Advances in Numerical Simulation in Physics and Engineering by Carlos Parés Carlos Vázquez & Frédéric Coquel

Author:Carlos Parés, Carlos Vázquez & Frédéric Coquel
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

The aim of this paragraph is to provide extensions of the above properties on the two first moments to more general integrands, under some suitable boundedness or integrability conditions.

3.5.1 Heuristic Arguments

In view of the previous construction, there is a natural candidate to be the stochastic integral . When h is piecewise constant process (called simple process), that is if s ∈ [t i , t i+1] for a given deterministic time grid (t i ) i , we set

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Without extra assumptions on the stochasticity of h, it is not clear why its expectation equals 0. This property should come from the centered Brownian increments and their independence to so that

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