$ the times of their maxima. Rogers and Shepp found the correlation of the two maximum values; we study the two times. As one planar path, the pair stays in the wedge with apex $(M_1, M_2)$ and angle $\alpha = \arccos(-\rho)$ and touches its two edges at $\theta_1$ and $\theta_2$. Cutting the path at the touches gives the joint density of $(\theta_1, \theta_2)$ as a double integral of two Bessel series of the wedge heat kernel, and the probability that both maxima occur before $t$ exactly through two independent copies of $(M_1, M_2)$. Near the diagonal the density behaves like $|t – s|^{\pi/\alpha – 2}$: the times repel for $\rho < 0$ and attract for $\rho > 0$, the argmax counterpart of a dichotomy for first-passage times. The correlation of the times is $\sum_n c_n \rho^n$ with $c_n \ge 0$, $\sum_n c_n = 1$ and $\sum_n (-1)^n c_n = (6 – 7\zeta(3))/4$, the value of Mori, Majumdar and Schehr at $\rho = -1$, and exactly $c_1 = 4 – 32/(3\pi)$, which gives two-sided bounds; the correlation itself has no closed form. At $\alpha = \pi/n$ the kernels are finite image sums. For correlated bridges $\theta_1$ is uniform and independent of the cyclic gap, whose density is elementary at $\alpha = \pi/n$. Quadrature, $12$ million simulated pairs, several motions and four laws of random-walk steps support and extend the results.