I'd like to express a uniform distribution $U[0,1]$ as a function of a Brownian motion $w(t)$.
Specifically, I am thinking about this problem in two versions of model setups.
Model setup version 1 is: The total income of the society follows a geometric Brownian motion: $d \delta_t = \delta_t \mu_\delta dt + \delta_t \sigma_\delta dw_t$. There are two agents in the society, and their share of the total income at each period is: $z_t \times \delta_t$ and $(1-z_t) \times \delta_t$, where $z_t \sim Uniform[0,1]$. However, I would like to express $z_t$ using a stochastic term (i.e., Brownian motion term) --- as opposed to a draw from a uniform distribution.
Model setup version 2 is: suppose there are two groups of income in the society -- $h$ and $l$. The transition probability from $h$ to $l$ is $p_{hl}$, from $h$ to $h$ is $p_{hh}$. And $p_{hl} + p_{hh}$ = 1. Suppose the agent's income at $t=0$ is $h$, then at $t=1$, his probability of arriving at $h$ is $p_{hl}$ and his probability of arriving at $l$ is $p_{hl}$. However, I would like to write his income at $t=1$ explicitly at $t=0$: something like $d y_t = y_t \mu_t dt + y_t \sigma_y dw_t$ (which is obviously not correct given the context), rather than two possible outcomes with probabilities.
I hope my question is clear. Please feel free to ask clarifying question, or to propose alternative model setups.
Thank you!
Best,
Darcy