Past exam of the martingales and financial mathematics course of the University of Oxford / 2007 / B / 5 / a / Solution

Let ${\Delta }_t$ be the number of stock units and ${\beta }_t$ the number of bank-account units, with predictable holdings satisfying the integrability conditions for the price integrals. The portfolio value is $V_t={\Delta }_t{S}_t+{\beta }_t{B}_t$. It is self-financing ifThus changes in value come only from gains on the assets held. Rebalancing transfers money between the stock and bank account; it does not inject or withdraw cash. The definition is an identity of stochastic integrals and automatically includes the bookkeeping needed when holdings themselves vary randomly.