Increasing-unbounded Simple Components

A one-counter MDP (OC-MDP) $M$ is an MDP where each transition is labeled by an integer which is added to the counter whenever the transition is taken. A counterless MD strategy (cMD) $\sigma$ on $M$ is a function assigning to each state a single action. Applying $\sigma$ to $M$ yields a one-counter Markov chain $M_\sigma$. We call a sub-OC-MDP of $N$ a simple component of $M$ if $N$ is exactly a BSCC of $M_\sigma$ for some cMD strategy $\sigma$. Given a simple component $N$, let $p$ be some state of $N$ and let $X$ be a random variable encoding the change of the counter of a run on $N$ initiated in $p$ and till $p$ is revisited for the first time. Let E(X) denote the expected value of $X$. We define the following (note it can be shown these are invariant to the choice of $p$): $N$ is bounded if $P(X=E(X))=1$, $N$ is unbounded if $P(X=E(X))<1$. and: $N$ is increasing if $E(X)>0$, $N$ is zero if $E(X)=0$, $N$ is decreasing if $E(X)<0$. For example a simple random walk (+-1 with equal probability) would have a single zero-unbounded simple component. The question is: Given a OC-MDP $M$, what is the complexity of deciding whether $M$ contains an increasing-unbounded simple component? What is the complexity of this problem for the subclass of OC-MDPs that contain no decreasing simple components? These are trivially in NP, but can they be solved in PTIME? Alternatively, is there a way to effectively encode all increasing-bounded (or increasing-unbounded) simple components? (for example for the class of OC-MDPs containing no decreasing simple components there exists a set $R$ of transitions such that $N$ is zero-bounded iff $N$ contains only transitions from $R$). Bonus problem (very difficult): is it decidable in PTIME whether a given OC-MDP contains a simple component with $P(X<0)=0$?