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__On the use of discrete forms of the Itô formula__

C. Kelly, G. Berkolaiko, E. Buckwar and A. Rodkina

**Abstract**

A discrete form of the Itô formula was introduced in a 2009 paper by Appleby et al [1] for the purpose of deriving sharp a.s. asymptotic stability conditions for a scalar stochastic difference equation of the form produced by an Euler-Maruyama discretisation of an SDE with nonlinear coefficients.
The formula has proved flexible, and has since found application in the a.s. asymptotic stability analysis of linear systems with a.s. stabilising and destabilising perturbations under $\theta$-Maruyama discretisation (see [2]).
We anticipate that versions of this discrete Itô formula will find wider application in the field of stochastic numerical analysis. However, the original 2009 proof contains an implicit assumption that must be carefully revisited each time the formula is adapted for a new discretisation method or test system. In this talk we explore the nature of this assumption and how it may be dealt with.

**Bibliography**

[1]
J. A. D. Appleby, G. Berkolaiko and A. Rodkina,
Non-exponential stability and decay rates in nonlinear stochastic difference equation with unbounded noise,
Stochastics: An International Journal of Probability and
Stochastic Processes, 81:2 (2009), pp. 99-127.

[2]
G. Berkolaiko, E. Buckwar, C. Kelly and A. Rodkina,
Almost sure asymptotic stability analysis of the $\theta$-Maruyama method applied to a test system with stabilising and destabilising perturbations.
LMS Journal of Computation and Mathematics, 15 (2012), pp. 71-83.