A Discrete-Time Itô's Formula

Nonlinear Stochastic Difference Equations Driven by Martingales

“…We conclude this article with a discrete-time Ito formula for the C m -martingale flow j k , m ≥ 2 (see also [5]). This concludes the proof (see [5] for some details).…”

“…Some groundwork for this study has been done in our earlier work [5,14]; we will appeal to some of the results from them. The stochastic difference equations (abbreviated SdE) that we consider are of the following forms: for k = 0 1 2 ,…”

The First Attempt on the Stochastic Calculus on Time Scale

“…It is used to describe the mathematical stochastic models in economy, physics, biology, medicine and social sciences Some of basic problems show concern for studying are stochastic integration, Doob-Meyer decomposition theorem, stochastic differential equation, Ito's formula which have been studied carefully for both discrete and continuous time (see for example [9,[11][12][13]). …”

Almost sure asymptotic stability analysis of the θ-Maruyama method applied to a test system with stabilising and destabilising stochastic perturbations

“…An introduction to discrete forms of the Itô formula may be found in Shiryaev [16, Chapter VII, p. 389]; note also the papers by Akahori [1], Kannan and Zhang [13] and the survey of discrete stochastic calculus presented in Gzyl [9]. Although [1] demonstrated the use of a discrete Itô formula to prove convergence of the Euler-Maruyama method, to the best of our knowledge such a formula has never been used to perform an almost sure linear stability analysis.…”