3
$\begingroup$

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?

$\endgroup$

2 Answers 2

2
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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 $\inf C$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.