Proses stokastik

Realisasi simulasi komputer bagi proses Wiener atau pergerakan Brownian pada permukaan sfera. Proses Wiener secara meluas dianggap sebagai proses stokastik yang paling banyak dikaji dan berpusat dalam teori kebarangkalian.^[1]^[2]^[3]

Dalam teori kebarangkalian dan bidang yang berkaitan, satu proses stokastik (/stoʊˈkæstɪk/) atau proses rawak ialah objek matematik biasanya ditakrifkan sebagai keluarga daripada pemboleh ubah rawak. Proses stokastik digunakan secara meluas sebagai model matematik untuk sistem dan fenomena yang kelihatan berbeza-beza secara rawak. Contohnya termasuk pertumbuhan populasi bakteria, arus elektrik turun naik disebabkan oleh bunyi haba, atau pergerakan molekul gas.^[1]^[4]^[5] Proses stokastik mempunyai aplikasi dalam banyak bidang seperti biologi,^[6] kimia,^[7] ekologi,^[8] neurosains,^[9] fizik,^[10] pemprosesan imej, pemprosesan isyarat,^[11] teori kawalan,^[12] teori maklumat,^[13] sains komputer,^[14] kriptografi^[15] dan telekomunikasi.^[16] Tambahan pula, perubahan yang kelihatan rawak dalam pasaran kewangan telah mendorong penggunaan meluas proses stokastik dalam kewangan.^[17]^[18]^[19]

Secara matematik, ia bermaksud sistem yang mana wujud pemboleh ubah rawak X (t) yang bersandar masa. Jika x₁, x₂, dsb adalah ukuran X (t) pada masa t₁, t₂, dst, maka sistem ini diperihalkan sepenuhnya oleh ketumpatan kebarangkalian bersama p (x₁, t₁; x₂, t₂; ...).

Di dalam kes di mana kebarangkalian mengukur x adalah bebas daripada nilai sebelumnya, dan bebas daripada masa, kita ada "cubaan Bernoull", di mana hukum kebarangkalian untuk x yang sama pakai. Satu lagi jenis proses stokastik ialah "proses Markov" di mana kebarangkalian tersebut hanya bergantung kepada keadaan x sebelumnya.

Rujukan

^ ^a ^b Joseph L. Doob (1990). Stochastic processes. Wiley. m/s. 46, 47.
^ L. C. G. Rogers; David Williams (2000). Diffusions, Markov Processes, and Martingales: Volume 1, Foundations. Cambridge University Press. m/s. 1. ISBN 978-1-107-71749-7.
^ J. Michael Steele (2012). Stochastic Calculus and Financial Applications. Springer Science & Business Media. m/s. 29. ISBN 978-1-4684-9305-4.
^ Emanuel Parzen (2015). Stochastic Processes. Courier Dover Publications. m/s. 7, 8. ISBN 978-0-486-79688-8.
^ Iosif Ilyich Gikhman; Anatoly Vladimirovich Skorokhod (1969). Introduction to the Theory of Random Processes. Courier Corporation. m/s. 1. ISBN 978-0-486-69387-3.
^ Paul C. Bressloff (2014). Stochastic Processes in Cell Biology. Springer. ISBN 978-3-319-08488-6.
^ N.G. Van Kampen (2011). Stochastic Processes in Physics and Chemistry. Elsevier. ISBN 978-0-08-047536-3.
^ Russell Lande; Steinar Engen; Bernt-Erik Sæther (2003). Stochastic Population Dynamics in Ecology and Conservation. Oxford University Press. ISBN 978-0-19-852525-7.
^ Carlo Laing; Gabriel J Lord (2010). Stochastic Methods in Neuroscience. OUP Oxford. ISBN 978-0-19-923507-0.
^ Wolfgang Paul; Jörg Baschnagel (2013). Stochastic Processes: From Physics to Finance. Springer Science & Business Media. ISBN 978-3-319-00327-6.
^ Edward R. Dougherty (1999). Random processes for image and signal processing. SPIE Optical Engineering Press. ISBN 978-0-8194-2513-3.
^ Dimitri P. Bertsekas (1996). Stochastic Optimal Control: The Discrete-Time Case. Athena Scientific. ISBN 1-886529-03-5.
^ Thomas M. Cover; Joy A. Thomas (2012). Elements of Information Theory. John Wiley & Sons. m/s. 71. ISBN 978-1-118-58577-1.
^ Michael Baron (2015). Probability and Statistics for Computer Scientists, Second Edition. CRC Press. m/s. 131. ISBN 978-1-4987-6060-7.
^ Jonathan Katz; Yehuda Lindell (2007). Introduction to Modern Cryptography: Principles and Protocols. CRC Press. m/s. 26. ISBN 978-1-58488-586-3.
^ François Baccelli; Bartlomiej Blaszczyszyn (2009). Stochastic Geometry and Wireless Networks. Now Publishers Inc. ISBN 978-1-60198-264-3.
^ J. Michael Steele (2001). Stochastic Calculus and Financial Applications. Springer Science & Business Media. ISBN 978-0-387-95016-7.
^ Marek Musiela; Marek Rutkowski (2006). Martingale Methods in Financial Modelling. Springer Science & Business Media. ISBN 978-3-540-26653-2.
^ Steven E. Shreve (2004). Stochastic Calculus for Finance II: Continuous-Time Models. Springer Science & Business Media. ISBN 978-0-387-40101-0.

Bacaan lanjut

Artikel

Applebaum, David (2004). "Lévy processes: From probability to finance and quantum groups". Notices of the AMS. 51 (11): 1336–1347.
Cramer, Harald (1976). "Half a Century with Probability Theory: Some Personal Recollections". The Annals of Probability. 4 (4): 509–546. doi:10.1214/aop/1176996025. ISSN 0091-1798.
Guttorp, Peter; Thorarinsdottir, Thordis L. (2012). "What Happened to Discrete Chaos, the Quenouille Process, and the Sharp Markov Property? Some History of Stochastic Point Processes". International Statistical Review. 80 (2): 253–268. doi:10.1111/j.1751-5823.2012.00181.x. ISSN 0306-7734.
Jarrow, Robert; Protter, Philip (2004). "A short history of stochastic integration and mathematical finance: the early years, 1880–1970". A Festschrift for Herman Rubin. Institute of Mathematical Statistics Lecture Notes - Monograph Series. m/s. 75–91. doi:10.1214/lnms/1196285381. ISBN 978-0-940600-61-4. ISSN 0749-2170.
Meyer, Paul-André (2009). "Stochastic Processes from 1950 to the Present". Electronic Journal for History of Probability and Statistics. 5 (1): 1–42.

Buku

Robert J. Adler (2010). The Geometry of Random Fields. SIAM. ISBN 978-0-89871-693-1.
Robert J. Adler; Jonathan E. Taylor (2009). Random Fields and Geometry. Springer Science & Business Media. ISBN 978-0-387-48116-6.
Pierre Brémaud (2013). Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues. Springer Science & Business Media. ISBN 978-1-4757-3124-8.
Joseph L. Doob (1990). Stochastic processes. Wiley.
Anders Hald (2005). A History of Probability and Statistics and Their Applications before 1750. John Wiley & Sons. ISBN 978-0-471-72517-6.
Crispin Gardiner (2010). Stochastic Methods. Springer. ISBN 978-3-540-70712-7.
Iosif I. Gikhman; Anatoly Vladimirovich Skorokhod (1996). Introduction to the Theory of Random Processes. Courier Corporation. ISBN 978-0-486-69387-3.
Emanuel Parzen (2015). Stochastic Processes. Courier Dover Publications. ISBN 978-0-486-79688-8.
Murray Rosenblatt (1962). Random Processes. Oxford University Press.

Pautan luar

Kategori berkenaan Proses stokastik di Wikimedia Commons

[doob1953stochasticP46to47-1] Joseph L. Doob (1990). Stochastic processes. Wiley. m/s. 46, 47.

[RogersWilliams2000page1-2] L. C. G. Rogers; David Williams (2000). Diffusions, Markov Processes, and Martingales: Volume 1, Foundations. Cambridge University Press. m/s. 1. ISBN 978-1-107-71749-7.

[Steele2012page29-3] J. Michael Steele (2012). Stochastic Calculus and Financial Applications. Springer Science & Business Media. m/s. 29. ISBN 978-1-4684-9305-4.

[Parzen1999-4] Emanuel Parzen (2015). Stochastic Processes. Courier Dover Publications. m/s. 7, 8. ISBN 978-0-486-79688-8.

[GikhmanSkorokhod1969page1-5] Iosif Ilyich Gikhman; Anatoly Vladimirovich Skorokhod (1969). Introduction to the Theory of Random Processes. Courier Corporation. m/s. 1. ISBN 978-0-486-69387-3.

[Bressloff2014-6] Paul C. Bressloff (2014). Stochastic Processes in Cell Biology. Springer. ISBN 978-3-319-08488-6.

[Kampen2011-7] N.G. Van Kampen (2011). Stochastic Processes in Physics and Chemistry. Elsevier. ISBN 978-0-08-047536-3.

[LandeEngen2003-8] Russell Lande; Steinar Engen; Bernt-Erik Sæther (2003). Stochastic Population Dynamics in Ecology and Conservation. Oxford University Press. ISBN 978-0-19-852525-7.

[LaingLord2010-9] Carlo Laing; Gabriel J Lord (2010). Stochastic Methods in Neuroscience. OUP Oxford. ISBN 978-0-19-923507-0.

[PaulBaschnagel2013-10] Wolfgang Paul; Jörg Baschnagel (2013). Stochastic Processes: From Physics to Finance. Springer Science & Business Media. ISBN 978-3-319-00327-6.

[Dougherty1999-11] Edward R. Dougherty (1999). Random processes for image and signal processing. SPIE Optical Engineering Press. ISBN 978-0-8194-2513-3.

[Bertsekas1996-12] Dimitri P. Bertsekas (1996). Stochastic Optimal Control: The Discrete-Time Case. Athena Scientific. ISBN 1-886529-03-5.

[CoverThomas2012page71-13] Thomas M. Cover; Joy A. Thomas (2012). Elements of Information Theory. John Wiley & Sons. m/s. 71. ISBN 978-1-118-58577-1.

[Baron2015-14] Michael Baron (2015). Probability and Statistics for Computer Scientists, Second Edition. CRC Press. m/s. 131. ISBN 978-1-4987-6060-7.

[15] Jonathan Katz; Yehuda Lindell (2007). Introduction to Modern Cryptography: Principles and Protocols. CRC Press. m/s. 26. ISBN 978-1-58488-586-3.

[BaccelliBlaszczyszyn2009-16] François Baccelli; Bartlomiej Blaszczyszyn (2009). Stochastic Geometry and Wireless Networks. Now Publishers Inc. ISBN 978-1-60198-264-3.

[Steele2001-17] J. Michael Steele (2001). Stochastic Calculus and Financial Applications. Springer Science & Business Media. ISBN 978-0-387-95016-7.

[MusielaRutkowski2006-18] Marek Musiela; Marek Rutkowski (2006). Martingale Methods in Financial Modelling. Springer Science & Business Media. ISBN 978-3-540-26653-2.

[Shreve2004-19] Steven E. Shreve (2004). Stochastic Calculus for Finance II: Continuous-Time Models. Springer Science & Business Media. ISBN 978-0-387-40101-0.

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