Discrete stochastic integration

Abstract EN

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 X and present it as a good integrator of a discrete stochastic integration ∫ C.DX , which is called the martingale transform of X by such that c 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.