I was brushing up some notes on stochastic processes, and I noticed that in many examples, we have that given a stochastic process $N_t$, we have that the event $$\{ N_t < k \} = \{ S_k > t\}$$ with $S_k$ being the time of the first arrival to the value $k$ given the process $N_t$. This construction basically denotes a dual stochastic process, I was wondering whether there was a name for such process. Furthermore, there's some deeper reason for this duality?
I put here some more examples since I'm no great writer:
- Consider a Poisson process $N_t$, i.e. increments are independents, identically distributed, and have distribution a Poisson distribution. Then if we define with $S_k = \inf \{t > 0 \mid N_t > k\}$ we have $$\{N_t < k\} = \{S_k > t\}$$
- Consider a one-dimensional Brownian motion $B_t$, then if we define $S_k = \inf \{t > 0 \mid\ B_t > k\}$ then $$\{B_t < k\} = \{S_k > t\}$$
