  A hypothetical risk-neutral agent who trades to maximize the expected profit
of the next trade will approximately exhibit long-term optimal behavior as long
as this agent uses the vector $p = \nabla V (t, x)$ as effective microstructure
alphas, where V is the Bellman value function for a smooth relaxation of the
problem. Effective microstructure alphas are the steepest-ascent direction of V
, equal to the generalized momenta in a dual Hamiltonian formulation. This
simple heuristics has wide-ranging practical implications; indeed, most
utility-maximization problems that require implementation via discrete
limit-order-book markets can be treated by our method.
