# Multivariate Stochastic Process

I am dealing with a multivariate Ornstein Uhlenbeck style SDE. Specifically $dx_{t,j}=\kappa_{j}(x_{t,j-1}-x_{t,j})dt+\sigma dW_{t,j}$ here j=1,2,...,6 , $x_{t,0}=\theta$ , $\kappa_{1}<\kappa_{2}<...<\kappa_{6}$ and $W_{j}$ are independent.

I need to restrict $x_{t,j}$'s to have decreasing concave or increasing convex shape at each time t. I couldn't figure out how I should change my SDE to have this kind of restriction. Or even if it is possible? Thank you for any help.

-
 @ bella : would you please give some precisions about what you mean exactly by concave decresasing (as a function of what)? Best regards – TheBridge Nov 15 '12 at 13:42 @TheBridge: It might work for me, for example, if at each time t, $x_{j,t}$ has a form such as $x_{t,j}=x_{t,j-1}+s_{t}\delta_{t}^{j-1}$ for j=2,...,6 and $\delta_{t}>1$. Whenver $s_{t}<0$, $x_{t,j}$ has decreasing concave shape and whenever $s_{t}>0$ , $x_{t,j}$ has increasing convex shape at time t. – bella Nov 16 '12 at 22:49 Sorry I still don't get what you mean by concave shape. Good luck and regards – TheBridge Nov 17 '12 at 13:01