It is common wisdom in the investment community that a long-term investor saving for his future would do well to invest in high-risk/high-return assets when he is young, slowly switching his portfolio over to low-risk/low-return assets as he grows older. I have never seen a mathematical demonstration of this, and would be interested in finding one.
This isn't obviously a mathematical question, but since econ.stackexchange.com doesn't exist yet, I'm asking it here.
We might begin to formulate the question along the following lines: for each year $t=0,1,\dots,T-1$ of his life an investor chooses to put a portion of his total wealth $W_t$ into either a risk-free bond, earning interest at rate $r$, or into a risky asset whose return is a random variable with mean $\mu>r$ and variance $\sigma^2$ (we could say it's a normal random variable, for simplicity). Let's say that at time $t$ he invests a fraction $\phi_t$ in the risky asset and $1-\phi_t$ in the riskless bond.
He is also able to invest an additional amount $P_t$ into his portfolio at time $t$ (which comes, for example, from his salary). We could take $P_t=P$ deterministically to begin with, and later generalize to non-homogeneous or stochastic $P_t$.
Then the question becomes: for a given level of risk-taking, what strategy $\{\phi_t\}$ maximises his expected wealth at time $T$? As proxies for the expected wealth and level of risk, we could take $E(W_T)$ and $\mathrm{Var}(W_T)$. A common approach is to introduce the Lagrange multiplier $\lambda$ and solve the unconstrained optimisation problem
$\max_{\phi} \, E(W_T) - \lambda\mathrm{Var}(W_T)$
to create an `optimal frontier' of strategies in $(E(W_T), \mathrm{Var}(W_T))$ space, although this isn't necessarily the best strategy for solving. I haven't taken this idea much further than this, and would welcome any comments or suggestions.
I have now given this some additional thought, and added my progress in an answer below. This is not complete though, and I'd welcome further input!