# Are there any continuous-time stochastic processes in which transition probabilities are discontinuous functions over time?

In stochastic processes, like homogeneous Markov processes, Poisson processes, Queueing systems etc., the functions that represent (transition) probabilities are continuous over time. This is also very convenient mathematically, since it allows further useful analysis to be done, such as transition rates in Markov processes.

I recently read a paper ( https://arxiv.org/abs/1811.07401 ) where a stochastic process is described, with probabilities being discontinuous at an infinite (!!) number of points. According to the paper, it shows the problematic nature of a certain class of algorithms with which the process is associated.

Are there any continuous-time stochastic processes with discontinuous transition probabilities or is it fundamentally incompatible with mathematical/physical reality?? I would like an answer for stochastic processes describing real phenomena/systems, not for theoretically/artificially constructed processes.

• Certainly there would be reasonable models of continuous time stochastic processes with discontinuous transition probabilities. Imagine (for example) a model of queuing to pay for ticket, where there is a completely standard service time unless tickets are being bought for tomorrow in which case you get to choose your seat. – Anthony Quas Dec 16 '18 at 22:01
• The transition kernel $k_t(\cdot,\cdot)$ of the Markov process defined by the (deterministic) right shift on the real line is given by $k_t(x,\cdot) = \delta_{t+x}$ for each time $t \ge 0$ and each real number $x$ (where $\delta_y$ denotes the Dirac measure at $y$ for each real $y$). Is that sufficienty discontinuous for your purposes? – Jochen Glueck Dec 17 '18 at 22:23
• @AnthonyQuas Thanks for your answer,Anthony! Would you be willing to provide more details for the example you gave? I would like to understand better how a function representing probabilities in your example is discontinuous over time. Thanks – Robert_Lewis Dec 18 '18 at 2:46
• It’s actually possible to build quite simple physically motivated discontinuous quantum Markov process. I suspect the same holds for the classical version. If you are interested I can post an answer. – lcv Dec 18 '18 at 7:28
• @JochenGlueck Thanks for your answer, Jochen! If I understand it correctly, it is indeed discontinuous. But isn't it a bit "constructed" given that the right shift is deterministic? – Robert_Lewis Dec 19 '18 at 2:33

One example type is a jump process that jumps at certain predetermined times, as in @AnthonyQuas' comment.

For instance, a stock price that can only make jumps when markets open, like New Zealand Stock Exchange at UTC+12 (+13), Australian Securities Exchange at UTC+10 (+11) etc.

This can be modeled by a process $$V$$ such that $$V_t=\begin{cases}S_t,&\quad 2n\le t<2n+1,\\ S_{2n+1},&\quad 2n+1\le t<2(n+1)\end{cases}$$ where $$S_t$$ is a geometric Brownian motion and $$n$$ is an integer.

• Hi Bjorn, thanks for your answer! You gave me the example of V(t)...since I am not very technically-familiar with geometric Brownian motion, I guess that the transition probability density function is discontinuous at t=2n+1...but it is continuous in each branch, right? – Robert_Lewis Dec 19 '18 at 3:02