In my modelling, I am considering a stochastic process quite like the Bernoulli process but with output variable $y\in[0,1]$ (i.e., a probabilistic outcome, rather than binary $\{0,1\}$ outcome). I guess there should be a well studied stochastic process modelling this, if yes then what it is? E.g., named after some guy with publications and application examples
For now, as a backup plan but not ideal, what I can think of is to use categorical distribution for the output variable $y$ by discretising $y$ into several categories... Thanks for your time.