2011, Courant lecture notes in mathematics, ISBN 0821872869, Volume 22, xi, 149

Book

2013, ISBN 9781470410544, viii, 151 pages

Book

Journal of Computational Physics, ISSN 0021-9991, 10/2015, Volume 298, pp. 254 - 265

This paper presents a computational method based on the Chebyshev wavelets for solving stochastic Itô–Volterra integral equations. First, a stochastic...

Stochastic Itô–Volterra integral equations | Itô integral | Chebyshev wavelets | Stochastic operational matrix | Stochastic Itô-Volterra integral equations | Ito integral | OPERATIONAL MATRIX | NUMERICAL-SOLUTION | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | RANDOM DIFFERENTIAL-EQUATIONS | PHYSICS, MATHEMATICAL | Stochastic Ito-Volterra integral equations | Wavelet | Error analysis | Computation | Integral equations | Chebyshev approximation | Mathematical models | Stochasticity | Convergence | POLYNOMIALS | ERRORS | STOCHASTIC PROCESSES | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | MATRICES | VOLTERRA INTEGRAL EQUATIONS | CONVERGENCE | ACCURACY

Stochastic Itô–Volterra integral equations | Itô integral | Chebyshev wavelets | Stochastic operational matrix | Stochastic Itô-Volterra integral equations | Ito integral | OPERATIONAL MATRIX | NUMERICAL-SOLUTION | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | RANDOM DIFFERENTIAL-EQUATIONS | PHYSICS, MATHEMATICAL | Stochastic Ito-Volterra integral equations | Wavelet | Error analysis | Computation | Integral equations | Chebyshev approximation | Mathematical models | Stochasticity | Convergence | POLYNOMIALS | ERRORS | STOCHASTIC PROCESSES | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | MATRICES | VOLTERRA INTEGRAL EQUATIONS | CONVERGENCE | ACCURACY

Journal Article

Stochastic Processes and their Applications, ISSN 0304-4149, 12/2019, Volume 129, Issue 12, pp. 4926 - 4964

For backward stochastic Volterra integral equations (BSVIEs, for short), under some mild conditions, the so-called adapted solutions or adapted M-solutions...

Representation of adapted solutions | Backward stochastic Volterra integral equations | Adapted solutions | Representation partial differential equations | SCHEME | THEOREMS | DIFFERENTIAL-EQUATIONS | SDES | STATISTICS & PROBABILITY | WELL-POSEDNESS | Differential equations

Representation of adapted solutions | Backward stochastic Volterra integral equations | Adapted solutions | Representation partial differential equations | SCHEME | THEOREMS | DIFFERENTIAL-EQUATIONS | SDES | STATISTICS & PROBABILITY | WELL-POSEDNESS | Differential equations

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 08/2014, Volume 270, pp. 402 - 415

In this paper, a new computational method based on the generalized hat basis functions is proposed for solving stochastic Itô–Volterra integral equations. In...

Brownian motion process | Stochastic Itô–Volterra integral equations | Generalized hat basis functions | Itô integral | Stochastic operational matrix | ItÔ integral | Stochastic ItÔ-Volterra integral equations | Ito integral | NUMERICAL-SOLUTION | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | RANDOM DIFFERENTIAL-EQUATIONS | INTEGRODIFFERENTIAL EQUATIONS | PHYSICS, MATHEMATICAL | Stochastic Ito Volterra integral equations | Basis functions | Computation | Integral equations | Mathematical analysis | Blocking | Texts | Mathematical models | Stochasticity | INTEGRALS | STOCHASTIC PROCESSES | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | MATRICES | VOLTERRA INTEGRAL EQUATIONS | CONVERGENCE | BROWNIAN MOVEMENT | RELIABILITY | COMPARATIVE EVALUATIONS | ACCURACY | PULSES

Brownian motion process | Stochastic Itô–Volterra integral equations | Generalized hat basis functions | Itô integral | Stochastic operational matrix | ItÔ integral | Stochastic ItÔ-Volterra integral equations | Ito integral | NUMERICAL-SOLUTION | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | RANDOM DIFFERENTIAL-EQUATIONS | INTEGRODIFFERENTIAL EQUATIONS | PHYSICS, MATHEMATICAL | Stochastic Ito Volterra integral equations | Basis functions | Computation | Integral equations | Mathematical analysis | Blocking | Texts | Mathematical models | Stochasticity | INTEGRALS | STOCHASTIC PROCESSES | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | MATRICES | VOLTERRA INTEGRAL EQUATIONS | CONVERGENCE | BROWNIAN MOVEMENT | RELIABILITY | COMPARATIVE EVALUATIONS | ACCURACY | PULSES

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 2012, Volume 63, Issue 1, pp. 133 - 143

The multidimensional Itô–Volterra integral equations arise in many problems such as an exponential population growth model with several independent white noise...

Brownian motion process | Stochastic Itô–Volterra integral equations | Block pulse functions | Itô integral | Stochastic operational matrix | Stochastic Itô-Volterra integral equations | It integral | Ito integral | MATHEMATICS, APPLIED | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | BLOCK-PULSE FUNCTIONS | KIND | DIFFERENTIAL-EQUATIONS | Stochastic Ito-Volterra integral equations | Confidence intervals | Approximation | Integral equations | Mathematical analysis | Blocking | Mathematical models | Stochasticity | Convergence

Brownian motion process | Stochastic Itô–Volterra integral equations | Block pulse functions | Itô integral | Stochastic operational matrix | Stochastic Itô-Volterra integral equations | It integral | Ito integral | MATHEMATICS, APPLIED | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | BLOCK-PULSE FUNCTIONS | KIND | DIFFERENTIAL-EQUATIONS | Stochastic Ito-Volterra integral equations | Confidence intervals | Approximation | Integral equations | Mathematical analysis | Blocking | Mathematical models | Stochasticity | Convergence

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 03/2018, Volume 330, pp. 574 - 585

In this study, a practical matrix method based on operational matrices of integration and collocation points is presented to find the approximate solution of...

Brownian motion process | Stochastic Itô–Volterra integral equations | Error analysis | Euler polynomials | Stochastic operational matrix | SYSTEM | MATHEMATICS, APPLIED | OPERATIONAL MATRICES | COEFFICIENTS | COMPUTATIONAL METHOD | BERNOULLI | FREDHOLM INTEGRODIFFERENTIAL EQUATIONS | Stochastic Ito-Volterra integral equations

Brownian motion process | Stochastic Itô–Volterra integral equations | Error analysis | Euler polynomials | Stochastic operational matrix | SYSTEM | MATHEMATICS, APPLIED | OPERATIONAL MATRICES | COEFFICIENTS | COMPUTATIONAL METHOD | BERNOULLI | FREDHOLM INTEGRODIFFERENTIAL EQUATIONS | Stochastic Ito-Volterra integral equations

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 06/2017, Volume 317, pp. 447 - 457

The Euler–Maruyama method is presented for linear stochastic Volterra integral equations. Then the strong convergence property is analyzed for convolution...

Strong convergence | Stochastic | The Euler–Maruyama method | Strong superconvergence | Volterra integral equations | ORDER | MATHEMATICS, APPLIED | EXPONENTIAL STABILITY | The Euler-Maruyama method | COMPUTATIONAL METHOD | CONVERGENCE | DELAY | Analysis | Methods | Differential equations

Strong convergence | Stochastic | The Euler–Maruyama method | Strong superconvergence | Volterra integral equations | ORDER | MATHEMATICS, APPLIED | EXPONENTIAL STABILITY | The Euler-Maruyama method | COMPUTATIONAL METHOD | CONVERGENCE | DELAY | Analysis | Methods | Differential equations

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 08/2013, Volume 219, Issue 24, pp. 11278 - 11290

We consider fuzzy stochastic integral equations with stochastic Lebesgue trajectory integrals and fuzzy stochastic Itô trajectory integrals. Some methods of...

Stochastic fuzzy differential equation | Uncertainty | Approximate solution | Fuzzy stochastic trajectory integral | Fuzzy stochastic integral equation | MATHEMATICS, APPLIED | DIFFERENTIAL-EQUATIONS | ITO TYPE | Fuzzy logic | Approximation | Integral equations | Mathematical analysis | Mathematical models | Fuzzy set theory | Stochasticity | Fuzzy

Stochastic fuzzy differential equation | Uncertainty | Approximate solution | Fuzzy stochastic trajectory integral | Fuzzy stochastic integral equation | MATHEMATICS, APPLIED | DIFFERENTIAL-EQUATIONS | ITO TYPE | Fuzzy logic | Approximation | Integral equations | Mathematical analysis | Mathematical models | Fuzzy set theory | Stochasticity | Fuzzy

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 05/2018, Volume 333, pp. 74 - 86

The main aim of this study is to propose a numerical iterative approach for obtaining approximate solutions of nonlinear stochastic Itô –Volterra integral...

Stochastic Volterra integral equations | Brownian motion process | Itô integral | Linear spline interpolation | Successive approximations method | Ito integral | OPERATIONAL MATRIX | MATHEMATICS, APPLIED | GALERKIN METHOD | COMPUTATIONAL METHOD

Stochastic Volterra integral equations | Brownian motion process | Itô integral | Linear spline interpolation | Successive approximations method | Ito integral | OPERATIONAL MATRIX | MATHEMATICS, APPLIED | GALERKIN METHOD | COMPUTATIONAL METHOD

Journal Article

2005, Mathematical and analytical techniques with applications to engineering, ISBN 9780387251752, xx, 434

Derivation of Ito's formulas, Girsanov's theorems and martingale representation theorem for stochastic DEs with jumpsApplications to population...

Stochastic differential equations | Mathematics | Engineering | Engineering Fluid Dynamics | Applications of Mathematics | Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | Financial Economics

Stochastic differential equations | Mathematics | Engineering | Engineering Fluid Dynamics | Applications of Mathematics | Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | Financial Economics

Book

Journal of Computational Physics, ISSN 0021-9991, 10/2017, Volume 346, pp. 49 - 70

A novel path integral (PI) based method for solution of the Fokker–Planck equation is presented. The proposed method, termed the transformed path integral...

Uncertainty propagation | Stochastic differential equations | Short-time propagator | Fokker–Planck equation | Path integral | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | NONLINEAR-SYSTEMS | FINITE-ELEMENT | Fokker-Planck equation | MONTE-CARLO-SIMULATION | PHYSICS, MATHEMATICAL | Monte Carlo method | Stochastic processes | Differential equations | FOKKER-PLANCK EQUATION | GRIDS | STOCHASTIC PROCESSES | MATHEMATICAL EVOLUTION | MONTE CARLO METHOD | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | ONE-DIMENSIONAL CALCULATIONS | COMPUTERIZED SIMULATION | MATHEMATICAL SOLUTIONS | PROBABILITY DENSITY FUNCTIONS | NONLINEAR PROBLEMS | PATH INTEGRALS | TRANSFORMATIONS | PROPAGATOR

Uncertainty propagation | Stochastic differential equations | Short-time propagator | Fokker–Planck equation | Path integral | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | NONLINEAR-SYSTEMS | FINITE-ELEMENT | Fokker-Planck equation | MONTE-CARLO-SIMULATION | PHYSICS, MATHEMATICAL | Monte Carlo method | Stochastic processes | Differential equations | FOKKER-PLANCK EQUATION | GRIDS | STOCHASTIC PROCESSES | MATHEMATICAL EVOLUTION | MONTE CARLO METHOD | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | ONE-DIMENSIONAL CALCULATIONS | COMPUTERIZED SIMULATION | MATHEMATICAL SOLUTIONS | PROBABILITY DENSITY FUNCTIONS | NONLINEAR PROBLEMS | PATH INTEGRALS | TRANSFORMATIONS | PROPAGATOR

Journal Article

Mathematical Methods in the Applied Sciences, ISSN 0170-4214, 03/2018, Volume 41, Issue 4, pp. 1410 - 1423

In this paper, an efficient numerical technique is applied to provide the approximate solution of nonlinear stochastic Itô‐Volterra integral equations driven...

collocation method | modification of hat functions | stochastic Itô‐Volterra integral equations | fractional Brownian motion process | operational matrix | stochastic Itô-Volterra integral equations | MATHEMATICS, APPLIED | HAT FUNCTIONS | APPROXIMATION | stochastic Ito-Volterra integral equations | DIFFERENTIAL-EQUATIONS | RADIAL BASIS FUNCTIONS | SCHEMES | Nonlinear equations | Error analysis | Numerical analysis | Integral equations | Collocation methods | Brownian movements | Nonlinear systems | Volterra integral equations

collocation method | modification of hat functions | stochastic Itô‐Volterra integral equations | fractional Brownian motion process | operational matrix | stochastic Itô-Volterra integral equations | MATHEMATICS, APPLIED | HAT FUNCTIONS | APPROXIMATION | stochastic Ito-Volterra integral equations | DIFFERENTIAL-EQUATIONS | RADIAL BASIS FUNCTIONS | SCHEMES | Nonlinear equations | Error analysis | Numerical analysis | Integral equations | Collocation methods | Brownian movements | Nonlinear systems | Volterra integral equations

Journal Article

Stochastic Processes and their Applications, ISSN 0304-4149, 02/2019, Volume 129, Issue 2, pp. 626 - 633

We present an explicit solution triplet (Y,Z,K) to the backward stochastic Volterra integral equation (BSVIE) of linear type, driven by a Brownian motion and a...

Brownian motion | Compensated Poisson random measure | Linear equation | Hida–Malliavin derivative | Explicit solution | Volterra type backward stochastic differential equation | Hida-Malliavin derivative | STATISTICS & PROBABILITY

Brownian motion | Compensated Poisson random measure | Linear equation | Hida–Malliavin derivative | Explicit solution | Volterra type backward stochastic differential equation | Hida-Malliavin derivative | STATISTICS & PROBABILITY

Journal Article

15.
Full Text
A collocation technique for solving nonlinear Stochastic Itô–Volterra integral equations

Applied Mathematics and Computation, ISSN 0096-3003, 11/2014, Volume 247, pp. 1011 - 1020

A numerical method for solving nonlinear Stochastic Itô–Volterra equations is proposed. The method is based on delta function (DF) approximations. The...

Delta functions | Error analysis | Stochastic | Collocation | Vector forms | Operational matrices | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | RANDOM DIFFERENTIAL-EQUATIONS | INTEGRODIFFERENTIAL EQUATIONS | Approximation | Computation | Mathematical analysis | Nonlinearity | Mathematical models | Stochasticity | Dynamical systems | Delta function

Delta functions | Error analysis | Stochastic | Collocation | Vector forms | Operational matrices | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | RANDOM DIFFERENTIAL-EQUATIONS | INTEGRODIFFERENTIAL EQUATIONS | Approximation | Computation | Mathematical analysis | Nonlinearity | Mathematical models | Stochasticity | Dynamical systems | Delta function

Journal Article

16.
Full Text
Symmetrical martingale solutions of backward doubly stochastic Volterra integral equations

Computers and Mathematics with Applications, ISSN 0898-1221, 03/2020, Volume 79, Issue 5, pp. 1435 - 1446

This paper aims to study a new class of integral equations called backward doubly stochastic Volterra integral equations (BDSVIEs, for short). The notion of...

Symmetrical martingale solution | Backward stochastic integral | Backward doubly stochastic Volterra integral equation

Symmetrical martingale solution | Backward stochastic integral | Backward doubly stochastic Volterra integral equation

Journal Article

Nonlinear Analysis: Real World Applications, ISSN 1468-1218, 08/2020, Volume 54, p. 103104

We prove some existence and uniqueness results for a nonlinear stochastic integral equation using fixed-point theory methods to ensure the convergence of the...

Integral equation | Renormalization of Banach space | Stochastic equation | Fixed point methods

Integral equation | Renormalization of Banach space | Stochastic equation | Fixed point methods

Journal Article

2013, ISBN 9789814447997, xiii, 159

Book

2005, ISBN 0521850150, x, 381

This book, first published in 2005, introduces measure and integration theory as it is needed in many parts of analysis and probability theory. The basic...

Measure theory | Integrals | Martingales (Mathematics) | Mathematics

Measure theory | Integrals | Martingales (Mathematics) | Mathematics

Book