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).


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


[1] Bohner, M.; Peterson, A., Dynamical Equations on Time Scales, An Introduction with Applications, (2001), Boston, Mass, USA: Birkhäuser Boston, Boston, Mass, USA · Zbl 0978.39001
[2] Bohner, M.; Stanzhytskyi, O. M.; Bratochkina, A. O., Stochastic dynamic equations on general time scales, Electronic Journal of Differential Equations, 57, 1-15, (2013) · Zbl 1292.34087
[3] Bohner, M.; Sanyal, S., The stochastic dynamic exponential and geometric Brownian motion on isolated time scales, Communications in Mathematical Analysis, 8, 3, 120-135, (2010) · Zbl 1202.60132
[4] Sanyal, S.; Grow, D., Existence and uniqueness for stochastic dynamic equations, International Journal of Statistics and Probability, 2, 2, (2013)
[5] Grow, D.; Sanyal, S., Brownian motion indexed by a time scale, Stochastic Analysis and Applications, 29, 3, 457-472, (2011) · Zbl 1217.60072
[6] Du, N. H.; Dieu, N. T., The first attempt on the stochastic calculus on time scale, Stochastic Analysis and Applications, 29, 6, 1057-1080, (2011) · Zbl 1238.60061
[7] Grow, D.; Sanyal, S., The quadratic variation of Brownian motion on a time scale, Statistics and Probability Letters, 82, 9, 1677-1680, (2012) · Zbl 1251.60061
[8] Bhamidi, S.; Evans, S. N.; Peled, R.; Ralph, P., Brownian motion on time scales, basic hypergrometric functions, and some continued fractions of Ramanujan, IMS Collections, 2, 42-75, (2008) · Zbl 1167.60347
[9] Kumagai, T., Short time asymptotic behaviour and large deviation for Brownian motion on some affine nested fractals, Publications of the Research Institute for Mathematical Sciences, 33, 2, 223-240, (1997) · Zbl 0888.60030
[10] Ben Arous, G.; Kumagai, T., Large deviations for Brownian motion on the Sierpinski gasket, Stochastic Processes and Their Applications, 85, 2, 225-235, (2000) · Zbl 0997.60018
[11] Cont, R.; Fournié, D.-A., Change of variable formulas for non-anticipative functionals on path space, Journal of Functional Analysis, 259, 4, 1043-1072, (2010) · Zbl 1201.60051
[12] Haven, E., Quantum calculus (\(q\)-calculus) and option pricing: a brief introduction, Quantum Interaction. Quantum Interaction, Lecture Notes in Computer Science, 5494, 308-314, (2009), Berlin, Germany: Springer, Berlin, Germany · Zbl 1229.91266
[13] Haven, E., Itô’s lemma with quantum calculus (\(q\)-calculus): some implications, Foundations of Physics, 41, 3, 529-537, (2011) · Zbl 1211.81083
[14] Bryc, W., On integration with respect to the q-Brownian motion, Statistics & Probability Letters, 94, 257-266, (2014) · Zbl 1321.60114
[15] Agarwal, R. P.; Bohner, M., Basic calculus on time scales and some of its applications, Results in Mathematics, 35, 1-2, 3-22, (1999) · Zbl 0927.39003
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