Stochastics and Dynamics, ISSN 0219-4937, 08/2018, Volume 18, Issue 4

The functional Ito formula, firstly introduced by Bruno Dupire for continuous semi-martingales, might be extended in two directions: different dynamics for the...

Functional Itô calculus | mollification | max-martingales | Meyer-Tanaka formula | running maximum | local time | MARTINGALES | VISCOSITY SOLUTIONS | Functional Ito calculus | CALCULUS | ITOS FORMULA | STATISTICS & PROBABILITY | SPACE | SEMIMARTINGALES | Mathematics - Probability

Functional Itô calculus | mollification | max-martingales | Meyer-Tanaka formula | running maximum | local time | MARTINGALES | VISCOSITY SOLUTIONS | Functional Ito calculus | CALCULUS | ITOS FORMULA | STATISTICS & PROBABILITY | SPACE | SEMIMARTINGALES | Mathematics - Probability

Journal Article

Probability Theory and Related Fields, ISSN 0178-8051, 6/2011, Volume 150, Issue 1, pp. 295 - 319

We prove the Itô–Wentzell formula for processes with values in the space of generalized functions by using the stochastic Fubini theorem and the Itô–Wentzell...

Mathematical and Computational Biology | Statistics for Business/Economics/Mathematical Finance/Insurance | Operations Research/Decision Theory | Theoretical, Mathematical and Computational Physics | Stochastic Fubini theorem | Probability Theory and Stochastic Processes | Mathematics | Itô–Wentzell formula | 60H05 | Quantitative Finance | 60H15 | Itô-Wentzell formula | STATISTICS & PROBABILITY | Ito-Wentzell formula | Studies | Probability | Theorems | Probability theory | Stochasticity

Mathematical and Computational Biology | Statistics for Business/Economics/Mathematical Finance/Insurance | Operations Research/Decision Theory | Theoretical, Mathematical and Computational Physics | Stochastic Fubini theorem | Probability Theory and Stochastic Processes | Mathematics | Itô–Wentzell formula | 60H05 | Quantitative Finance | 60H15 | Itô-Wentzell formula | STATISTICS & PROBABILITY | Ito-Wentzell formula | Studies | Probability | Theorems | Probability theory | Stochasticity

Journal Article

ANNALS OF PROBABILITY, ISSN 0091-1798, 01/2019, Volume 47, Issue 1, pp. 1 - 60

Given a Gaussian process X, its canonical geometric rough path lift X, and a solution Y to the rough differential equation (RDE) dY(t) = V (Y-t) circle dX(t),...

Rough paths theory | RESPECT | generalized Ito-Stratonovich correction formulas | CALCULUS | STOCHASTIC INTEGRATION | STATISTICS & PROBABILITY | DIFFERENTIAL-EQUATIONS DRIVEN | Malliavin calculus

Rough paths theory | RESPECT | generalized Ito-Stratonovich correction formulas | CALCULUS | STOCHASTIC INTEGRATION | STATISTICS & PROBABILITY | DIFFERENTIAL-EQUATIONS DRIVEN | Malliavin calculus

Journal Article

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Full Text
Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation

Stochastic Processes and their Applications, ISSN 0304-4149, 2008, Volume 118, Issue 12, pp. 2223 - 2253

We develop a notion of nonlinear expectation– -expectation–generated by a nonlinear heat equation with infinitesimal generator . We first study...

[formula omitted]-expectation | Itô’s formula | Jensen’s inequality | Itô’s integral | BSDE | Nonlinear probability theory | Quadratic variation process | Nonlinear expectation | Brownian motion | Gaussian process | SDE | Itô’s stochastic calculus | [formula omitted]-convexity | [formula omitted]-normal distribution | G-convexity | Itô's stochastic calculus | G-normal distribution | Itô's formula | Jensen's inequality | Itô's integral | g-expectation | Ito's stochastic calculus | THEOREM | DIFFERENTIAL-EQUATIONS | RISK | STATISTICS & PROBABILITY | NONLINEAR EXPECTATIONS | Ito's formula | JENSENS INEQUALITY | Ito's integral | g-expectation G-expectation G-normal distribution BSDE SDE Nonlinear probability theory Nonlinear expectation Brownian motion Ito's stochastic calculus Ito's integral Ito's formula Gaussian process Quadratic variation process Jensen's inequality G-convexity

[formula omitted]-expectation | Itô’s formula | Jensen’s inequality | Itô’s integral | BSDE | Nonlinear probability theory | Quadratic variation process | Nonlinear expectation | Brownian motion | Gaussian process | SDE | Itô’s stochastic calculus | [formula omitted]-convexity | [formula omitted]-normal distribution | G-convexity | Itô's stochastic calculus | G-normal distribution | Itô's formula | Jensen's inequality | Itô's integral | g-expectation | Ito's stochastic calculus | THEOREM | DIFFERENTIAL-EQUATIONS | RISK | STATISTICS & PROBABILITY | NONLINEAR EXPECTATIONS | Ito's formula | JENSENS INEQUALITY | Ito's integral | g-expectation G-expectation G-normal distribution BSDE SDE Nonlinear probability theory Nonlinear expectation Brownian motion Ito's stochastic calculus Ito's integral Ito's formula Gaussian process Quadratic variation process Jensen's inequality G-convexity

Journal Article

Discrete and Continuous Dynamical Systems - Series B, ISSN 1531-3492, 08/2018, Volume 23, Issue 6, pp. 2245 - 2263

The mild Ito formula proposed in Theorem 1 in [Da Prato, G., Jentzen, A., & Rockner, M., A mild Ito formula for SPDEs, arXiv : 1009.3526 (2012), To appear in...

MATHEMATICS, APPLIED | Stochastic analysis | MODELS | Ito formula | mild solution | EQUATIONS | CONVERGENCE | Nemytskii operators | Banach spaces | EXPONENTIAL INTEGRATORS

MATHEMATICS, APPLIED | Stochastic analysis | MODELS | Ito formula | mild solution | EQUATIONS | CONVERGENCE | Nemytskii operators | Banach spaces | EXPONENTIAL INTEGRATORS

Journal Article

Stochastic Processes and their Applications, ISSN 0304-4149, 08/2019

Journal Article

Physica A: Statistical Mechanics and its Applications, ISSN 0378-4371, 11/2014, Volume 414, pp. 154 - 162

In the paper, we first extend the discrete time quantum random walk (DQRW) to continuous time quantum random walk (CQRW). Then we establish an Itô formula for...

DQRW | Tanaka’s formula | Itô formula | CQRW | Tanaka's formula | STOCHASTIC CALCULUS | PHYSICS, MULTIDISCIPLINARY | Ito formula

DQRW | Tanaka’s formula | Itô formula | CQRW | Tanaka's formula | STOCHASTIC CALCULUS | PHYSICS, MULTIDISCIPLINARY | Ito formula

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 10/2015, Volume 430, Issue 2, pp. 1163 - 1174

Extending Itô's formula to non-smooth functions is important both in theory and applications. One of the fairly general extensions of the formula, known as...

Itô's formula | Finite variation Lévy process | Weak derivative | PIDE | ItÔ's formula | MATHEMATICS | MATHEMATICS, APPLIED | Finite variation Levy process | Ito's formula

Itô's formula | Finite variation Lévy process | Weak derivative | PIDE | ItÔ's formula | MATHEMATICS | MATHEMATICS, APPLIED | Finite variation Levy process | Ito's formula

Journal Article

Mathematische Nachrichten, ISSN 0025-584X, 12/2016, Volume 289, Issue 17-18, pp. 2192 - 2222

The paper studies a class of Ornstein–Uhlenbeck processes on the classical Wiener space. These processes are associated with a diffusion type Dirichlet form...

Infinite dimensional Ornstein–Uhlenbeck process | weak approximation | Itô formula | quadratic variation | Secondary: 60G15 | Primary: 60J60 | MATHEMATICS | SPACE VALUED PROCESSES | PATH SPACE | Ito formula | Infinite dimensional Ornstein-Uhlenbeck process | RIEMANNIAN MANIFOLD | COVARIATION | Naturvetenskap | Mathematics | Natural Sciences | Matematik

Infinite dimensional Ornstein–Uhlenbeck process | weak approximation | Itô formula | quadratic variation | Secondary: 60G15 | Primary: 60J60 | MATHEMATICS | SPACE VALUED PROCESSES | PATH SPACE | Ito formula | Infinite dimensional Ornstein-Uhlenbeck process | RIEMANNIAN MANIFOLD | COVARIATION | Naturvetenskap | Mathematics | Natural Sciences | Matematik

Journal Article

Statistics and Probability Letters, ISSN 0167-7152, 04/2014, Volume 87, Issue 1, pp. 48 - 53

We prove an Itô's formula for Walsh's Brownian motion in the plane with angles according to a probability measure μ on [0, 2π [. This extends Freidlin-Sheu...

Harmonic functions | Stochastic flows | Itô's formula | Walsh's Brownian motion | STATISTICS & PROBABILITY | FLOWS | Ito's formula

Harmonic functions | Stochastic flows | Itô's formula | Walsh's Brownian motion | STATISTICS & PROBABILITY | FLOWS | Ito's formula

Journal Article

PROBABILITY AND MATHEMATICAL STATISTICS-POLAND, ISSN 0208-4147, 2019, Volume 39, Issue 1, pp. 39 - 60

For symmetric Levy processes, if the local times exist, the Tanaka formula has already been constructed via the techniques in the potential theory by Salminen...

stable process | Ito's stochastic calculus | STATISTICS & PROBABILITY | Local time

stable process | Ito's stochastic calculus | STATISTICS & PROBABILITY | Local time

Journal Article

Stochastic Processes and their Applications, ISSN 0304-4149, 08/2015, Volume 125, Issue 8, pp. 2989 - 3022

We derive a generalised Itō formula for stochastic processes which are constructed by a convolution of a deterministic kernel with a centred Lévy process. This...

Skorokhod integral | [formula omitted]-transform | Fractional Lévy process | Stochastic convolution | Itō formula | Ito formula | S-transform

Skorokhod integral | [formula omitted]-transform | Fractional Lévy process | Stochastic convolution | Itō formula | Ito formula | S-transform

Journal Article

Stochastic Processes and their Applications, ISSN 0304-4149, 11/2014, Volume 124, Issue 11, pp. 3869 - 3885

In this paper, we show that the integration of a stochastic differential equation driven by -Brownian motion ( -SDE for short) in can be reduced to the...

[formula omitted]-Itô’s formula | Comparison theorem | [formula omitted]-Brownian motion | [formula omitted]-SDE | G-SDE | G-Itô's formula | G-Brownian motion | THEOREM | STATISTICS & PROBABILITY | FORMULA | G-Ito's formula | Analysis | Differential equations

[formula omitted]-Itô’s formula | Comparison theorem | [formula omitted]-Brownian motion | [formula omitted]-SDE | G-SDE | G-Itô's formula | G-Brownian motion | THEOREM | STATISTICS & PROBABILITY | FORMULA | G-Ito's formula | Analysis | Differential equations

Journal Article

Random Operators and Stochastic Equations, ISSN 0926-6364, 06/2017, Volume 25, Issue 2, pp. 79 - 105

We use the Yosida approximation to find an Itô formula for mild solutions of SPDEs with Gaussian and non-Gaussian colored noise, with the non-Gaussian noise...

mild solutions | Stochastic partial differential equations | Itô formula | Yosida approximation | 60G51 | 47A58 | 20Mxx | 60H15 | generator of a semigroup | exponential stability | 60G15 | 34D05 | Generator of a semigroup | Mild solutions | Exponential stability | Itôformula | Random noise theory | Mathematics | Formulae | Research | Differential equations, Partial | Mathematical research | Mean square values | Random noise | Approximation | Noise | Mathematical analysis | Normal distribution

mild solutions | Stochastic partial differential equations | Itô formula | Yosida approximation | 60G51 | 47A58 | 20Mxx | 60H15 | generator of a semigroup | exponential stability | 60G15 | 34D05 | Generator of a semigroup | Mild solutions | Exponential stability | Itôformula | Random noise theory | Mathematics | Formulae | Research | Differential equations, Partial | Mathematical research | Mean square values | Random noise | Approximation | Noise | Mathematical analysis | Normal distribution

Journal Article

Alea, ISSN 1980-0436, 2018, Volume 15, Issue 1, pp. 703 - 753

We investigate Bochner integrabilities of generalized Wiener functionals. We further formulate an Ito formula for a diffusion in a distributional setting, and...

Distributional Itô's formula | Hölder continuity of diffusion local times | Malliavin calculus | Smoothing effect brought by time-integral | SOBOLEV SPACES | SMOOTHNESS | LOCAL TIME | STATISTICS & PROBABILITY | Holder continuity of diffusion local times | Distributional Ito's formula

Distributional Itô's formula | Hölder continuity of diffusion local times | Malliavin calculus | Smoothing effect brought by time-integral | SOBOLEV SPACES | SMOOTHNESS | LOCAL TIME | STATISTICS & PROBABILITY | Holder continuity of diffusion local times | Distributional Ito's formula

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 10/2012, Volume 64, Issue 7, pp. 2302 - 2311

We apply a variant of a discretised Itô formula to develop sharp conditions for the global a.s. asymptotic stability of the equilibrium solution of a...

a.s. asymptotic stability | Random sequence | Stochastic difference equations | Discrete Itô formula multiplicative noise | a.s. asymptotic stability | MATHEMATICS, APPLIED | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | Discrete Ito formula multiplicative noise | Differential equations

a.s. asymptotic stability | Random sequence | Stochastic difference equations | Discrete Itô formula multiplicative noise | a.s. asymptotic stability | MATHEMATICS, APPLIED | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | Discrete Ito formula multiplicative noise | Differential equations

Journal Article

The Annals of Probability, ISSN 0091-1798, 1/2013, Volume 41, Issue 1, pp. 109 - 133

We develop a nonanticipative calculus for functionals of a continuous semimartingale, using an extension of the Itô formula to path-dependent functionals which...

Differential calculus | Mathematical theorems | Approximation | Integrands | Mathematical integrals | Directional derivatives | Calculus | Martingales | Mathematical integration | functional calculus | functional Ito formula | Malliavin derivative | Wiener functionals | Stochastic calculus | semimartingale | Clark-Ocone formula | DIFFERENTIAL-EQUATIONS | STATISTICS & PROBABILITY | martingale representation | FORMULA | MALLIAVIN CALCULUS | Functional Analysis | Probability | Mathematics | 60G44 | Clark–Ocone formula | 60H07 | 60H05 | 60H25 | functional Itô formula

Differential calculus | Mathematical theorems | Approximation | Integrands | Mathematical integrals | Directional derivatives | Calculus | Martingales | Mathematical integration | functional calculus | functional Ito formula | Malliavin derivative | Wiener functionals | Stochastic calculus | semimartingale | Clark-Ocone formula | DIFFERENTIAL-EQUATIONS | STATISTICS & PROBABILITY | martingale representation | FORMULA | MALLIAVIN CALCULUS | Functional Analysis | Probability | Mathematics | 60G44 | Clark–Ocone formula | 60H07 | 60H05 | 60H25 | functional Itô formula

Journal Article

Stochastic Processes and their Applications, ISSN 0304-4149, 2009, Volume 119, Issue 10, pp. 3356 - 3382

We study pathwise properties and homeomorphic property with respect to the initial values for stochastic differential equations driven by -Brownian motion. We...

Homeomorphic flow | Itô’s-formula | Moment estimate | Hölder continuity | [formula omitted]-Brownian motion | [formula omitted]-stochastic differential equation | BDG inequality | Itô's-formula | G-Brownian motion | G-stochastic differential equation | Holder continuity | STATISTICS & PROBABILITY | Ito's-formula | G-Brownian motion G-stochastic differential equation BDG inequality Ito's-formula Moment estimate Holder continuity Homeomorphic flow

Homeomorphic flow | Itô’s-formula | Moment estimate | Hölder continuity | [formula omitted]-Brownian motion | [formula omitted]-stochastic differential equation | BDG inequality | Itô's-formula | G-Brownian motion | G-stochastic differential equation | Holder continuity | STATISTICS & PROBABILITY | Ito's-formula | G-Brownian motion G-stochastic differential equation BDG inequality Ito's-formula Moment estimate Holder continuity Homeomorphic flow

Journal Article

Mathematical and Computer Modelling, ISSN 0895-7177, 2011, Volume 54, Issue 1, pp. 221 - 232

In this paper, we explore a stochastic SIR model and show that this model has a unique global positive solution. Furthermore, we investigate the asymptotic...

Brownian motion | Stochastic SIR model | It [formula omitted]’s formula | Global positive solution | Asymptotic behavior | Itô's formula | MATHEMATICS, APPLIED | Ito's formula

Brownian motion | Stochastic SIR model | It [formula omitted]’s formula | Global positive solution | Asymptotic behavior | Itô's formula | MATHEMATICS, APPLIED | Ito's formula

Journal Article

SIAM Journal on Financial Mathematics, ISSN 1945-497X, 2018, Volume 9, Issue 3, pp. 1074 - 1101

We use pathwise Ito calculus to prove two strictly pathwise versions of the mater formula in Fernholz' stochastic portfolio theory. Our first version is set...

Portfolio-generating functionals | Functional Itô formula | Portfolio analysis | Entropy weighting | Functional master formula on path space | Market portfolio | Pathwise Itô calculus | Föllmer integral | Follmer integral | entropy weighting | RELATIVE ARBITRAGE | portfolio analysis | market portfolio | OPTIONS | BUSINESS, FINANCE | functional Ito formula | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | EQUITY MARKETS | portfolio-generating functionals | functional master formula on path space | SOCIAL SCIENCES, MATHEMATICAL METHODS | pathwise Ito calculus | DIVERSITY | PROBABILITIES

Portfolio-generating functionals | Functional Itô formula | Portfolio analysis | Entropy weighting | Functional master formula on path space | Market portfolio | Pathwise Itô calculus | Föllmer integral | Follmer integral | entropy weighting | RELATIVE ARBITRAGE | portfolio analysis | market portfolio | OPTIONS | BUSINESS, FINANCE | functional Ito formula | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | EQUITY MARKETS | portfolio-generating functionals | functional master formula on path space | SOCIAL SCIENCES, MATHEMATICAL METHODS | pathwise Ito calculus | DIVERSITY | PROBABILITIES

Journal Article

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