We present in this article a game of chance (Saint Petersburg Paradox) and
generalize it on a probability space as an example of a previsible (predictable) process,
from which we get a discrete stochastic integration (DSI). Then we define a marting
ale
and present it as a good integrator of a discrete stochastic integration ∫ , which is
called the martingale transform of by such that is a previsible process.
After that we present the most important properties of the DSI, which include that the
DSI is also a martingale , the theorem of stability for it, the definition of the covariation of
two given martingales and the proof that the DSI is centered with a specific given variance.
Finally, we define Doob-decomposition and the quadratic variation and present Itȏformula
as a certain sort of it.
