Path integral method for stochastic equations of financial engineering

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dc.citation.epage177
dc.citation.issue1
dc.citation.spage166
dc.contributor.affiliationНаціональний університет “Львівська політехніка”
dc.contributor.affiliationLviv Polytechnic National University
dc.contributor.authorЯнішевський, В. С.
dc.contributor.authorБарановська, С. П.
dc.contributor.authorYanishevskyi, V. S.
dc.contributor.authorBaranovska, S. P.
dc.coverage.placenameЛьвів
dc.coverage.placenameLviv
dc.date.accessioned2023-12-13T09:11:03Z
dc.date.available2023-12-13T09:11:03Z
dc.date.created2021-03-01
dc.date.issued2021-03-01
dc.description.abstractМетод функціонального інтегрування застосовано для визначення деяких середніх випадкових величини, що зустрічаються в задачах фінансової інженерії. Випадкова величина задається стохастичним рівнянням, де дрейф та волатильність є функціями випадкової величини. В результаті для густини умовної ймовірності побудовано функціональний інтеграл шляхом заміни змінних у функціональному інтегралі Вінера (мірі Вінера). Для стохастичного рівняння використано правило Іто для інтерпретації стохастичного інтегралу. Функціональний інтеграл для густини умовної ймовірності знайдено також у результаті розв’язку рівняння Фоккера–Планка, що відповідає стохастичному рівнянню. Показано, що два підходи дають еквівалентні результати.
dc.description.abstractThe integral path method was applied to determine certain stochastic variables which occur in problems of financial engineering. A stochastic variable was defined by a stochastic equation where drift and volatility are functions of a stochastic variable. As a result, for transition probability density, a path integral was built by substituting variables Wiener’s path integral (Wiener’s measure). For the stochastic equation, Ito rule was applied in order to interpret a stochastic integral. The path integral for transition probability density was also found as a result of the Fokker–Planck equation solution, corresponding to the stochastic equation. It was shown that these two approaches give equivalent results.
dc.format.extent166-177
dc.format.pages12
dc.identifier.citationYanishevskyi V. S. Path integral method for stochastic equations of financial engineering / V. S. Yanishevskyi, S. P. Baranovska // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2022. — Vol 9. — No 1. — P. 166–177.
dc.identifier.citationenYanishevskyi V. S. Path integral method for stochastic equations of financial engineering / V. S. Yanishevskyi, S. P. Baranovska // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2022. — Vol 9. — No 1. — P. 166–177.
dc.identifier.doi10.23939/mmc2022.01.166
dc.identifier.urihttps://ena.lpnu.ua/handle/ntb/60547
dc.language.isoen
dc.publisherВидавництво Львівської політехніки
dc.publisherLviv Politechnic Publishing House
dc.relation.ispartofMathematical Modeling and Computing, 1 (9), 2022
dc.relation.references[1] Gardiner C. W. Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer Series in Synergetics (2004).
dc.relation.references[2] Øksendal B. K. Stochastic differential equations: an introduction with applications. Springer–Verlag, Berlin, Heidelberg (2003).
dc.relation.references[3] Chaichian M., Demichev A. Path integrals in physics. Stochastic processes and quantum mechanics. Taylor & Francis (2001).
dc.relation.references[4] Kleinert H. Path integrals in quantum mechanics, statistics, polymer physics and financial markets. World Scientific Publishing Co. (2004).
dc.relation.references[5] Chaichian M., Demichev A. Path integrals in physics. QFT, statistical physics and modern applications. Taylor & Francis (2001).
dc.relation.references[6] Linetsky V. The Path Integral Approach to Financial Modeling and Options Pricing. Computational Economics. 11, 129–163 (1998).
dc.relation.references[7] Goovaerts M., De Schepper A., Decamps M. Closed-form approximations for diffusion densities: a path integral approach. Journal of Computational and Applied Mathematics. 164–165, 337–364 (2004).
dc.relation.references[8] Yanishevskyi V. S., Nodzhak L. S. The path integral method in interest rate models. Mathematical Modeling and Computing. 8 (1), 125–136 (2021).
dc.relation.references[9] Baaquie B. E. Quantum finance. Path Integrals and Hamiltonians for Options and Interest Rates. Cambridge University Press, New York (2004).
dc.relation.references[10] Jurisch A. Lagrangian dynamics in inhomogeneous and thermal environments. An application of the Onsager–Machlup theory I. Preprint arXiv:2012.06820v1 (2020).
dc.relation.references[11] Arnold P. Symmetric path integrals for stochastic equations with multiplicative noise. Physical Review E. 61, 6099 (2000).
dc.relation.references[12] Bressloff P. C. Construction of stochastic hybrid path integrals using operator methods. Journal of Physics A: Mathematical and Theoretical. 54 (18), 185001 (2021).
dc.relation.references[13] Cugliandolo L. F., Lecomte V., van Wijland F. Building a path-integral calculus: a covariant discretization approach. Journal of Physics A: Mathematical and Theoretical. 52, 50LT01 (2019).
dc.relation.references[14] Cugliandolo L. F., Lecomte V. Rules of calculus in the path integral representation of white noise Langevin equations: the Onsager–Machlup approach. Journal of Physics A: Mathematical and Theoretical. 50, 345001 (2017).
dc.relation.references[15] Arenas Z. G., Barci D. G. Functional integral approach for multiplicative stochastic processes. Physical Review E. 81, 051113 (2010).
dc.relation.references[16] Lyuu Y.-D. Financial Engineering and Computation: Principles, Mathematics, and Algorithms. Cambridge University Press (2004).
dc.relation.references[17] Yanishevsky V. S. Application of the path integral method to some stochastic models of financial engineering. Journal of Physical Studies. 25 (2), 2801 (2021).
dc.relation.references[18] Nazajkinsky B. J., Sternin B. Y., Shalatov В. J. The methods of non-commutive analysis. Moscov, Tehnosfera (2002), (in Russian).
dc.relation.references[19] Blazhievskii L. F. Path integrals and ordering of operators. Theoretical and Mathematical Physics. 40, 596–604 (1979).
dc.relation.references[20] Blazhyevskyi L. F., Yanishevsky V. S. The path integral representation kernel of evolution operator in Merton-Garman model. Condensed Matter Physics. 14 (2), 23001 (2011).
dc.relation.references[21] Grosche C., Steiner F. Handbook of Feynman Path Integrals. Springer, Berlin, Heidelberg (1998).
dc.relation.referencesen[1] Gardiner C. W. Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer Series in Synergetics (2004).
dc.relation.referencesen[2] Øksendal B. K. Stochastic differential equations: an introduction with applications. Springer–Verlag, Berlin, Heidelberg (2003).
dc.relation.referencesen[3] Chaichian M., Demichev A. Path integrals in physics. Stochastic processes and quantum mechanics. Taylor & Francis (2001).
dc.relation.referencesen[4] Kleinert H. Path integrals in quantum mechanics, statistics, polymer physics and financial markets. World Scientific Publishing Co. (2004).
dc.relation.referencesen[5] Chaichian M., Demichev A. Path integrals in physics. QFT, statistical physics and modern applications. Taylor & Francis (2001).
dc.relation.referencesen[6] Linetsky V. The Path Integral Approach to Financial Modeling and Options Pricing. Computational Economics. 11, 129–163 (1998).
dc.relation.referencesen[7] Goovaerts M., De Schepper A., Decamps M. Closed-form approximations for diffusion densities: a path integral approach. Journal of Computational and Applied Mathematics. 164–165, 337–364 (2004).
dc.relation.referencesen[8] Yanishevskyi V. S., Nodzhak L. S. The path integral method in interest rate models. Mathematical Modeling and Computing. 8 (1), 125–136 (2021).
dc.relation.referencesen[9] Baaquie B. E. Quantum finance. Path Integrals and Hamiltonians for Options and Interest Rates. Cambridge University Press, New York (2004).
dc.relation.referencesen[10] Jurisch A. Lagrangian dynamics in inhomogeneous and thermal environments. An application of the Onsager–Machlup theory I. Preprint arXiv:2012.06820v1 (2020).
dc.relation.referencesen[11] Arnold P. Symmetric path integrals for stochastic equations with multiplicative noise. Physical Review E. 61, 6099 (2000).
dc.relation.referencesen[12] Bressloff P. C. Construction of stochastic hybrid path integrals using operator methods. Journal of Physics A: Mathematical and Theoretical. 54 (18), 185001 (2021).
dc.relation.referencesen[13] Cugliandolo L. F., Lecomte V., van Wijland F. Building a path-integral calculus: a covariant discretization approach. Journal of Physics A: Mathematical and Theoretical. 52, 50LT01 (2019).
dc.relation.referencesen[14] Cugliandolo L. F., Lecomte V. Rules of calculus in the path integral representation of white noise Langevin equations: the Onsager–Machlup approach. Journal of Physics A: Mathematical and Theoretical. 50, 345001 (2017).
dc.relation.referencesen[15] Arenas Z. G., Barci D. G. Functional integral approach for multiplicative stochastic processes. Physical Review E. 81, 051113 (2010).
dc.relation.referencesen[16] Lyuu Y.-D. Financial Engineering and Computation: Principles, Mathematics, and Algorithms. Cambridge University Press (2004).
dc.relation.referencesen[17] Yanishevsky V. S. Application of the path integral method to some stochastic models of financial engineering. Journal of Physical Studies. 25 (2), 2801 (2021).
dc.relation.referencesen[18] Nazajkinsky B. J., Sternin B. Y., Shalatov V. J. The methods of non-commutive analysis. Moscov, Tehnosfera (2002), (in Russian).
dc.relation.referencesen[19] Blazhievskii L. F. Path integrals and ordering of operators. Theoretical and Mathematical Physics. 40, 596–604 (1979).
dc.relation.referencesen[20] Blazhyevskyi L. F., Yanishevsky V. S. The path integral representation kernel of evolution operator in Merton-Garman model. Condensed Matter Physics. 14 (2), 23001 (2011).
dc.relation.referencesen[21] Grosche C., Steiner F. Handbook of Feynman Path Integrals. Springer, Berlin, Heidelberg (1998).
dc.rights.holder© Національний університет “Львівська політехніка”, 2022
dc.subjectстохастичне рівняння
dc.subjectгустина умовної ймовірності
dc.subjectрівняння Фоккера–Планка
dc.subjectфункціональний інтеграл
dc.subjectstochastic equation
dc.subjectFokker–Plank equation
dc.subjecttransition probability density
dc.subjectpath integrals
dc.titlePath integral method for stochastic equations of financial engineering
dc.title.alternativeМетод функціонального інтегрування в стохастичних рівняннях фінансової інженерії
dc.typeArticle

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Mathematical Modeling And Computing. – 2022. – Vol. 9, No. 1