# Help guessing the solution to an optimal control problem

I am considering an investor facing a discrete-time multi-period minimization problem $$\min_{\{v_t\}_{t=0}^\infty}\Bigg[\sum_{t=0}^\infty(1-\rho)^{t+1}\bigg(\frac{1}{2}v_{t}\Omega_{t+1}v_{t}'\bigg)+\frac{(1-\rho)^t}{2}\bigg(\frac{1}{2}\Delta v_t'\Lambda\Delta v_t\bigg) \Bigg] \quad \text{s.t.} \quad v_t'\textbf{1}=1$$ Let $$v_t$$ be a vector of weights attached to each asset, $$\Omega_t$$ be the time-varying covariance matrix and $$\Lambda$$ be a symmetric matrix of trading cost. Finally, $$\rho\in(0,1)$$ be the discount factor and $$\textbf{1}$$ being a vector of 1's. This problem has a corresponding value function $$V(v_{t-1})=\min_{v_t}\Bigg[\frac{1}{2}\Delta v_t'\Lambda\Delta v_t+(1-\rho)\bigg(\frac{1}{2}v_{t}\Omega_{t+1} v_{t}' +\mathbb{E}_t[V(v_{t})] \bigg)\Bigg] - \lambda(v_{t-1}'\textbf{1}-1)$$

I am looking to find the Bellman equation via the 'guess and verify' method (similar to Gârleanu and Pedersen, 2013).

Without the constraint ($$v_{t-1}'\textbf{1}=1$$), I can verify that $$V(v_t)=v_t'A_{vv}v_t+A_0$$ is a solution with $$A_{vv}$$ a symmetric matrix of parameters. But with the constraint, I have been unable to find a guess that solves the problem. Can you find a suitable guess that includes the constraint?

Your constraint of positions summing up to one would be strange unless you imply a long-only constraint as well. There is no closed-form solution to mean-variance optimization with non-negativity (or box) constraints. This is a standard quadratic programming problem, which can be efficiently handled numerically. In a long/short context, position constraints are imposed in terms of $$L_1$$ or $$L_2$$ norm. The former case wouldn't allow a closed-form solution either; the latter just adds to your $$\Omega$$ matrix. Further generalizations, including non-local impact costs, slippage, continuous time, etc., are described in my recent book.

I thought that the standard mean-variance dynamic problem is actually time inconsistent. See for example: Basak and Chabakauri (2010)