Let $(X_t)$, $t \in [0, \infty)$ be a stochastic process on a measurable space $(E, \mathcal{E})$. Then $X_t$ is said to

  • have the Markov property (or be a Markov process) if $P(X_t \in B \mid \mathcal{F}_s) = P(X_t \in B \mid X_s)$ a.s. for all $B \in \mathcal{E}$ and $0 \leq s \leq t$ where $\mathcal{F_s}$ is the filtration generated by $X_s$
  • have a transition function if for each $0 \leq s \leq t$ there exists a stochastic kernel $p_{st}(x,B)$ on $(E, \mathcal{E})$ satisfying $p_{su} = p_{st} p_{tu}$ for $0 \leq s \leq t \leq u$ (composition of kernels) and such that $P(X_t \in B \mid X_s) = p_{st}(X_s,B)$ a.s.

I am interested in examples for stochastic processes $X_t$ such that

  1. $X_t$ has the Markov property but no transition function
  2. $X_t$ has a transition function but not the Markov property

Remark: It is known that on a standard Borel space $(E, \mathcal{E})$ any stochastic process with the Markov property has a transition function. Hence, for the example in 1. the space $(E, \mathcal{E})$ must be necessarily not Borel. If possible, I would like to see in both cases also examples where $\mathcal{E}$ is still "nice enough" and not too pathological (e.g. contains all singleton sets and is countably generated).

Note that if one is given a Markov process $X_t$ (in the sense that it satisfies only the Markov property) then I often see in the literature expressions like $p_{s,t}(x,B) := P(X_t \in B \mid X_s = x)$. But the right-hand side may not be defined since $\{ X_s = x \}$ need not be measurable (independent of whether $\{ x \}$ is contained in $\mathcal{E}$ or not) and if it is measurable then $P(X_s = x) = 0$ is possible.

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