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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.
1
vote
Getting $B_t$ from its local times $L^x_t$
This paper seems to answer (something very close to) your question :
Warren, J. and Yor, M. (1998), The Brownian burglar: conditioning Brownian motion by its local time process. Seminaire de Probabil …
1
vote
infimum of a set of stopping times
This is not true. Fix some $t>0$, and define $t^n_k = \frac{k}{n}.t$ for $0 \leq k \leq n$. Define $\Lambda$ as the set of deterministic, $\{-1,1\}$-valued processes and $\Lambda_n$ as the subset of $ …