Are there stochastic processes with convex sample paths? Suppose $C$ is a given convex set. Is there a real valued stochastic process $X\_t, t \in C$ such that the sample path $f:C \rightarrow R $ given by $f(t)=X\_{t}$ is convex almost surely?



Samples paths are convex
Let $X\_t:=\xi\, g(t)$ for $t\in C$, where $g$ is any convex function from $C$ to $R$ and $\xi$ is any nonnegative random variable. Then all sample paths of the stochastic process $(X\_t)\_{t\in C}$ are convex. 


All the sample paths of the sum $Y\_t:=\sum\_{k=1}^n\xi\_k\, g\_k(t)$ of processes such as the one described above will also be convex. 


