Computation of the "Enrichment" of a Value Functions of an Optimization Problem on Cumulated Transaction-Costs through a Generalized Lax-Hopf Formula

The Lax-Hopf formula simplifies the value function of an intertemporal optimization (infinite dimensional) problem associated with a convex transaction-cost function which depends only on the transactions (velocities) of a commodity evolution: it states that the value function is equal to the marginal fonction of a finite dimensional problem with respect to durations and average ransactions, much simpler to solve. The average velocity of the value function on a investment temporal window is regarded as an enrichment, proportional to the profit and inversely proportional to the investment duration. At optimum, the Lax-Hopf formula implies that the enrichment is equal to the cost of the average transaction on the investment temporal window. In this study, we generalize the Lax-Hopf formula when the transaction-cost function depends also on time and commodity, for reducing the infinite dimensional problem to a finite dimensional problem. For that purpose, we introduce the moderated ansaction-cost function which depends only on the duration and on a commodity. Here again, the generalized Lax-Hopf formula reduces the computation of the value function to the marginal fonction of an optimization problem on durations and commodities involving the moderated transaction cost function. At optimum, the enrichment of the value function is still equal to the moderated transition cost-function of average transaction.