Independence of increments of some processes

Question

I am stuck on this question:

Let $(B_t)$ be a standard Brownian motion. Define $$ (\tau_1)_t := \inf \{s \geq 0 : B_s = t \} ; \quad (\tau_2)_t := \inf \{s \geq 0 : B_s > t \}. $$

Any ideas how to prove that $(\tau_2)_t$ has independent increments? Also, I am not sure whether $(\tau_1)_t$ has independent increments.

Answer 1

Note that both processes are non-decreasing. Can be shown that $\tau_2$ is cadlag (right continuous, with left limits) while the $\tau_1$ is the opposite.

Moreover, $\tau_2$ is Levy process, more specifically Lévy subordinator.

For more details consider the book “David Applebaum - Levy processes and stochastic calculus”, Theorem 2.2.9