Self-financing condition ($d{V}_t={\varphi }_{t}d{S}_t+{\psi }_{t}d{B}_t$)

For holdings with value $V_t={\varphi }_t{S}_t+{\psi }_t{B}_t$, the condition says that all gains and losses arise from the held assets, with no external cash transfer. The stochastic-integrability and admissibility requirements on the holdings are part of a valid trading strategy. Differentiating the value identity alone does not establish self-financing, since changes of holdings can contribute additional terms.