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

Differential Equations, ISSN 0012-2661, 4/2018, Volume 54, Issue 4, pp. 557 - 561

For a stochastic differential equation of the heat equation type, we obtain a Feynman–Kac formula to which the method of analytic continuation with respect to...

Difference and Functional Equations | Mathematics | Ordinary Differential Equations | Partial Differential Equations | MATHEMATICS | Formulas (mathematics) | Schroedinger equation | Differential equations

Difference and Functional Equations | Mathematics | Ordinary Differential Equations | Partial Differential Equations | MATHEMATICS | Formulas (mathematics) | Schroedinger equation | Differential equations

Journal Article

Stochastics and Dynamics, ISSN 0219-4937, 08/2016, Volume 16, Issue 4

Dupire [16] introduced a notion of smoothness for functionals of paths and arrived at a generalization of Ito's formula that applies to functionals with a...

Pathwise stochastic calculus | functional Itō formula | VISCOSITY SOLUTIONS | STATISTICS & PROBABILITY | functional Ito formula

Pathwise stochastic calculus | functional Itō formula | VISCOSITY SOLUTIONS | STATISTICS & PROBABILITY | functional Ito formula

Journal Article

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

Journal of Differential Equations, ISSN 0022-0396, 2008, Volume 245, Issue 1, pp. 30 - 58

Using the theory of stochastic integration for processes with values in a UMD Banach space developed recently by the authors, an Itô formula is proved which is...

Itô formula | Stochastic integration in Banach spaces | UMD spaces | Wong–Zakai approximation | Stochastic evolution equations | Zakai equation | Non-autonomous equations | Wong-Zakai approximation | MATHEMATICS | stochastic integration in Banach spaces | Ito formula | stochastic evolution equations | PARTIAL-DIFFERENTIAL EQUATIONS | STOCHASTIC INTEGRATION | VALUES | EVOLUTION-EQUATIONS | non-autonomous equations

Itô formula | Stochastic integration in Banach spaces | UMD spaces | Wong–Zakai approximation | Stochastic evolution equations | Zakai equation | Non-autonomous equations | Wong-Zakai approximation | MATHEMATICS | stochastic integration in Banach spaces | Ito formula | stochastic evolution equations | PARTIAL-DIFFERENTIAL EQUATIONS | STOCHASTIC INTEGRATION | VALUES | EVOLUTION-EQUATIONS | non-autonomous equations

Journal Article

Stochastic Processes and their Applications, ISSN 0304-4149, 03/2016, Volume 126, Issue 3, pp. 735 - 766

This paper introduces path derivatives, in the spirit of Dupire’s functional Itô calculus, for controlled rough paths in rough path theory with possibly...

Functional Itô calculus | Rough path | Itô–Ventzell formula | Path derivatives | Characteristics | Rough differential equations | Rough PDEs | Stochastic PDEs | 60H10 | MSC 60H05 | 60G17 | 60G05 | 60H15 | STOCHASTIC VISCOSITY SOLUTIONS | Functional Ito calculus | PARTIAL-DIFFERENTIAL-EQUATIONS | Ito-Ventzell formula | STATISTICS & PROBABILITY | DRIVEN

Functional Itô calculus | Rough path | Itô–Ventzell formula | Path derivatives | Characteristics | Rough differential equations | Rough PDEs | Stochastic PDEs | 60H10 | MSC 60H05 | 60G17 | 60G05 | 60H15 | STOCHASTIC VISCOSITY SOLUTIONS | Functional Ito calculus | PARTIAL-DIFFERENTIAL-EQUATIONS | Ito-Ventzell formula | STATISTICS & PROBABILITY | DRIVEN

Journal Article

Stochastic Processes and their Applications, ISSN 0304-4149, 07/2015, Volume 125, Issue 7, pp. 2820 - 2855

In this paper we establish the pathwise Taylor expansions for random fields that are “regular” in terms of Dupire’s path-derivatives [6]. Using the language of...

Stochastic partial differential equations | Pathwise Taylor expansion | Path derivatives | Functional Itô formula | Itô–Wentzell formula | Ito-Wentzell formula | STATISTICS & PROBABILITY | Functional Ito formula

Stochastic partial differential equations | Pathwise Taylor expansion | Path derivatives | Functional Itô formula | Itô–Wentzell formula | Ito-Wentzell formula | STATISTICS & PROBABILITY | Functional Ito 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

2016, Advanced Courses in Mathematics - CRM Barcelona, ISBN 9783319271279, Issue DOI: 10.1007/978-3-319-27128-6, 213

This volume contains lecture notes from the courses given by Vlad Bally and Rama Cont at the Barcelona Summer School on Stochastic Analysis (July 2012).The...

Probability Theory and Stochastic Processes | Mathematics | Ordinary Differential Equations | Partial Differential Equations | Probability

Probability Theory and Stochastic Processes | Mathematics | Ordinary Differential Equations | Partial Differential Equations | Probability

eBook

The Annals of Probability, ISSN 0091-1798, 3/2014, Volume 42, Issue 2, pp. 497 - 526

We characterize the asymptotic independence between blocks consisting of multiple Wiener–Itô integrals. As a consequence of this characterization, we derive...

Integers | Brownian motion | Cauchy Schwarz inequality | Tensors | Mathematical theorems | Covariance | Skis | Mathematical integrals | Articles | Mathematical moments | Random variables | Multiple Wiener-Itô integral | Limit theorems | Multiplication formula | multiplication formula | Multiple Wiener-Ito integral | THEOREMS | CONVERGENCE | STATISTICS & PROBABILITY | limit theorems | FUNCTIONALS | Probability | Mathematics | 60H07 | 60H05 | 60F05 | 60G15 | Multiple Wiener–Itô integral

Integers | Brownian motion | Cauchy Schwarz inequality | Tensors | Mathematical theorems | Covariance | Skis | Mathematical integrals | Articles | Mathematical moments | Random variables | Multiple Wiener-Itô integral | Limit theorems | Multiplication formula | multiplication formula | Multiple Wiener-Ito integral | THEOREMS | CONVERGENCE | STATISTICS & PROBABILITY | limit theorems | FUNCTIONALS | Probability | Mathematics | 60H07 | 60H05 | 60F05 | 60G15 | Multiple Wiener–Itô integral

Journal Article

Bernoulli, ISSN 1350-7265, 11/2012, Volume 18, Issue 4, pp. 1150 - 1171

Chen, Fitzsimmons, Kuwae and Zhang (Ann. Probab. 36 (2008) 931-970) have established an Itô formula consisting in the development of F(u(X)) for a symmetric...

Mathematical theorems | Mathematical monotonicity | Real numbers | Markov processes | Mathematical functions | Martingales | Continuous functions | Perceptron convergence procedure | Mathematical integration | Fukushima decomposition | Symmetric Markov process | Additive functional | Itô formula | Stochastic calculus | Zero energy process | stochastic calculus | symmetric Markov process | LEVY PROCESSES | Ito formula | TIME-SPACE CALCULUS | additive functional | LOCAL TIME | STATISTICS & PROBABILITY | zero energy process | FORMULA

Mathematical theorems | Mathematical monotonicity | Real numbers | Markov processes | Mathematical functions | Martingales | Continuous functions | Perceptron convergence procedure | Mathematical integration | Fukushima decomposition | Symmetric Markov process | Additive functional | Itô formula | Stochastic calculus | Zero energy process | stochastic calculus | symmetric Markov process | LEVY PROCESSES | Ito formula | TIME-SPACE CALCULUS | additive functional | LOCAL TIME | STATISTICS & PROBABILITY | zero energy process | FORMULA

Journal Article

Infinite Dimensional Analysis, Quantum Probability and Related Topics, ISSN 0219-0257, 12/2016, Volume 19, Issue 4

Functional Ito calculus was introduced in order to expand a functional F(t, X.+(t), X-t) depending on time t, past and present values of the process X. Another...

Functional Itô calculus | strict solutions | Banach space valued stochastic calculus | calculus via regularization | path-dependent partial differential equation | MATHEMATICS, APPLIED | QUANTUM SCIENCE & TECHNOLOGY | BACKWARD | Functional Ito calculus | VALUED PROCESSES | REPRESENTATION | STATISTICS & PROBABILITY | FORMULA | PHYSICS, MATHEMATICAL | GENERALIZED COVARIATION

Functional Itô calculus | strict solutions | Banach space valued stochastic calculus | calculus via regularization | path-dependent partial differential equation | MATHEMATICS, APPLIED | QUANTUM SCIENCE & TECHNOLOGY | BACKWARD | Functional Ito calculus | VALUED PROCESSES | REPRESENTATION | STATISTICS & PROBABILITY | FORMULA | PHYSICS, MATHEMATICAL | GENERALIZED COVARIATION

Journal Article

13.
Full Text
Mean-square exponential input-to-state stability of stochastic delayed neural networks

Neurocomputing, ISSN 0925-2312, 05/2014, Volume 131, pp. 157 - 163

In this paper, we focus on the stability problem for a class of stochastic delayed recurrent neural networks. Different from the traditional stability...

Itô's formula | Stochastic delayed neural network | Exponential stability | Lyapunov–Krasovskii functional | Mean-square exponential input-to-state stability | ItÔ's formula | Lyapunov-Krasovskii functional | ROBUST STABILITY | GLOBAL ASYMPTOTIC STABILITY | Ito's formula | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | MARKOVIAN JUMP PARAMETERS | Neural networks | Analysis

Itô's formula | Stochastic delayed neural network | Exponential stability | Lyapunov–Krasovskii functional | Mean-square exponential input-to-state stability | ItÔ's formula | Lyapunov-Krasovskii functional | ROBUST STABILITY | GLOBAL ASYMPTOTIC STABILITY | Ito's formula | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | MARKOVIAN JUMP PARAMETERS | Neural networks | Analysis

Journal Article

IEEE Transactions on Automatic Control, ISSN 0018-9286, 8/2019, pp. 1 - 1

Given an unstable hybrid stochastic differential equation (SDE), can we design a feedback control, based on the discrete-time observations of the state at...

Markov chain | Asymptotic stability | Symmetric matrices | Lyapunov functional | Highly nonlinear | Stability criteria | Ito formula | Differential equations | Markov processes | Control systems | Feedback control

Markov chain | Asymptotic stability | Symmetric matrices | Lyapunov functional | Highly nonlinear | Stability criteria | Ito formula | Differential equations | Markov processes | Control systems | Feedback control

Journal Article

Journal of Functional Analysis, ISSN 0022-1236, 2010, Volume 259, Issue 4, pp. 1043 - 1072

We derive a change of variable formula for non-anticipative functionals defined on the space of R d -valued right-continuous paths with left limits. The...

Itô formula | Functional derivative | Dirichlet process | Malliavin calculus | Semimartingale | Cadlag functions | Functional calculus | Stochastic integral | Quadratic variation | MATHEMATICS | Ito formula | DIRICHLET PROCESSES | Functional Analysis | Probability | Mathematics

Itô formula | Functional derivative | Dirichlet process | Malliavin calculus | Semimartingale | Cadlag functions | Functional calculus | Stochastic integral | Quadratic variation | MATHEMATICS | Ito formula | DIRICHLET PROCESSES | Functional Analysis | Probability | Mathematics

Journal Article

Potential Analysis, ISSN 0926-2601, 1/2019, Volume 50, Issue 1, pp. 1 - 42

Stochastic integration wrt Gaussian processes has raised strong interest in recent years, motivated in particular by its applications in Internet traffic...

Varying regularity processes | 60G22 | Wick-Itô integrals | 60H40 | Probability Theory and Stochastic Processes | Mathematics | White noise theory | Geometry | Stochastic analysis | Itô formula | Potential Theory | Functional Analysis | Gaussian processes | 60H05 | 60G15 | MATHEMATICS | INTEGRATION | Wick-Ito integrals | Ito formula | FRACTIONAL BROWNIAN-MOTION | DIVERGENCE OPERATOR | Mathematics - Probability | Probability

Varying regularity processes | 60G22 | Wick-Itô integrals | 60H40 | Probability Theory and Stochastic Processes | Mathematics | White noise theory | Geometry | Stochastic analysis | Itô formula | Potential Theory | Functional Analysis | Gaussian processes | 60H05 | 60G15 | MATHEMATICS | INTEGRATION | Wick-Ito integrals | Ito formula | FRACTIONAL BROWNIAN-MOTION | DIVERGENCE OPERATOR | Mathematics - Probability | Probability

Journal Article

17.
Full Text
Almost sure exponential stability of hybrid stochastic functional differential equations

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 02/2018, Volume 458, Issue 2, pp. 1390 - 1408

This paper is concerned with the almost sure exponential stability of the n-dimensional nonlinear hybrid stochastic functional differential equation (SFDE)...

Brownian motion | Markov chain | Stability | Itô formula | Hybrid stochastic differential functional equations | MATHEMATICS | MATHEMATICS, APPLIED | Ito formula | STABILIZATION | DESTABILIZATION | SYSTEMS | DELAY EQUATIONS | NOISE

Brownian motion | Markov chain | Stability | Itô formula | Hybrid stochastic differential functional equations | MATHEMATICS | MATHEMATICS, APPLIED | Ito formula | STABILIZATION | DESTABILIZATION | SYSTEMS | DELAY EQUATIONS | NOISE

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 2009, Volume 359, Issue 2, pp. 482 - 498

In this paper, we consider a predator–prey model with modified Leslie–Gower and Holling-type II schemes with stochastic perturbation. We show there is a unique...

Globally stable in time average | Itô's formula | Persistent in mean | Extinction | COMPLEX DYNAMICS | MATHEMATICS, APPLIED | STABILITY | BEHAVIOR | Ito's formula | FUNCTIONAL-RESPONSE | MATHEMATICS | IMPULSIVE PERTURBATIONS | LOTKA-VOLTERRA MODEL | SYSTEMS

Globally stable in time average | Itô's formula | Persistent in mean | Extinction | COMPLEX DYNAMICS | MATHEMATICS, APPLIED | STABILITY | BEHAVIOR | Ito's formula | FUNCTIONAL-RESPONSE | MATHEMATICS | IMPULSIVE PERTURBATIONS | LOTKA-VOLTERRA MODEL | SYSTEMS

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 09/2015, Volume 266, pp. 860 - 873

•We discuss exponential p-convergence for stochastic BAM neural networks.•The delays here are multiple time-varying and infinite distributed delays.•We...

[formula omitted]-operator differential-integral inequality | Stochastic BAM neural networks | It[formula omitted]’s formula | Exponential p-convergence | Infinite distributed delays | L-operator differential-integral inequality | It o ^ 's formula | EXISTENCE | MATHEMATICS, APPLIED | NEUTRAL-TYPE | Stochastic BAM neural netvvorks | Z-operator differential-integral inequality | Ito's formula | FUNCTIONAL-DIFFERENTIAL EQUATIONS | ATTRACTIVE SETS | POSITIVE INVARIANT | IMPULSES | DISCRETE | ASSOCIATIVE MEMORY NETWORKS | QUASI-INVARIANT SETS | ROBUST ASYMPTOTIC STABILITY | Neural networks | Analysis

[formula omitted]-operator differential-integral inequality | Stochastic BAM neural networks | It[formula omitted]’s formula | Exponential p-convergence | Infinite distributed delays | L-operator differential-integral inequality | It o ^ 's formula | EXISTENCE | MATHEMATICS, APPLIED | NEUTRAL-TYPE | Stochastic BAM neural netvvorks | Z-operator differential-integral inequality | Ito's formula | FUNCTIONAL-DIFFERENTIAL EQUATIONS | ATTRACTIVE SETS | POSITIVE INVARIANT | IMPULSES | DISCRETE | ASSOCIATIVE MEMORY NETWORKS | QUASI-INVARIANT SETS | ROBUST ASYMPTOTIC STABILITY | Neural networks | Analysis

Journal Article

JOURNAL OF STATISTICAL PHYSICS, ISSN 0022-4715, 02/2020, Volume 178, Issue 3, pp. 666 - 681

We consider the Dean-Kawasaki equation with smooth drift interaction potential and show that measure-valued solutions exist only in certain parameter regimes...

Wasserstein diffusion | BROWNIAN PARTICLES | Dean-Kawasaki equation | Langevin dynamics | Ito formula for measure-valued processes | MODEL | DENSITY-FUNCTIONAL THEORY | PHYSICS, MATHEMATICAL

Wasserstein diffusion | BROWNIAN PARTICLES | Dean-Kawasaki equation | Langevin dynamics | Ito formula for measure-valued processes | MODEL | DENSITY-FUNCTIONAL THEORY | PHYSICS, MATHEMATICAL

Journal Article

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