How to describe such a stochastic process in continuous time

Question

Below an intuitive description of a stochastic process I want to construct. But I do not know how to define it formally in continuous time. Any hint, help or reference will be appreciated.

Consider a process $\{X_t\}_{t \geq 0}$ that only takes two values, $0$ and $1$. The initial state is $0$. At each time $t > 0$, the state changes to $1$ with probability $\lambda(t) \in [0, 1]$. State $1$ is absorbing so that once it becomes $1$, it remains there.

In discrete time, this is rather straightforward, as it is just a non-homogeneous Markov chain. I have no idea how to formally define such a process in continuous time. I do not even know whether such process exists for arbitrary $\lambda$ function. Especially, given $\lambda$ function, how can I calculate quantities like $\mathbb{E}(f(\tau))$ where $\tau$, for instant, is the stopping time that the process jumps to $1$ and $f$ is a function (e.g., linear, quadratic).

Thanks.

Answer 1

If we want to say that at each time $t > 0$, the state changes to $1$ with probability $\lambda(t) \in [0, 1]$, then what does it mean? There's no "the nearest moment of time" in the future. And if we mean something like $P(X(t+0) = 1|X(t) = 0) = \lambda(t)$, where $X(t+0)$ is a right limit, then we should suppose, that $X(t+0)$ exists, smth. like cad-lag case, but in this case the process will be right-continious in probability, but your construction looks like it's not the case you need.

Maybe it's better to speak about intensity function $\lambda(t)$, as in case with Poisson process.