# Samples paths are convex

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?

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.
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 $$\inf C$$.