I am reading Bass's book on Stochastic processes on my own. I cam across the following exercise in Chapter 17.
17.8 Suppose $X_t$ is an increasing process with paths that are right continuous with left limits, $X_0=0$ a.s., $X$ is purely discontinuous, and all jumps are of size $+1$ only. Suppose that $\lim\limits_{t\to \infty}X_t=\infty,$ a.s. Then prove that $X_t$ is a time change Poisson process.
I am feeling uneasy about this exercise. Suppose, I have a (deterministic) process which jumps at every integer, and jump sizes are one. Then this process satisfies all the hypothesis in the question, but how can it be a Poisson process? Jumps in Poisson process are completely unpredictable. I can not wrap my head around this problem.
I would appreciate if somebody can elucidate the issue and explain to me what I am missing?