Discrete Stochastic Integration
Abstract
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 martingale 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 Itȏ-formula as a certain sort of it.
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