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Articles

The Itô integral and near-martingales in Riesz spaces

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Pages 5068-5081 | Received 28 Jul 2020, Accepted 01 Nov 2021, Published online: 29 Nov 2021

References

  • Aliprantis, C. D., and O. Burkinshaw. 1985. Positive operator. Orlando: Academic Press.
  • Ayed, W., and H.-H. Kuo. 2008. An extension of the Itô integral. Communications on Stochastic Analysis 2 (3):323–33. doi:10.31390/cosa.2.3.05.
  • Ayed, W., and H.-H. Kuo. 2010. An extension of the Itô integral: Toward a general theory of stochastic integration. Theory of Stochastic Processes 16 (32):17–28.
  • Black, F., and M. Scholes. 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81 (3):637–54. doi:10.1086/260062.
  • Buckdahn, R. 1991. Anticipative Girsanov transformations. Probability Theory and Related Fields 89 (2):211–38. doi:10.1007/BF01366907.
  • DeMarr, R. 1966. A martingale convergence theorem in vector lattices. Canadian Journal of Mathematics 18:424–32. doi:10.4153/CJM-1966-045-x.
  • Dorogovtsev, A. A. 1990. Itô-Voltera equations with an anticipating right-hand side in the absence of moments. In Infinite-dimensional stochastic analysis (Russian), 41–50. Kiev: Akad. Nauk Ukrain SSR, Inst. Mat.
  • Grobler, J. J. 2010. Continuous stochastic processes in Riesz spaces: The Doob-Meyer decomposition. Positivity 14 (4):731–51. doi:10.1007/s11117-010-0088-2.
  • Grobler, J. J. 2011. Doob’s optional sampling theorem in Riesz spaces. Positivity 15 (4):617–37. doi:10.1007/s11117-011-0114-z.
  • Grobler, J. J. 2014a. Corrigendum to” The Kolmogorov-Centsov theorem and Brownian motion in vector lattices. Journal of Mathematical Analysis and Applications 420 (1):878.
  • Grobler, J. J. 2014b. The Kolmogorov-Centsov theorem in vector lattices. Journal of Mathematical Analysis and Applications 410 (2):891–901. doi:10.1016/j.jmaa.2013.08.056.
  • Grobler, J. J. 2021. Stopped processes and Doob’s optional sampling theorem. Journal of Mathematical Analysis and Applications 497 (1):124875. doi:10.1016/j.jmaa.2020.124875.
  • Grobler, J. J., and C. A. Labuschagne. 2015. The Itô integral for Brownian motion in vector lattices: Part 1. Journal of Mathematical Analysis and Applications 423 (1):797–819. doi:10.1016/j.jmaa.2014.08.013.
  • Hwang, C.-R., H.-H. Kuo, H. Ouerdiane, and B. Szozda. 2013. Linear stochastic differential equations with anticipating initial conditions. Communication on Stochastic Analysis 7 (2):245–53.
  • Hwang, C.-R., H.-H. Kuo, K. Saitô, and J. Zhai. 2016. A general Itô formula for adapted and instantly independent stochastic processes. Communication on Stochastic Analysis 10 (3):341–62.
  • Itô, K. 1944. Stochastic integral. Proc. Imp. Acad. Tokyo 20:519–524.
  • Itô, K. 1978. Extension of stochastic integrals. Proc.Intern. Symp. Stochastic on Differential Equations, K. Itô (eds.):95–109.
  • Kuo, H.-H. 2014. The Itô calculus and white noise theory: A brief survey toward general stochastic integration. Communication on Stochastic Analysis 8:1:111–139.
  • Kuo, H.-H., and K. Saito. 2015. Doob’s decomposition theorem for near-submartingales. Communications on Stochastic Analysis 9 (4):467–76. doi:10.31390/cosa.9.4.03.
  • Kuo, H.-H., Y. Pend, and B. Szozda. 2013. Itô formula and Girsanov theorem for anticipating stochastic integrals. Communication on Stochastic Analysis 7 (3):441–58.
  • Kuo, W.-C. 2006. Stochastic processes on vector lattices. Thesis, University of the Witwatersrand.
  • Kuo, W.-C., C. C. A. Labuschagne, and B. A. Watson. 2004a. Discrete time stochastic processes on Riesz spaces. Indagationes Mathematicae 15 (3):435–51. doi:10.1016/S0019-3577(04)80010-7.
  • Kuo, W.-C., C. C. A. Labuschagne, and B. A. Watson. 2004b. An upcrossing theorem for martingale on Riesz spaces. Soft methodology and random information systems, 101–8. Berlin: Springer.
  • Kuo, W.-C., C. C. A. Labuschagne, and B. A. Watson. 2005. Conditonal expectation on Riesz spaces. Journal of Mathematical Analysis and Applications. 03:509–21.
  • Kuo, W.-C., C. C. A. Labuschagne, and B. A. Watson. 2006. Convergence of Riesz space martingales. Indagationes Mathematicae 17 (2):271–83. doi:10.1016/S0019-3577(06)80021-2.
  • Labuschagne, C. C. A., and B. A. Watson. 2010. Discrete time stochastic integrals in Riesz spaces. Positivity 14 (4):859–75. doi:10.1007/s11117-010-0089-1.
  • Leon, J. A., and P. Protter. 1994. Some formulas for anticipative Girsanov transformations. In Chaos expansions, Multiple Wiener-Itô integrals and their applications, eds. C. Houdre and V. Perez-Abreu. CRC Press.
  • Merton, R. C. 1974. On the pricing of corporate debt: The risk structure of interest rates. The Journal of Finance 29:449–70.
  • Sadeghi, G., and A. Talebi. 2019. A trick for investigation of near-martingales in quantum probability spaces. Advances in Operator Theory 4 (4):784–92. doi:10.15352/aot.1903-1484.
  • Stoica, G. 1990. Martingales in vector lattices. Bulletin mathématique de la Société des Sciences Mathématiques 34 (82):357–62.
  • Troitsky, V. 2012. Martingales processes and martingale generalizations on Riesz spaces. PhD thesis, University of the Witwatersrand, Johannesburg, South Africa.
  • Vardy, J. J., and B. A. Watson. 2012. Markov processes on Riesz spaces. Positivity 16 (2):373–91. doi:10.1007/s11117-011-0121-0.

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