Weak time-derivatives and no arbitrage pricing

The risk-neutral pricing formula provides the valuation of random payoffs in continuous-time markets. Despite the variety of payoffs, no arbitrage price dynamics are driven by the same (possibly stochastic) interest rate. We formalize this intuition by showing that no arbitrage prices constitute the solution of a differential equation, where interest rates are prominent. To achieve this goal, we introduce the notion of weak time-derivative, which permits to differentiate adapted processes. This instrument isolates drifts of semimartingales and it is null for martingales. Finally, we reformulate the eigenvalue problem of Hansen and Scheinkman (2009) by employing weak time-derivatives.