We show that the initial value problem for Hamilton-Jacobi equations with
multiplicative rough time dependence, typically stochastic, and convex
Hamiltonians satisfies finite speed of propagation. We prove that in general
the range of dependence is bounded by a multiple of the length of the
"skeleton" of the path, that is a piecewise linear path obtained by connecting
the successive extrema of the original one. When the driving path is a Brownian
motion, we prove that its skeleton has almost surely finite length. We also
discuss the optimality of the estimate
