Random walks and stochastic differential equations
The Wiener random walk is often studied in books on probability. It is a discrete process and classically, we cannot ask any question about the continuity or derivability of the trajectories or about the density of the process. The two results below are obtained by using discrete methods, but we are interested in the case of an infinitely large number of steps. With this hypothesis on the size of the variables, we prove some results which are similar to theorems on classical brownian motion. We want to take advantage of the simplicity of discrete concepts together with the simplicity of the analytical calculus. The link between the two classical methods is provided by nonstandard analysis.
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