Let $W_{1,t},W_{2,t},...,W_{n,t}$ be $n$ independent geometric Brownian motions.
Now let's say I construct the following processes:
$$ X_1 = \frac{W_1}{\sum_i^n W_{i,t}} $$
$$ X_2 = \frac{W_2}{\sum_i^n W_{i,t}} $$
$$ \vdots $$
$$ X_n = \frac{W_n}{\sum_i^n W_{i,t}} $$
Surely, these have (at least) the following properties:
- $ 0 \leq X_i \leq 1$ for all $i$
- Independent increments
Does this process have a name? What would be its other properties? I'm stuck here.
It would be very convenient if someone is able to define this process. I will reward extra reputation to the most helpful answer. Especially if someone is able to link the properties of this stochastic process to known processes such as e.g. an Ornstein–Uhlenbeck process or other well-known stochastic processes. Many thanks in advance.