I am trying to evaluate an issue regarding convexity in continuous mathematical finance. But I am not sure if (1) my general conclusion is right and (2) if so, how to prove it.
Suppose we have a 'return' process that is Brownian. At any time $t$, its instantaneous value is $r_t$.
If we start with a value of 1.0 and compound continuously, the value after a time $t_f$ has passed is, I think,
$$\int_0^{t_f}e^{r_t}dt $$
Now, there is a fixed value $\hat r$ such that
$$\int_0^{t_f}e^{\hat r}dt $$
has the same value - i.e., it is the fixed rate that gives the same result.
Intuitively, it seems like $\hat r<E[r_t]$ due to convexity (as well as extending the discrete time results to continuous time).
Is that right, and how would one most readily start a proof?