A stochastic step function? Is there a way to have a step function the heights and widths of whose constant portions are random? Said another way, can I define a "random" step function $f$ on $[0,1]$ using some parameters so that it doesn't always look like a complete mess? 
Ideally, I would like it to be constant on sub-intervals whose lengths are "suitable" for my purposes. Assuming that the independent variable is (rightward-moving) time, I would like the values the function takes on "next" to lie within some interval determined by the function's "current" value. I'm trying to model a discrete phenomenon in which some quantity always has a small (in my case .005) chance of changing. 
I know very little probability theory except by a brief encounter I had with it in measure theory, so this is a perplexing question for me. Any help would be greatly appreciated. 
Thanks.
 A: One might try
$$
f(t,\omega)=\sum_{k=1}^nX_k(\omega)\mathbf 1_{t\geqslant T_{k-1}(\omega)},
$$
where the sequence of random variables $(T_k)_{0\leqslant k\leqslant n}$ is increasing from $T_0=0$ to $T_n=1$ and $(X_k)_{1\leqslant k\leqslant n}$ is a sequence of independent random variables with the same distribution.
In words, $f(t)=X_1+\cdots+X_k$ for every $t$ in $[T_{k-1},T_k)$ and, when $t$ crosses the value $T_k$, $f(t)$ jumps by $X_{k+1}$. Hence, $(T_k)_{1\leqslant k\leqslant n-1}$ describes the times of the jumps and $(X_k)_{2\leqslant k\leqslant n}$ describes their amplitudes (while $T_0$ and $T_n$ are fixed and $X_1$ describes the level $f$ starts at).
The choice of the structure of $(T_k)_{0\leqslant k\leqslant n}$ and of the common distribution of the jumps $X_k$ is free. From the description of the variations of $f$ you are interested in, it is not quite clear whether the times $T_k$ should be deterministic, for example $P[T_k=k/n]=1$ for every $k$, or random. And the common distribution of the random variables $X_k$ might be such that $P[X_k=0]=1-.005$ and $X_k$ uniform on some interval $(-x,x)$ with probability $.005$.
As always, these choices depend on the phenomenon you want to describe and on the questions about this phenomenon you want the model to address.
