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Itô's stochastic calculus: Its surprising power for applications

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  • Kunita, Hiroshi

Abstract

We trace Itô's early work in the 1940s, concerning stochastic integrals, stochastic differential equations (SDEs) and Itô's formula. Then we study its developments in the 1960s, combining it with martingale theory. Finally, we review a surprising application of Itô's formula in mathematical finance in the 1970s. Throughout the paper, we treat Itô's jump SDEs driven by Brownian motions and Poisson random measures, as well as the well-known continuous SDEs driven by Brownian motions.

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  • Kunita, Hiroshi, 2010. "Itô's stochastic calculus: Its surprising power for applications," Stochastic Processes and their Applications, Elsevier, vol. 120(5), pages 622-652, May.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:5:p:622-652
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    1. Ishikawa, Yasushi & Kunita, Hiroshi, 2006. "Malliavin calculus on the Wiener-Poisson space and its application to canonical SDE with jumps," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 1743-1769, December.
    2. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
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    2. Jorge Sanchez-Ortiz & Omar U. Lopez-Cresencio & Francisco J. Ariza-Hernandez & Martin P. Arciga-Alejandre, 2021. "Cauchy Problem for a Stochastic Fractional Differential Equation with Caputo-Itô Derivative," Mathematics, MDPI, vol. 9(13), pages 1-10, June.
    3. Ishikawa, Yasushi & Kunita, Hiroshi & Tsuchiya, Masaaki, 2018. "Smooth density and its short time estimate for jump process determined by SDE," Stochastic Processes and their Applications, Elsevier, vol. 128(9), pages 3181-3219.
    4. Kang, Yuanbao & Wang, Caishi, 2014. "Itô formula for one-dimensional continuous-time quantum random walk," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 414(C), pages 154-162.
    5. Biane, Philippe, 2010. "Itô's stochastic calculus and Heisenberg commutation relations," Stochastic Processes and their Applications, Elsevier, vol. 120(5), pages 698-720, May.
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    12. Tan, Li & Jin, Wei & Hou, Zhenting, 2013. "Weak convergence of functional stochastic differential equations with variable delays," Statistics & Probability Letters, Elsevier, vol. 83(11), pages 2592-2599.
    13. Liu, Meng & Bai, Chuanzhi, 2016. "Optimal harvesting of a stochastic mutualism model with Lévy jumps," Applied Mathematics and Computation, Elsevier, vol. 276(C), pages 301-309.
    14. Gao, Miaomiao & Jiang, Daqing & Hayat, Tasawar & Alsaedi, Ahmed, 2019. "Threshold behavior of a stochastic Lotka–Volterra food chain chemostat model with jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 191-203.
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    17. Liu, Meng & Deng, Meiling & Du, Bo, 2015. "Analysis of a stochastic logistic model with diffusion," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 169-182.

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