OK this isn’t homework. I’m much to old for that.
The diffusion process dx = sx(1-x)dw keeps x confined to (0,1)
The diffusion process:
dx[sub]1[/sub] = sx[sub]1/subdw = sx[sub]1[/sub]x[sub]2[/sub]dw
dx[sub]2[/sub] = -sx[sub]2/subdw = sx[sub]2[/sub]x[sub]1[/sub]dw
and starting with x[sub]1[/sub] + x[sub]2[/sub] = 1 keeps (x[sub]1[/sub], x[sub]2[/sub]) confined to the unit simplex.
Is there any simple (and ideally symmetric) way to extend this to N x’s?