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The Lax-Hopf formula simplifies the value function of an intertemporaloptimization (infinite dimensional) problem associated with a convextransaction-cost function which depends only on the transactions (velocities)of a commodity evolution: it states that the value function is equal to themarginal fonction of a finite dimensional problem with respect to durations andaverage ransactions, much simpler to solve. The average velocity of the valuefunction on a investment temporal window is regarded as an enrichment,proportional to the profit and inversely proportional to the investmentduration. At optimum, the Lax-Hopf formula implies that the enrichment is equalto the cost of the average transaction on the investment temporal window. Inthis study, we generalize the Lax-Hopf formula when the transaction-costfunction depends also on time and commodity, for reducing the infinitedimensional problem to a finite dimensional problem. For that purpose, weintroduce the moderated ansaction-cost function which depends only on theduration and on a commodity. Here again, the generalized Lax-Hopf formulareduces the computation of the value function to the marginal fonction of anoptimization problem on durations and commodities involving the moderatedtransaction cost function. At optimum, the enrichment of the value function isstill equal to the moderated transition cost-function of average transaction.

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