# How to describe such a stochastic process in continuous time

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.

• I would think that this process jumps to $1$ in any small interval $[0,t_0]$ unless the probability of staying at zero is $=1$ for almost all $t$ Commented Oct 22, 2021 at 8:34

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.
• Thanks. Yes, my description of the model'' is problematic. I will try to understand the intensity way. Commented Oct 26, 2021 at 2:09