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Journal articles on the topic 'Itô calculus'

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Published: 4 June 2021

Last updated: 26 July 2025

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1

Dupire, Bruno. "Functional Itô calculus." Quantitative Finance 19, no. 5 (2019): 721–29. http://dx.doi.org/10.1080/14697688.2019.1575974.

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2

Cont, Rama, and Ruhong Jin. "Fractional Ito calculus." Transactions of the American Mathematical Society, Series B 11, no. 22 (2024): 727–61. http://dx.doi.org/10.1090/btran/185.

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We derive Itô–type change of variable formulas for smooth functionals of irregular paths with nonzero p p th variation along a sequence of partitions, where p ≥ 1 p \geq 1 is arbitrary, in terms of fractional derivative operators. Our results extend the results of the Föllmer–Itô calculus to the general case of paths with ‘fractional’ regularity. In the case where p p is not an integer, we show that the change of variable formula may sometimes contain a nonzero ‘fractional’ Itô remainder term and provide a representation for this remainder term. These results are then extended to functionals o

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3

Cosso, Andrea, and Francesco Russo. "Functional Itô versus Banach space stochastic calculus and strict solutions of semilinear path-dependent equations." Infinite Dimensional Analysis, Quantum Probability and Related Topics 19, no. 04 (2016): 1650024. http://dx.doi.org/10.1142/s0219025716500247.

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Functional Itô calculus was introduced in order to expand a functional [Formula: see text] depending on time [Formula: see text], past and present values of the process [Formula: see text]. Another possibility to expand [Formula: see text] consists in considering the path [Formula: see text] as an element of the Banach space of continuous functions on [Formula: see text] and to use Banach space stochastic calculus. The aim of this paper is threefold. (1) To reformulate functional Itô calculus, separating time and past, making use of the regularization procedures which match more naturally the

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4

Duong, Dam Ton, and Hao Ngoc Duong. "ITÔ – HERMITE RANDOM PROCESS." Science and Technology Development Journal 13, no. 3 (2010): 13–18. http://dx.doi.org/10.32508/stdj.v13i3.2149.

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5

Akahori, Jirô, Corina Constantinescu, and Kei Miyagi. "Itô calculus for Cramér-Lundberg model." JSIAM Letters 12 (2020): 25–28. http://dx.doi.org/10.14495/jsiaml.12.25.

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6

Aletti, Giacomo, and Diane Saada. "Set-Indexed Itô Calculus Along Paths." Stochastic Analysis and Applications 22, no. 4 (2004): 1027–66. http://dx.doi.org/10.1081/sap-120037630.

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7

PICKEN, R. F., and J. N. WEBB. "DERIVATION OF ANOMALIES USING THE ITÔ STOCHASTIC CALCULUS." International Journal of Modern Physics A 04, no. 13 (1989): 3179–91. http://dx.doi.org/10.1142/s0217751x89001291.

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Itô’s stochastic calculus is used to derive a number of anomalous Ward identities. The general features of the method are described and similarities with Fujikawa’s path integral approach are pointed out. The stochastic calculus provides an easily understood origin for anomalies within the framework of the stochastic quantization scheme.

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8

LENCZEWSKI, ROMUALD. "FILTERED STOCHASTIC CALCULUS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 04, no. 03 (2001): 309–46. http://dx.doi.org/10.1142/s0219025701000553.

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By introducing a color filtration to the multiplicity space [Formula: see text], we extend the quantum Itô calculus on multiple symmetric Fock space [Formula: see text] to the framework of filtered adapted biprocesses. In this new notion of adaptedness, "classical" time filtration makes the integrands similar to adapted processes, whereas "quantum" color filtration produces their deviations from adaptedness. An important feature of this calculus, which we call filtered stochastic calculus, is that it provides an explicit interpolation between the main types of calculi, regardless of the type o

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9

Walsh, Alexander. "Extended Itô calculus for symmetric Markov processes." Bernoulli 18, no. 4 (2012): 1150–71. http://dx.doi.org/10.3150/11-bej377.

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10

HUDSON, R. "Itô calculus and quantisation of Lie bialgebras." Annales de l'Institut Henri Poincare (B) Probability and Statistics 41, no. 3 (2005): 375–90. http://dx.doi.org/10.1016/j.anihpb.2004.09.008.

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11

DI NUNNO, GIULIA, THILO MEYER-BRANDIS, BERNT ØKSENDAL, and FRANK PROSKE. "MALLIAVIN CALCULUS AND ANTICIPATIVE ITÔ FORMULAE FOR LÉVY PROCESSES." Infinite Dimensional Analysis, Quantum Probability and Related Topics 08, no. 02 (2005): 235–58. http://dx.doi.org/10.1142/s0219025705001950.

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We introduce the forward integral with respect to a pure jump Lévy process and prove an Itô formula for this integral. Then we use Mallivin calculus to establish a relationship between the forward integral and the Skorohod integral and apply this to obtain an Itô formula for the Skorohod integral.

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12

HU, YAOZHONG, and BERNT ØKSENDAL. "FRACTIONAL WHITE NOISE CALCULUS AND APPLICATIONS TO FINANCE." Infinite Dimensional Analysis, Quantum Probability and Related Topics 06, no. 01 (2003): 1–32. http://dx.doi.org/10.1142/s0219025703001110.

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The purpose of this paper is to develop a fractional white noise calculus and to apply this to markets modeled by (Wick–) Itô type of stochastic differential equations driven by fractional Brownian motion BH(t); 1/2 < H < 1. We show that if we use an Itô type of stochastic integration with respect to BH(t) (as developed in Ref. 8), then the corresponding Itô fractional Black–Scholes market has no arbitrage, contrary to the situation when the pathwise integration is used. Moreover, we prove that our Itô fractional Black–Scholes market is complete and we compute explicitly the price and re

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13

ES-SEBAIY, KHALIFA, and CIPRIAN A. TUDOR. "MULTIDIMENSIONAL BIFRACTIONAL BROWNIAN MOTION: ITÔ AND TANAKA FORMULAS." Stochastics and Dynamics 07, no. 03 (2007): 365–88. http://dx.doi.org/10.1142/s0219493707002050.

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14

Graczyk, P., and L. Vostrikova. "The Moments of Wishart Processes via Itô Calculus." Theory of Probability & Its Applications 51, no. 4 (2007): 609–25. http://dx.doi.org/10.1137/s0040585x97982682.

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15

Veeravalli, Tanya, and Maxim Raginsky. "Revisiting Stochastic Realization Theory using Functional Itô Calculus." IFAC-PapersOnLine 58, no. 17 (2024): 326–31. http://dx.doi.org/10.1016/j.ifacol.2024.10.190.

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16

Braumann, Carlos A. "Itô versus Stratonovich calculus in random population growth." Mathematical Biosciences 206, no. 1 (2007): 81–107. http://dx.doi.org/10.1016/j.mbs.2004.09.002.

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17

Vovk, Vladimir. "Itô Calculus without Probability in Idealized Financial Markets*." Lithuanian Mathematical Journal 55, no. 2 (2015): 270–90. http://dx.doi.org/10.1007/s10986-015-9280-1.

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18

Graham, Robert. "Covariant stochastic calculus in the sense of Itô." Physics Letters A 109, no. 5 (1985): 209–12. http://dx.doi.org/10.1016/0375-9601(85)90304-4.

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19

Ebrahimi-Fard, Kurusch, Simon J. A. Malham, Frédéric Patras, and Anke Wiese. "Flows and stochastic Taylor series in Itô calculus." Journal of Physics A: Mathematical and Theoretical 48, no. 49 (2015): 495202. http://dx.doi.org/10.1088/1751-8113/48/49/495202.

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20

SOBEHART, J. R., and S. C. KEENAN. "A PARADOX OF INTUITION: HEDGING THE LIMIT OR HEDGING IN THE LIMIT?" International Journal of Theoretical and Applied Finance 05, no. 07 (2002): 729–36. http://dx.doi.org/10.1142/s0219024902001705.

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Here we review the notion of covergence in Itô calculus and its application to the Black-Scholes options pricing model and its extensions. The concept of covergence is fundamental to the development of the differential calculus of stochastic processes. It is also the key to understanding the validity of the no arbitrage condition imposed by Black and Scholes (1973) that leads to their options pricing equation.

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21

MANNELLA, RICCARDO, and PETER V. E. McCLINTOCK. "ITÔ VERSUS STRATONOVICH: 30 YEARS LATER." Fluctuation and Noise Letters 11, no. 01 (2012): 1240010. http://dx.doi.org/10.1142/s021947751240010x.

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The Itô versus Stratonovich controversy, about the "correct" calculus to use for integration of Langevin equations, was settled to general satisfaction some 30 years ago. Recently, however, it has started to re-emerge, following the advent of new experimental techniques. We briefly review the historical background and discuss critically some of the most recent contributions. We show that some of the new findings are not well based.

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22

Alshanskiy, M. A. "Wiener-Itô Chaos Expansion of Hilbert Space Valued Random Variables." Journal of Probability 2014 (April 7, 2014): 1–9. http://dx.doi.org/10.1155/2014/786854.

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The notion of n-fold iterated Itô integral with respect to a cylindrical Hilbert space valued Wiener process is introduced and the Wiener-Itô chaos expansion is obtained for a square Bochner integrable Hilbert space valued random variable. The expansion can serve a basis for developing the Hilbert space valued analog of Malliavin calculus of variations which can then be applied to the study of stochastic differential equations in Hilbert spaces and their solutions.

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23

Yor, M. "DIFFUSIONS, MARKOV PROCESSES AND MARTINGALES: Volume 2: Itô Calculus." Bulletin of the London Mathematical Society 21, no. 1 (1989): 106–7. http://dx.doi.org/10.1112/blms/21.1.106.

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24

Leão, Dorival, Alberto Ohashi, and Alexandre B. Simas. "A weak version of path-dependent functional Itô calculus." Annals of Probability 46, no. 6 (2018): 3399–441. http://dx.doi.org/10.1214/17-aop1250.

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25

Braumann, Carlos A. "Harvesting in a random environment: Itô or Stratonovich calculus?" Journal of Theoretical Biology 244, no. 3 (2007): 424–32. http://dx.doi.org/10.1016/j.jtbi.2006.08.029.

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26

Cont, Rama, and David-Antoine Fournié. "Functional Itô calculus and stochastic integral representation of martingales." Annals of Probability 41, no. 1 (2013): 109–33. http://dx.doi.org/10.1214/11-aop721.

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27

Patrascioiu, A., та J. L. Richard. "Itô calculus for σ-models and Yang-Mills theories". Letters in Mathematical Physics 9, № 3 (1985): 191–94. http://dx.doi.org/10.1007/bf00402828.

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28

Ben Makhlouf, Abdellatif, Lassaad Mchiri, Hakeem A. Othman, Hafedh M. S. Rguigui, and Salah Boulaaras. "Proportional Itô–Doob Stochastic Fractional Order Systems." Mathematics 11, no. 9 (2023): 2049. http://dx.doi.org/10.3390/math11092049.

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In this article, we discuss the existence and uniqueness of proportional Itô–Doob stochastic fractional order systems (PIDSFOS) by using the Picard iteration method. The paper provides new results using the proportional fractional integral and stochastic calculus techniques. We have shown the convergence of the solution of the averaged PIDSFOS to that of the standard PIDSFOS in the context of the mean square and also in probability. One example is given to illustrate our results.

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29

Carr, Peter. "First-order calculus and option pricing." Journal of Financial Engineering 01, no. 01 (2014): 1450009. http://dx.doi.org/10.1142/s2345768614500093.

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The modern theory of option pricing rests on Itô calculus, which is a second-order calculus based on the quadratic variation of a stochastic process. One can instead develop a first-order stochastic calculus, which is based on the running minimum of a stochastic process, rather than its quadratic variation. We focus here on the analog of geometric Brownian motion (GBM) in this alternative stochastic calculus. The resulting stochastic process is a positive continuous martingale whose laws are easy to calculate. We show that this analog behaves locally like a GBM whenever its running minimum dec

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30

Ji, Shaolin, and Shuzhen Yang. "Classical Solutions of Path-Dependent PDEs and Functional Forward-Backward Stochastic Systems." Mathematical Problems in Engineering 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/423101.

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In this paper we study the relationship between functional forward-backward stochastic systems and path-dependent PDEs. In the framework of functional Itô calculus, we introduce a path-dependent PDE and prove that its solution is uniquely determined by a functional forward-backward stochastic system.

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31

Ruge-Leiva, Diego Iván. "The impact of Kiyoshi Ito´s stochastic calculus of financial economics." ODEON, no. 10 (October 6, 2016): 157. http://dx.doi.org/10.18601/17941113.n10.07.

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We discuss the direct or indirect incorporation into financial economics of Kiyoshi Itô´s work on stochastic calculus, particularly the Itô formula, the relevance of his findings for option pricing theory and the way his work has been used to find a unique option pricing function in a competitive and non-arbitrage market. On that basis, we discuss how the option pricing theory may be linked with the general equilibrium theory and other aspects of conventional economics, and finally, Itô’s role in econophysics.

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32

Framstad, Nils Chr. "Continuous-time (Ross-type) portfolio separation, (almost) without Itô calculus." Stochastics 89, no. 1 (2016): 38–64. http://dx.doi.org/10.1080/17442508.2015.1132218.

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33

Jazaerli, Samy, and Yuri F. Saporito. "Functional Itô calculus, path-dependence and the computation of Greeks." Stochastic Processes and their Applications 127, no. 12 (2017): 3997–4028. http://dx.doi.org/10.1016/j.spa.2017.03.015.

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34

Holm, Darryl D. "Stochastic modelling in fluid dynamics: Itô versus Stratonovich." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2237 (2020): 20190812. http://dx.doi.org/10.1098/rspa.2019.0812.

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Suppose the observations of Lagrangian trajectories for fluid flow in some physical situation can be modelled sufficiently accurately by a spatially correlated Itô stochastic process (with zero mean) obtained from data which is taken in fixed Eulerian space. Suppose we also want to apply Hamilton’s principle to derive the stochastic fluid equations for this situation. Now, the variational calculus for applying Hamilton’s principle requires the Stratonovich process, so we must transform from Itô noise in the data frame to the equivalent Stratonovich noise. However, the transformation from the I

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35

Zayed, Elsayed M. E., Mohamed E. M. Alngar, Reham M. A. Shohib, et al. "Dispersive Optical Solitons with Differential Group Delay Having Multiplicative White Noise by Itô Calculus." Electronics 12, no. 3 (2023): 634. http://dx.doi.org/10.3390/electronics12030634.

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The current paper recovers dispersive optical solitons in birefringent fibers that are modeled by the Schrödinger–Hirota equation with differential group delay and white noise. Itô Calculus conducts the preliminary analysis. The G′/G-expansion approach and the enhanced Kudryashov’s scheme gave way to a wide spectrum of soliton solutions with the white noise component reflected in the phase of the soliton.

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36

Håkansson, P., M. Mella, Dario Bressanini, Gabriele Morosi, and Marta Patrone. "Improved diffusion Monte Carlo propagators for bosonic systems using Itô calculus." Journal of Chemical Physics 125, no. 18 (2006): 184106. http://dx.doi.org/10.1063/1.2371077.

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37

Dorogovtsev, Andrey A. "An approach to the stochastic calculus in the non-Gaussian case." Journal of Applied Mathematics and Stochastic Analysis 8, no. 4 (1995): 361–70. http://dx.doi.org/10.1155/s1048953395000323.

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38

Almulhem, Munerah, Samia Z. Hassan, Alanwood Al-buainain, Mohammed A. Sohaly, and Mahmoud A. E. Abdelrahman. "Characteristics of Solitary Stochastic Structures for Heisenberg Ferromagnetic Spin Chain Equation." Symmetry 15, no. 4 (2023): 927. http://dx.doi.org/10.3390/sym15040927.

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The impact of Stratonovich integrals on the solutions of the Heisenberg ferromagnetic spin chain equation using the unified solver approach is examined in this study. In particular, using arbitrary parameters, the traveling wave arrangements of rational, trigonometric, and hyperbolic functions are developed. The detailed arrangements are exceptionally critical for clarifying diverse complex wonders in plasma material science, optical fiber, quantum mechanics, super liquids and so on. Here, the Itô stochastic calculus and the Stratonovich stochastic calculus are considered. To describe the dyna

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39

Kendall, Wilfrid S. "A remark on the proof of Itô's formula for C2 functions of continuous semimartingales." Journal of Applied Probability 29, no. 1 (1992): 216–21. http://dx.doi.org/10.2307/3214807.

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The Itô formula is the fundamental theorem of stochastic calculus. This short note presents a new proof of Itô's formula for the case of continuous semimartingales. The new proof is more geometric than previous approaches, and has the particular advantage of generalizing immediately to the multivariate case without extra notational complexity.

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40

Kendall, Wilfrid S. "A remark on the proof of Itô's formula for C2 functions of continuous semimartingales." Journal of Applied Probability 29, no. 01 (1992): 216–21. http://dx.doi.org/10.1017/s0021900200106783.

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The Itô formula is the fundamental theorem of stochastic calculus. This short note presents a new proof of Itô's formula for the case of continuous semimartingales. The new proof is more geometric than previous approaches, and has the particular advantage of generalizing immediately to the multivariate case without extra notational complexity.

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41

Barndorff-Nielsen, Ole E., José Manuel Corcuera, Mark Podolskij, and Jeannette H. C. Woerner. "Bipower Variation for Gaussian Processes with Stationary Increments." Journal of Applied Probability 46, no. 1 (2009): 132–50. http://dx.doi.org/10.1239/jap/1238592121.

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Convergence in probability and central limit laws of bipower variation for Gaussian processes with stationary increments and for integrals with respect to such processes are derived. The main tools of the proofs are some recent powerful techniques of Wiener/Itô/Malliavin calculus for establishing limit laws, due to Nualart, Peccati, and others.

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42

Barndorff-Nielsen, Ole E., José Manuel Corcuera, Mark Podolskij, and Jeannette H. C. Woerner. "Bipower Variation for Gaussian Processes with Stationary Increments." Journal of Applied Probability 46, no. 01 (2009): 132–50. http://dx.doi.org/10.1017/s0021900200005271.

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Convergence in probability and central limit laws of bipower variation for Gaussian processes with stationary increments and for integrals with respect to such processes are derived. The main tools of the proofs are some recent powerful techniques of Wiener/Itô/Malliavin calculus for establishing limit laws, due to Nualart, Peccati, and others.

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43

Navickas, Zenonas, Inga Timofejeva, Tadas Telksnys, Romas Marcinkevicius, and Minvydas Ragulskis. "Construction of special soliton solutions to the stochastic Riccati equation." Open Mathematics 20, no. 1 (2022): 829–44. http://dx.doi.org/10.1515/math-2022-0051.

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Abstract A scheme for the analytical stochastization of ordinary differential equations (ODEs) is presented in this article. Using Itô calculus, an ODE is transformed into a stochastic differential equation (SDE) in such a way that the analytical solutions of the obtained equation can be constructed. Furthermore, the constructed stochastic trajectories remain bounded in the same interval as the deterministic solutions. The proposed approach is in a stark contrast to methods based on the randomization of solution trajectories and is not focused on the analysis of martingales. This article exten

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44

Ladde, G. S., and Hu Bijin. "Ornstein–Uhlenbeck operator and Wiener functionals generated by Itô- and Mcshane–calculus." Stochastic Analysis and Applications 5, no. 1 (1987): 27–51. http://dx.doi.org/10.1080/07362998708809106.

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45

Ayed, Wided. "Module Free White Noise Flows." Open Systems & Information Dynamics 25, no. 04 (2018): 1850018. http://dx.doi.org/10.1142/s123016121850018x.

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The main result of this paper is to extend to Hilbert module level the proof of the inclusion of (non-Hamiltonian) stochastic differential equations based on free noise into the class of Hamiltonian equations driven by free white noise. To achieve this goal, free white noise calculus is extended to a trivial Hilbert module. The white noise formulation of the Ito table is radically different from the usual Itô tables, both classical and quantum and, combined with the Accardi–Boukas approach to Ito algebra, allows to drastically simplify calculations. Infinitesimal generators of Hilbert module f

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46

Fredericks, E., and F. M. Mahomed. "Symmetries of th-Order Approximate Stochastic Ordinary Differential Equations." Journal of Applied Mathematics 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/263570.

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Symmetries of th-order approximate stochastic ordinary differential equations (SODEs) are studied. The determining equations of these SODEs are derived in an Itô calculus context. These determining equations are not stochastic in nature. SODEs are normally used to model nature (e.g., earthquakes) or for testing the safety and reliability of models in construction engineering when looking at the impact of random perturbations.

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47

Kendall, Wilfrid S. "Symbolic computation and the diffusion of shapes of triads." Advances in Applied Probability 20, no. 4 (1988): 775–97. http://dx.doi.org/10.2307/1427360.

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This paper introduces the use of symbolic computation (also known as computer algebra) in stochastic analysis and particularly in the Itô calculus. Two related examples are considered: the Clifford-Green theorem on random Gaussian triangles, and a generalization of the D. G. Kendall theorem on the kinematics of shape.The Clifford–Green theorem gives a remarkable characterization of the joint distribution of the squared-side-lengths of n independent Gaussian points in n-space, namely that this distribution is that of n independent exponential random variables conditioned to satisfy all the ineq

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48

Kendall, Wilfrid S. "Symbolic computation and the diffusion of shapes of triads." Advances in Applied Probability 20, no. 04 (1988): 775–97. http://dx.doi.org/10.1017/s0001867800018371.

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This paper introduces the use of symbolic computation (also known as computer algebra) in stochastic analysis and particularly in the Itô calculus. Two related examples are considered: the Clifford-Green theorem on random Gaussian triangles, and a generalization of the D. G. Kendall theorem on the kinematics of shape. The Clifford–Green theorem gives a remarkable characterization of the joint distribution of the squared-side-lengths of n independent Gaussian points in n-space, namely that this distribution is that of n independent exponential random variables conditioned to satisfy all the ine

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49

Hodyss, Daniel, Justin G. McLay, Jon Moskaitis, and Efren A. Serra. "Inducing Tropical Cyclones to Undergo Brownian Motion: A Comparison between Itô and Stratonovich in a Numerical Weather Prediction Model." Monthly Weather Review 142, no. 5 (2014): 1982–96. http://dx.doi.org/10.1175/mwr-d-13-00299.1.

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Abstract Stochastic parameterization has become commonplace in numerical weather prediction (NWP) models used for probabilistic prediction. Here a specific stochastic parameterization will be related to the theory of stochastic differential equations and shown to be affected strongly by the choice of stochastic calculus. From an NWP perspective the focus will be on ameliorating a common trait of the ensemble distributions of tropical cyclone (TC) tracks (or position); namely, that they generally contain a bias and an underestimate of the variance. With this trait in mind the authors present a

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50

Adler, Stephen L. "Derivation of the Lindblad generator structure by use of the Itô stochastic calculus." Physics Letters A 265, no. 1-2 (2000): 58–61. http://dx.doi.org/10.1016/s0375-9601(99)00847-6.

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