Abstract
Stochastic integration theory is developed by axiomatizing the concept of semi-martingale in terms of a continuity property of integrals of simple functions. Using this approach, stochastic integration for left-continuous integrands, the change of variables formula and properties of the quadratic variation process are established in an elementary way. Submartingale decomposition theorems are introduced at a late stage in order to extend the results to general predictable integrands.
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