Optimal control / Portoflio optimization: Maximize expected utility of total consumption

Question

I came across a portoflio optimization problem, where I need to solve for optimal investment and consumption processes, such that the expected utility of total consumption and terminal wealth is maximized. Namely

$max_{\pi,c}\;\mathbb{E}\bigg[\,U_{1}\big(\int_{0}^{T}c(t)\,dt\big) + U_{2}\big(V^{\pi,c}(T)\big)\;\bigg]$

where $\pi$ (investment) and $c$ (consumption) are two control processes subject to technical conditions, $V^{\pi,c}(t)$ is the investors wealth at time $t\in[0,T]$, and $U_{1}$ and $U_{2}$ are (deterministic, utility) functions.

Has somebody come across a similar problem? Can someone point me in the right direction towards a solution? Any comments, ideas, suggestions are highly appreciated! Thank you very much.

P.S.: Note the difference to the standard optimization problem of Merton, where one solves the related problem

$max_{\pi,c}\;\mathbb{E}\bigg[\,\int_{0}^{T}U_{1}\big(c(t)\big)\,dt + U_{2}\big(V^{\pi,c}(T)\big)\;\bigg]$

which can be solved using the (so-called) Martingale Method or the associated Hamilton-Jacobi-Bellman equation.

Answer 1

The problem as stated has no solution except in the special case where it is optimal not to consume at all.

To see this, note that the payoff depends on the consumption rate process $c(t)$ only through the integral $\int_0^T c(s) \, ds$. Assume that there exists some optimal solution $(\pi^*, c^*)$ with $c^* \neq 0$. Then we can find a consumption rate $\tilde{c}$ such that $\int_0^T \tilde c(s) \, ds = \int_0^T c^*(s) \, ds$, $\int_0^t \tilde c(s) \, ds \leq \int_0^t c^*(s) \, ds$ for all $t \in [0,T]$ and $\int_0^t \tilde c(s) \, ds < \int_0^t c^*(s) \, ds$ on some set of $dt \otimes \mathbb{P}$ positive measure. The amount consumed less on this set can instead be invested in the riskless asset and earn thus additional money, leading to an admissible strategy satisfying $V^{\tilde{\pi}_T, \tilde{c}} > V^{\pi^*, c^*}_T$ and thus contradicting the optimality of $(\pi^*, c^*)$.

Intuitively, this means the optimal consumption should happen not cumulatively over time but singularily in the last moment. To achieve this, you will have to give up the assumption of an absolutely continuous consumption and allow that the consumption process $C_t$ is an arbitrary non-decreasing process (instead assuming $C_t = \int_0^t c(s) \, ds$ for some consumption rate $c$), which will effectively satisfy $C_t= 0$ for $t < T$ by the argumen. Then the problem consists of an optimal investment problem where the final wealth $V_T = \bar{V}_T + \bar{C}_T$ is then split optimally in to nonnegative random variables $\bar{V}_T$ and $\bar{C}_T$.

Finally, note (may be a bit pedantically) that the above proof requires a riskfree asset that grows at a positive rate. With some care (and careful investment in the risky asset) this can be generalized to the case of zero rate. However, it does not work in an environment of negative interest rate. The obvious change in the proof (shorting the riskless asset and overconsuming before terminal time) might not be an admissible strategy if one applies the usual constraints of nonnegative wealth- and consumption processes.