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Itô’s formula, the stochastic exponential, and change of measure on general time scales. (English) Zbl 1448.81413

Summary: We provide an Itô formula for stochastic dynamical equation on general time scales. Based on this Itô’s formula we give a closed-form expression for stochastic exponential on general time scales. We then demonstrate Girsanov’s change of measure formula in the case of general time scales. Our result is being applied to a Brownian motion on the quantum time scale (\(q\)-time scale).

MSC:

81S25 Quantum stochastic calculus
60G65 Nonlinear processes (e.g., \(G\)-Brownian motion, \(G\)-Lévy processes)
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