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ON THE CONVERGENCE OF DISCRETE PROCESSES WITH MULTIPLE INDEPENDENT VARIABLES

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  06 March 2017

N. ISHIMURA Open the ORCID record for N. ISHIMURA [Opens in a new window]  and
N. YOSHIDA
N. ISHIMURA*
Affiliation:
Faculty of Commerce, Chuo University, Hachioji, Tokyo 192-0393, Japan email naoyuki@tamacc.chuo-u.ac.jp
N. YOSHIDA
Affiliation:
Graduate School of Economics, Hitotsubashi University, Tokyo 186-8601, Japan email ed141003@g.hit-u.ac.jp
*
✉naoyuki@tamacc.chuo-u.ac.jp
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Abstract

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We discuss discrete stochastic processes with two independent variables: one is the standard symmetric random walk, and the other is the Poisson process. Convergence of discrete stochastic processes is analysed, such that the symmetric random walk tends to the standard Brownian motion. We show that a discrete analogue of Ito’s formula converges to the corresponding continuous formula.

Keywords

discrete processesconvergenceIto’s formula

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60J75: Jump processes
Type
Research Article
Information
The ANZIAM Journal , Volume 58 , Issue 3-4 , April 2017 , pp. 379 - 385
Copyright
© 2017 Australian Mathematical Society 

References

Cont, R. and Tankov, P., Financial modelling with jump processes (CRC Press, Boca Raton, FL, 2003).Google Scholar
Cox, J. C., Ross, S. A. and Rubinstein, M., “Option pricing: a simplified approach”, J. Financial Econ. 7 (1979) 229–263; doi:10.1016/0304-405X(79)90015-1.CrossRefGoogle Scholar
Fujita, T., Ishimura, N. and Kawai, N., “Discrete stochastic calculus and its applications: an expository note”, Adv. Math. Econ. 16 (2012) 119–131; doi:10.1007/978-4-431-54114-1_6.CrossRefGoogle Scholar
Fujita, T. and Kawanishi, Y., “A proof of Ito’s formula using a discrete Ito’s formula”, Stud. Sci. Math. Hungar. 45 (2008) 125–134; doi:10.1556/SScMath.2007.1043.Google Scholar
Ishimura, N. and Mita, Y., “A note on the optimal portfolio problem in discrete processes”, Kybernetika 45 (2009) 681–688; http://www.kybernetika.cz/content/2009/4/681.Google Scholar
Merton, R. C., “Option pricing when underlying stock returns are discontinuous”, J. Financial Econ. 3 (1976) 125–144; doi:10.1016/0304-405X(76)90022-2.CrossRefGoogle Scholar
Revuz, D. and Yor, M., Continuous martingales and Brownian motion, 3rd edn (Springer, New York, 2005).Google Scholar
Szabados, T., “An elemntary introduction to the Wiener process and stochastic integrals”, Stud. Sci. Math. Hungar. 31 (1996) 249–297; arXiv:1008.1510v1.Google Scholar
Yoshida, N., “Remarks on the transformation of Ito’s formula for jump-diffusion processes”, JSIAM Lett. 7 (2015) 29–32; https://www.jstage.jst.go.jp/article/jsiaml/7/0/7_29/_article.CrossRefGoogle Scholar
Yoshida, N. and Ishimura, N., “Remarks on the optimal portfolio problem in discrete variables with multiple stochastic processes”, Int. J. Model. Optim. 6 (2016) 96–99; doi:10.7763/IJMO.2016.V6.511.CrossRefGoogle Scholar