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Stochastic calculus provides a consistent theory of integration for stochastic processes and is used to model random systems. Its applications range from statistical physics to quantitative finance.
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Wiener process related counterexample
To continue on Byron's answer, properties 1) and 3) specify the law of the process, but not the topological features of a given trajectory $\omega$. In a sense, 1) and 3) are enough to define the Wien …