On strong causal binomial approximation for stochastic processes
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This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Discrete and Continuous Dynamical Systems Series B following peer review. The definitive publisher-authenticated version Dokuchaev, N. 2014. On strong causal binomial approximation for stochastic processes. Discrete and Continuous Dynamical Systems Series B. 19 (6): pp. 1549-1562 is available online at: http://doi.org/10.3934/dcdsb.2014.19.1549
This paper considers binomial approximation of continuous time stochastic processes. It is shown that, under some mild integrability conditions, a process can be approximated in mean square sense and in other strong metrics by binomial processes, i.e., by processes with fixed size binary increments at sampling points. Moreover, this approximation can be causal, i.e., at every time it requires only past historical values of the underlying process. In addition, possibility of approximation of solutions of stochastic differential equations by solutions of ordinary equations with binary noise is established. Some consequences for the financial modelling and options pricing models are discussed.
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