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Journal articles on the topic 'Discretization of stochastic integrals'

Author: Grafiati
Published: 16 December 2023

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1

Fukasawa, Masaaki. "Efficient discretization of stochastic integrals." Finance and Stochastics 18, no. 1 (October 4, 2013): 175–208. http://dx.doi.org/10.1007/s00780-013-0215-6.

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2

Fukasawa, Masaaki. "Discretization error of stochastic integrals." Annals of Applied Probability 21, no. 4 (August 2011): 1436–65. http://dx.doi.org/10.1214/10-aap730.

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3

Gobet, Emmanuel, and Uladzislau Stazhynski. "Model-adaptive optimal discretization of stochastic integrals." Stochastics 91, no. 3 (October 29, 2018): 321–51. http://dx.doi.org/10.1080/17442508.2018.1539087.

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4

MARAZZINA, DANIELE, OLEG REICHMANN, and CHRISTOPH SCHWAB. "hp-DGFEM FOR KOLMOGOROV–FOKKER–PLANCK EQUATIONS OF MULTIVARIATE LÉVY PROCESSES." Mathematical Models and Methods in Applied Sciences 22, no. 01 (January 2012): 1150005. http://dx.doi.org/10.1142/s0218202512005897.

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We analyze the discretization of nonlocal degenerate integrodifferential equations arising as so-called forward equations for jump-diffusion processes. Such equations arise in option pricing problems when the stochastic dynamics of the markets is modeled by Lévy driven stochastic volatility models. Well-posedness of the arising equations is addressed. We develop and analyze stable discretization schemes, in particular the discontinuous Galerkin Finite Element Methods (DG-FEM). In the DG-FEM, a new regularization of hypersingular integrals in the Dirichlet form of the pure jump part of infinite variation processes is proposed, allowing in particular a stable DG discretization of hypersingular integral operators. Robustness of the stabilized discretization with respect to various degeneracies in the characteristic triple of the stochastic process is proved. We provide in particular an hp-error analysis of the DG-FEM. Numerical experiments for model equations confirm the theoretical results.
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5

Zhou, Li-kai, and Zhong-gen Su. "Discretization error of irregular sampling approximations of stochastic integrals." Applied Mathematics-A Journal of Chinese Universities 31, no. 3 (August 26, 2016): 296–306. http://dx.doi.org/10.1007/s11766-016-3426-8.

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6

Gobet, Emmanuel, and Uladzislau Stazhynski. "Optimal discretization of stochastic integrals driven by general Brownian semimartingale." Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 54, no. 3 (August 2018): 1556–82. http://dx.doi.org/10.1214/17-aihp848.

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7

Kloeden, P. E., E. Platen, H. Schurz, and M. Sørensen. "On effects of discretization on estimators of drift parameters for diffusion processes." Journal of Applied Probability 33, no. 4 (December 1996): 1061–76. http://dx.doi.org/10.2307/3214986.

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In this paper statistical properties of estimators of drift parameters for diffusion processes are studied by modern numerical methods for stochastic differential equations. This is a particularly useful method for discrete time samples, where estimators can be constructed by making discrete time approximations to the stochastic integrals appearing in the maximum likelihood estimators for continuously observed diffusions. A review is given of the necessary theory for parameter estimation for diffusion processes and for simulation of diffusion processes. Three examples are studied.
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8

Kloeden, P. E., E. Platen, H. Schurz, and M. Sørensen. "On effects of discretization on estimators of drift parameters for diffusion processes." Journal of Applied Probability 33, no. 04 (December 1996): 1061–76. http://dx.doi.org/10.1017/s0021900200100488.

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In this paper statistical properties of estimators of drift parameters for diffusion processes are studied by modern numerical methods for stochastic differential equations. This is a particularly useful method for discrete time samples, where estimators can be constructed by making discrete time approximations to the stochastic integrals appearing in the maximum likelihood estimators for continuously observed diffusions. A review is given of the necessary theory for parameter estimation for diffusion processes and for simulation of diffusion processes. Three examples are studied.
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9

Salmhofer, Manfred. "Functional Integral and Stochastic Representations for Ensembles of Identical Bosons on a Lattice." Communications in Mathematical Physics 385, no. 2 (March 11, 2021): 1163–211. http://dx.doi.org/10.1007/s00220-021-04010-4.

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AbstractRegularized coherent-state functional integrals are derived for ensembles of identical bosons on a lattice, the regularization being a discretization of Euclidian time. Convergence of the time-continuum limit is proven for various discretized actions. The focus is on the integral representation for the partition function and expectation values in the canonical ensemble. The connection to the grand-canonical integral is exhibited and some important differences are discussed. Uniform bounds for covariances are proven, which simplify the analysis of the time-continuum limit and can also be used to analyze the thermodynamic limit. The relation to a stochastic representation by an ensemble of interacting random walks is made explicit, and its modifications in presence of a condensate are discussed.
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10

Tynda, Aleksandr, Samad Noeiaghdam, and Denis Sidorov. "Polynomial Spline Collocation Method for Solving Weakly Regular Volterra Integral Equations of the First Kind." Bulletin of Irkutsk State University. Series Mathematics 39 (2022): 62–79. http://dx.doi.org/10.26516/1997-7670.2022.39.62.

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The polynomial spline collocation method is proposed for solution of Volterra integral equations of the first kind with special piecewise continuous kernels. The Gausstype quadrature formula is used to approximate integrals during the discretization of the proposed projection method. The estimate of accuracy of approximate solution is obtained. Stochastic arithmetics is also used based on the Controle et Estimation Stochastique des Arrondis de Calculs (CESTAC) method and the Control of Accuracy and Debugging for Numerical Applications (CADNA) library. Applying this approach it is possible to find optimal parameters of the projective method. The numerical examples are included to illustrate the efficiency of proposed novel collocation method.
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11

Geiss, Stefan, and Anni Toivola. "Weak convergence of error processes in discretizations of stochastic integrals and Besov spaces." Bernoulli 15, no. 4 (November 2009): 925–54. http://dx.doi.org/10.3150/09-bej197.

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12

Cervenka, Johann, Robert Kosik, and Mihail Nedjalkov. "A deterministic Wigner approach for superposed states." Journal of Computational Electronics 20, no. 6 (October 20, 2021): 2104–10. http://dx.doi.org/10.1007/s10825-021-01801-9.

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AbstractThe Wigner formalism is a convenient way of describing quantum mechanical effects through a framework of distribution functions in phase space. Currently, there are stochastic and deterministic approaches in use. In our deterministic method, the critical discretization of the diffusion term is done through the utilization of an integral formulation of the Wigner equation. This deterministic method is studied in the context of superposed quantum states as a precursor to simulations of entangled states.
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13

Anderson, Ross P., and Dejan Milutinović. "Stochastic optimal enhancement of distributed formation control using Kalman smoothers." Robotica 32, no. 2 (January 31, 2014): 305–24. http://dx.doi.org/10.1017/s0263574714000022.

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SUMMARYBeginning with a deterministic distributed feedback control for nonholonomic vehicle formations, we develop a stochastic optimal control approach for agents to enhance their non-optimal controls with additive correction terms based on the Hamilton–Jacobi–Bellman equation, making them optimal and robust to uncertainties. In order to avoid discretization of the high-dimensional cost-to-go function, we exploit the stochasticity of the distributed nature of the problem to develop an equivalent Kalman smoothing problem in a continuous state space using a path integral representation. Our approach is illustrated by numerical examples in which agents achieve a formation with their neighbors using only local observations.
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14

TAMAGNO, Pierre, and Elias VANDERMEERSCH. "Comprehensive stochastic sensitivities to resonance parameters." EPJ Web of Conferences 239 (2020): 13008. http://dx.doi.org/10.1051/epjconf/202023913008.

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Integral experiments in reactors or critical configurations claim to have very small experimental and technological uncertainties. Therefore these latter can be considered valuable experimental information in nuclear data evaluation. Because in the evaluation process the information is carried by model parameters, to perform a rigorous feedback on a nuclear model parameters p - for instance using a measured reactivity ρ-sensitivities S =∂ρ/ρ⁄∂p/p are needed. In usual integral feedbacks, sensitivity to multi-group cross sections are first obtained with deterministic code using perturbation theory. Then these multi-group cross section sensitivities are “convoluted” with parameter sensitivities in order to provide the sensitivity on nuclear model parameter. Recently stochastic approaches have been elaborated in order to obtain continuous cross-section sensitivities thus avoiding the multi-group discretization. In the present work we used the recent Iterated Fission Probability method of the TRIPOLI4 code [1] in order to obtain directly the sensitivity to nuclear physics parameters. We focus here on the sensitivity on resonance parameters and exemplified the method on the computation of sensitivities for 239Pu and 16O resonance parameters one the ICSBEP benchmark PST001. The underlying nuclear model describing resonant cross sections are based in the R-matrix formalism [2] that provides not only the interaction cross sections but also the angular distribution of the scattered neutrons i.e. differential cross sections. The method has thus been updated in order to compute parameter sensitives that include both contributions: cross section and angular distributions. This extension of the method was tested with exact perturbation of angular distribution and fission spectrum.
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15

Bolin, David, Kristin Kirchner, and Mihály Kovács. "Numerical solution of fractional elliptic stochastic PDEs with spatial white noise." IMA Journal of Numerical Analysis 40, no. 2 (December 20, 2018): 1051–73. http://dx.doi.org/10.1093/imanum/dry091.

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Abstract The numerical approximation of solutions to stochastic partial differential equations with additive spatial white noise on bounded domains in $\mathbb{R}^d$ is considered. The differential operator is given by the fractional power $L^\beta $, $\beta \in (0,1)$ of an integer-order elliptic differential operator $L$ and is therefore nonlocal. Its inverse $L^{-\beta }$ is represented by a Bochner integral from the Dunford–Taylor functional calculus. By applying a quadrature formula to this integral representation the inverse fractional-order operator $L^{-\beta }$ is approximated by a weighted sum of nonfractional resolvents $( I + \exp(2 y_\ell) \, L )^{-1}$ at certain quadrature nodes $t_j> 0$. The resolvents are then discretized in space by a standard finite element method. This approach is combined with an approximation of the white noise, which is based only on the mass matrix of the finite element discretization. In this way an efficient numerical algorithm for computing samples of the approximate solution is obtained. For the resulting approximation the strong mean-square error is analyzed and an explicit rate of convergence is derived. Numerical experiments for $L=\kappa ^2-\Delta $, $\kappa> 0$ with homogeneous Dirichlet boundary conditions on the unit cube $(0,1)^d$ in $d=1,2,3$ spatial dimensions for varying $\beta \in (0,1)$ attest to the theoretical results.
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16

Ali, Ishtiaq, and Sami Ullah Khan. "A Dynamic Competition Analysis of Stochastic Fractional Differential Equation Arising in Finance via Pseudospectral Method." Mathematics 11, no. 6 (March 9, 2023): 1328. http://dx.doi.org/10.3390/math11061328.

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This research focuses on the analysis of the competitive model used in the banking sector based on the stochastic fractional differential equation. For the approximate solution, a pseudospectral technique is utilized for the proposed model based on the stochastic Lotka–Volterra equation using a wide range of fractional order parameters in simulations. Conditions for stable and unstable equilibrium points are provided using the Jacobian. The Lotka–Volterra equation is unstable in the long term and can produce highly fluctuating dynamics, which is also one of the reasons that this equation is used to model the problems arising in finance, where fluctuations are important. For this reason, the conventional analytical and numerical methods are not the best choices. To overcome this difficulty, an automatic procedure is used to solve the resultant algebraic equation after the discretization of the operator. In order to fully use the properties of orthogonal polynomials, the proposed scheme is applied to the equivalent integral form of stochastic fractional differential equations under consideration. This also helps in the analysis of fractional differential equations, which mostly fall in the framework of their integrated form. We demonstrate that this fractional approach may be considered as the best tool to model such real-world data situations with very reasonable accuracy. Our numerical simulations further demonstrate that the use of the fractional Atangana–Baleanu operator approach produces results that are more precise and flexible, allowing individuals or companies to use it with confidence to model such real-world situations. It is shown that our numerical simulation results have a very good agreement with the real data, further showing the efficiency and effectiveness of our numerical scheme for the proposed model.
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17

STOEV, STILIAN, and MURAD S. TAQQU. "SIMULATION METHODS FOR LINEAR FRACTIONAL STABLE MOTION AND FARIMA USING THE FAST FOURIER TRANSFORM." Fractals 12, no. 01 (March 2004): 95–121. http://dx.doi.org/10.1142/s0218348x04002379.

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We present efficient methods for simulation, using the Fast Fourier Transform (FFT) algorithm, of two classes of processes with symmetric α-stable (SαS) distributions. Namely, (i) the linear fractional stable motion (LFSM) process and (ii) the fractional autoregressive moving average (FARIMA) time series with SαS innovations. These two types of heavy-tailed processes have infinite variances and long-range dependence and they can be used in modeling the traffic of modern computer telecommunication networks. We generate paths of the LFSM process by using Riemann-sum approximations of its SαS stochastic integral representation and paths of the FARIMA time series by truncating their moving average representation. In both the LFSM and FARIMA cases, we compute the involved sums efficiently by using the Fast Fourier Transform algorithm and provide bounds and/or estimates of the approximation error. We discuss different choices of the discretization and truncation parameters involved in our algorithms and illustrate our method. We include MATLAB implementations of these simulation algorithms and indicate how the practitioner can use them.
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18

Guterding, Daniel. "Sparse Modeling Approach to the Arbitrage-Free Interpolation of Plain-Vanilla Option Prices and Implied Volatilities." Risks 11, no. 5 (April 28, 2023): 83. http://dx.doi.org/10.3390/risks11050083.

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We present a method for the arbitrage-free interpolation of plain-vanilla option prices and implied volatilities, which is based on a system of integral equations that relates terminal density and option prices. Using a discretization of the terminal density, we write these integral equations as a system of linear equations. We show that the kernel matrix of this system is, in general, ill-conditioned, so that it cannot be solved for the discretized density using a naive approach. Instead, we construct a sparse model for the kernel matrix using singular value decomposition (SVD), which allows us not only to systematically improve the condition number of the kernel matrix, but also determines the computational effort and accuracy of our method. In order to allow for the treatment of realistic inputs that may contain arbitrage, we reformulate the system of linear equations as an optimization problem, in which the SVD-transformed density minimizes the error between the input prices and the arbitrage-free prices generated by our method. To further stabilize the method in the presence of noisy input prices or arbitrage, we apply an L1-regularization to the SVD-transformed density. Our approach, which is inspired by recent progress in theoretical physics, offers a flexible and efficient framework for the arbitrage-free interpolation of plain-vanilla option prices and implied volatilities, without the need to explicitly specify a stochastic process, expansion basis functions or any other kind of model. We demonstrate the capabilities of our method in a number of artificial and realistic test cases.
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19

Samson, J. H. "Time discretization of functional integrals." Journal of Physics A: Mathematical and General 33, no. 16 (April 28, 2000): 3111–20. http://dx.doi.org/10.1088/0305-4470/33/16/305.

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20

Steinhaus, Sebastian. "Perfect discretization of path integrals." Journal of Physics: Conference Series 360 (May 16, 2012): 012025. http://dx.doi.org/10.1088/1742-6596/360/1/012025.

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21

Barnett, Chris, J. M. Lindsay, and Ivan F. Wilde. "Quantum stochastic integrals as belated integrals." Glasgow Mathematical Journal 34, no. 2 (May 1992): 165–73. http://dx.doi.org/10.1017/s0017089500008685.

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Quantum stochastic integrals have been constructed in various contexts [2, 3, 4, 5, 9] by adapting the construction of the classical L2-Itô-integral with respect to Brownian motion. Thus, the integral is first defined for simple integrands as a finite sum, then one establishes certain isometry relations or suitable bounds to allow the extension, by continuity, to more general integrands. The integrator is typically operator-valued, the integrand is vector-valued or operator-valued and the quantum stochastic integral is then given as a vector in a Hilbert space, or as an operator on the Hilbert space determined by its action on suitable vectors.
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22

Berezanskii, Yu M., N. V. Zhernakov, and G. F. Us. "Stochastic operator integrals." Ukrainian Mathematical Journal 39, no. 2 (1987): 120–24. http://dx.doi.org/10.1007/bf01057489.

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23

Fečkan, Michal, and Michal Pospíšil. "Discretization of dynamical systems with first integrals." Discrete and Continuous Dynamical Systems 33, no. 8 (January 2013): 3543–54. http://dx.doi.org/10.3934/dcds.2013.33.3543.

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24

Brockwell, Peter J., and P. E. Kopp. "Martingales and Stochastic Integrals." Journal of the American Statistical Association 82, no. 397 (March 1987): 346. http://dx.doi.org/10.2307/2289180.

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25

Privault, Nicolas, and Qihao She. "Conditionally Gaussian stochastic integrals." Comptes Rendus Mathematique 353, no. 12 (December 2015): 1153–58. http://dx.doi.org/10.1016/j.crma.2015.09.022.

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26

Buhmann, Martin D., Feng Dai, and Yeli Niu. "Discretization of integrals on compact metric measure spaces." Advances in Mathematics 381 (April 2021): 107602. http://dx.doi.org/10.1016/j.aim.2021.107602.

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27

Habibullin, Ismagil, and Natalya Zheltukhina. "Discretization of Liouville type nonautonomous equations preserving integrals." Journal of Nonlinear Mathematical Physics 23, no. 4 (October 2016): 620–42. http://dx.doi.org/10.1080/14029251.2016.1248159.

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28

McLachlan, R. I. "Spatial discretization of partial differential equations with integrals." IMA Journal of Numerical Analysis 23, no. 4 (October 1, 2003): 645–64. http://dx.doi.org/10.1093/imanum/23.4.645.

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29

Meyer, Fabian, Christian Rohde, and Jan Giesselmann. "A posteriori error analysis for random scalar conservation laws using the stochastic Galerkin method." IMA Journal of Numerical Analysis 40, no. 2 (February 15, 2019): 1094–121. http://dx.doi.org/10.1093/imanum/drz004.

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Abstract In this article we present an a posteriori error estimator for the spatial–stochastic error of a Galerkin-type discretization of an initial value problem for a random hyperbolic conservation law. For the stochastic discretization we use the stochastic Galerkin method and for the spatial–temporal discretization of the stochastic Galerkin system a Runge–Kutta discontinuous Galerkin method. The estimator is obtained using smooth reconstructions of the discrete solution. Combined with the relative entropy stability framework of Dafermos (2016, Hyperbolic Conservation Laws in Continuum Physics, 4th edn., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Berlin, Springer, pp. xxxviii+826), this leads to computable error bounds for the space–stochastic discretization error. Moreover, it turns out that the error estimator admits a splitting into one part representing the spatial error, and a remaining term, which can be interpreted as the stochastic error. This decomposition allows us to balance the errors arising from spatial and stochastic discretization. We conclude with some numerical examples confirming the theoretical findings.
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30

Sykora, Henrik T., Daniel Bachrathy, and Gabor Stepan. "Stochastic semi‐discretization for linear stochastic delay differential equations." International Journal for Numerical Methods in Engineering 119, no. 9 (April 30, 2019): 879–98. http://dx.doi.org/10.1002/nme.6076.

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31

GRUNDBERG, J., U. LINDSTRÖM, and H. NORDSTRÖM. "DISCRETIZATION OF THE SUPERPARTICLE PATH INTEGRAL." Modern Physics Letters A 08, no. 14 (May 10, 1993): 1323–29. http://dx.doi.org/10.1142/s0217732393001057.

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We investigate the hitherto unexplored relation between the superparticle path integral and superfield theory. Requiring that the path integral has the global symmetries of the classical action and obeys the natural composition property of path integrals, and also that the discretized action has the correct naive continuum limit, we find a viable discretization of the (D = 3, N = 2) superparticle action.
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32

Charalambous, C. D., R. J. Elliott, and V. Krishnamurthy. "Conditional Moment Generating Functions for Integrals and Stochastic Integrals." SIAM Journal on Control and Optimization 42, no. 5 (January 2003): 1578–603. http://dx.doi.org/10.1137/s036301299833327x.

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33

Artikis, Panagiotis T., and Constantinos T. Artikis. "Stochastic integrals in strategic operations." Journal of Statistics and Management Systems 24, no. 6 (August 18, 2021): 1363–70. http://dx.doi.org/10.1080/09720510.2021.1968133.

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34

Henstock. "STOCHASTIC AND OTHER FUNCTIONAL INTEGRALS." Real Analysis Exchange 16, no. 2 (1990): 460. http://dx.doi.org/10.2307/44153722.

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35

Osękowski, Adam. "Maximal Inequalities for Stochastic Integrals." Bulletin of the Polish Academy of Sciences Mathematics 58, no. 3 (2010): 273–87. http://dx.doi.org/10.4064/ba58-3-10.

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36

Szyszkowski, Ireneusz, and Ireneusz Szyszkowski. "Weak convergence of stochastic integrals." Teoriya Veroyatnostei i ee Primeneniya 41, no. 4 (1996): 942–46. http://dx.doi.org/10.4213/tvp3286.

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37

Nasyrov, F. S. "Symmetric Integrals and Stochastic Analysis." Theory of Probability & Its Applications 51, no. 3 (January 2007): 486–503. http://dx.doi.org/10.1137/s0040585x97982499.

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38

Rota, Gian-Carlo, and Timothy C. Wallstrom. "Stochastic integrals: a combinatorial approach." Annals of Probability 25, no. 3 (July 1997): 1257–83. http://dx.doi.org/10.1214/aop/1024404513.

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39

Jeanblanc, Monique, and Frédéric Vrins. "Conic martingales from stochastic integrals." Mathematical Finance 28, no. 2 (April 18, 2017): 516–35. http://dx.doi.org/10.1111/mafi.12147.

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40

Norin, N. V. "Stochastic Integrals and Differential Measures." Theory of Probability & Its Applications 32, no. 1 (January 1988): 107–16. http://dx.doi.org/10.1137/1132010.

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41

Tocino, A. "Multiple stochastic integrals with Mathematica." Mathematics and Computers in Simulation 79, no. 5 (January 2009): 1658–67. http://dx.doi.org/10.1016/j.matcom.2008.08.005.

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42

Łuczak, Andrzej. "Quantum Stochastic Integrals as Operators." International Journal of Theoretical Physics 49, no. 12 (May 4, 2010): 3176–84. http://dx.doi.org/10.1007/s10773-010-0367-5.

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43

Ubøe, Jan. "Complex valued multiparameter stochastic integrals." Journal of Theoretical Probability 8, no. 3 (July 1995): 601–24. http://dx.doi.org/10.1007/bf02218046.

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44

Lin, Zhengyan, and Hanchao Wang. "On Convergence to Stochastic Integrals." Journal of Theoretical Probability 29, no. 3 (January 31, 2015): 717–36. http://dx.doi.org/10.1007/s10959-015-0598-8.

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45

Caithamer, Peter. "Decoupled double stochastic fractional integrals." Stochastics 77, no. 3 (June 15, 2005): 205–10. http://dx.doi.org/10.1080/10451120500138556.

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46

Sato, Ken-Iti. "Additive processes and stochastic integrals." Illinois Journal of Mathematics 50, no. 1-4 (2006): 825–51. http://dx.doi.org/10.1215/ijm/1258059494.

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47

Stein, Michael L. "Predicting Integrals of Stochastic Processes." Annals of Applied Probability 5, no. 1 (February 1995): 158–70. http://dx.doi.org/10.1214/aoap/1177004834.

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48

Jung, Eun Ju, and Jai Heui Kim. "On Set-Valued Stochastic Integrals." Stochastic Analysis and Applications 21, no. 2 (January 4, 2003): 401–18. http://dx.doi.org/10.1081/sap-120019292.

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49

Sion, Maurice. "Outer measures and stochastic integrals." Archiv der Mathematik 59, no. 4 (October 1992): 383–95. http://dx.doi.org/10.1007/bf01197056.

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50

Kuznetsov, Dmitriy Feliksovich. "Mean-square Approximation of Iterated Ito and Stratonovich Stochastic Integrals: Method of Generalized Multiple Fourier Series. Application to Numerical Integration of Ito Sdes and Semilinear Spdes (third Edition)." Differential Equations and Control Processes, no. 1 (2023): 151–1097. http://dx.doi.org/10.21638/11701/spbu35.2023.110.

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This is the third edition of the monograph (first edition 2020, second edition 2021) devoted to the problem of mean-square approximation of iterated Ito and Stratonovich stochastic integrals with respect to components of the multidimensional Wiener process. The mentioned problem is considered in the book as applied to the numerical integration of non-commutative Ito stochastic differential equations and semilinear stochastic partial differential equations with nonlinear non-commutative trace class noise. The book opens up a new direction in researching of iterated stochastic integrals. For the first time we use the generalized multiple Fourier series converging in the sense of norm in Hilbert space for the expansion of iterated Ito stochastic integrals of arbitrary multiplicity k with respect to components of the multidimensional Wiener process (Chapter 1). Sections 1.11-1.13 (Chapter 1) are new and generalize the results of Chapter 1 obtained earlier by the author and are also closely related to the multiple Wiener stochastic integral introduced by Ito in 1951. The convergence with probability 1 as well as the convergence in the sense of n-th (n=2, 3,...) moment for the expansion of iterated Ito stochastic integrals have been proved (Chapter 1). Moreover, the rate of both types of convergence has been established. The main difference between the third and second editions of the book is that the third edition includes original material (Chapter 2, Sections 2.10-2.19) on a new approach to the series expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k with respect to components of the multidimensional Wiener process. The above approach allowed us to generalize some of the author's earlier results and also to make significant progress in solving the problem of series expansion of iterated Stratonovich stochastic integrals. In particular, for iterated Stratonovich stochastic integrals of the fifth and sixth multiplicity, series expansions based on multiple Fourier-Legendre series and multiple trigonometric Fourier series are obtained. In addition, expansions of iterated Stratonovich stochastic integrals of multiplicities 2 to 4 were generalized. These results (Chapter 2) adapt the results of Chapter 1 for iterated Stratonovich stochastic integrals. Two theorems on expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k based on generalized iterated Fourier series with pointwise convergence are formulated and proved (Chapter 2). The results of Chapters 1 and 2 can be considered from the point of view of the Wong-Zakai approximation for the case of a multidimensional Wiener process and the Wiener process approximation based on its series expansion using Legendre polynomials and trigonometric functions. The integration order replacement technique for iterated Ito stochastic integrals has been introduced (Chapter 3). Exact expressions are obtained for the mean-square approximation error of iterated Ito stochastic integrals of arbitrary multiplicity k (Chapter 1) and iterated Stratonovich stochastic integrals of multiplicities 1 to 4 (Chapter 5). Furthermore, we provided a significant practical material (Chapter 5) devoted to the expansions and mean-square approximations of specific iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 from the unified Taylor-Ito and Taylor-Stratonovich expansions (Chapter 4). These approximations were obtained using Legendre polynomials and trigonometric functions. The methods constructed in the book have been compared with some existing methods (Chapter 6). The results of Chapter 1 were applied (Chapter 7) to the approximation of iterated stochastic integrals with respect to the finite-dimensional approximation of the Q-Wiener process (for integrals of multiplicity k) and with respect to the infinite-dimensional Q-Wiener process (for integrals of multiplicities 1 to 3).
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