Title: On Natural and Predictable Processes

Author(s): Victor M. Kruglov
Issue: Volume 78 Series A Part 1 Year 2016
Pages: 43 -- 51
We give short, nontechnical proofs of the equivalence of predictability and naturality of stochastic processes of integrable variation, as well as of the properties that any natural martingale of integrable variation is indistinguishable from zero, any local predictable martingale of locally integrable variation is constant a.s., any local martingale is predictable if and only if it is continuous a.s. These statements are well known, but their proofs are very complicated since they depend on deep results from the advanced theory of stochastic integration and local martingales. Our proofs use only basic concepts of stochastic calculus.
AMS (2000) subject classification. Primary 60G05, Secondary 60G07.
Keywords and phrases: Martingale, Stopping time, Natural process, Predictable process.