Samples paths are convex

Question

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?

Answer 1

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.

Answer 2

How about if we let $f(t)=\int_0^t W_s^2ds$ where $W$ is a standard Wiener process, and $g(t)=\int_0^tf(s)ds$. Then $g''(t)=W_t^2\ge 0$ so $g$ is convex.

For general $C$ replace 0 by $\inf C$.