Computers and Mathematics with Applications, ISSN 0898-1221, 03/2016, Volume 71, Issue 6, pp. 1248 - 1258

The multiple exp-function algorithm, as a generalization of Hirota’s perturbation scheme, is used to construct multiple wave solutions to the generalized...

Multiple exp-function algorithm | The generalized ([formula omitted])-dimensional and ([formula omitted])-dimensional Ito equations | Multiple wave solutions | The generalized (1+1)-dimensional and (2+1)-dimensional Ito equations | INTEGRABLE MODELS | MATHEMATICS, APPLIED | SOLITON-SOLUTIONS | MADELUNG FLUID DESCRIPTION | NONLINEAR SCHRODINGER-EQUATION | BACKLUND TRANSFORMATION | KINK SOLUTIONS | Algorithms | Construction | Mathematical models | Perturbation methods | Phase shift | Mathematical analysis

Multiple exp-function algorithm | The generalized ([formula omitted])-dimensional and ([formula omitted])-dimensional Ito equations | Multiple wave solutions | The generalized (1+1)-dimensional and (2+1)-dimensional Ito equations | INTEGRABLE MODELS | MATHEMATICS, APPLIED | SOLITON-SOLUTIONS | MADELUNG FLUID DESCRIPTION | NONLINEAR SCHRODINGER-EQUATION | BACKLUND TRANSFORMATION | KINK SOLUTIONS | Algorithms | Construction | Mathematical models | Perturbation methods | Phase shift | Mathematical analysis

Journal Article

Applicable Analysis, ISSN 0003-6811, 03/2014, Volume 93, Issue 3, pp. 539 - 550

The paper introduces a novel Itô's formula for time- dependent tempered generalized functions. As an application, we study the heat equation when initial...

Itô's formula | Hermite expansions | generalized functions | Hermite functions | COLOMBEAU ALGEBRAS | MATHEMATICS, APPLIED | TOPOLOGICAL STRUCTURES | Ito's formula | Initial conditions | Proving | Heat equations

Itô's formula | Hermite expansions | generalized functions | Hermite functions | COLOMBEAU ALGEBRAS | MATHEMATICS, APPLIED | TOPOLOGICAL STRUCTURES | Ito's formula | Initial conditions | Proving | Heat equations

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 02/2019, Volume 77, Issue 4, pp. 947 - 966

In this paper, the N-soliton solution is constructed for the (2+1)-dimensional generalized Hirota–Satsuma–Ito equation, from which some localized waves such as...

Generalized Hirota–Satsuma–Ito equation | [formula omitted]-soliton solution | Lump soliton | Periodic soliton | N-soliton solution | RATIONAL SOLUTIONS | MATHEMATICS, APPLIED | DARBOUX TRANSFORMATION | BOUSSINESQ EQUATION | ORDER ROGUE WAVE | LUMP-KINK SOLUTIONS | RESONANCE STRIPE SOLITONS | BACKLUND TRANSFORMATION | Generalized Hirota-Satsuma-Ito equation | RESIDUAL SYMMETRIES | (3+1)-DIMENSIONAL JIMBO-MIWA | VARIABLE SEPARATION SOLUTIONS | Nonlinear equations | Wave interaction | Interaction parameters | Mathematical analysis | Solitary waves | Wave equations

Generalized Hirota–Satsuma–Ito equation | [formula omitted]-soliton solution | Lump soliton | Periodic soliton | N-soliton solution | RATIONAL SOLUTIONS | MATHEMATICS, APPLIED | DARBOUX TRANSFORMATION | BOUSSINESQ EQUATION | ORDER ROGUE WAVE | LUMP-KINK SOLUTIONS | RESONANCE STRIPE SOLITONS | BACKLUND TRANSFORMATION | Generalized Hirota-Satsuma-Ito equation | RESIDUAL SYMMETRIES | (3+1)-DIMENSIONAL JIMBO-MIWA | VARIABLE SEPARATION SOLUTIONS | Nonlinear equations | Wave interaction | Interaction parameters | Mathematical analysis | Solitary waves | Wave equations

Journal Article

Bulletin of the Georgian National Academy of Sciences, ISSN 0132-1447, 2011, Volume 5, Issue 3, pp. 11 - 16

Journal Article

Stochastic Analysis and Applications, ISSN 0736-2994, 11/2015, Volume 33, Issue 6, pp. 1020 - 1049

Given a (conservative) symmetric Markov process on a metric space we consider related bilinear forms that generalize the energy form for a particle in an...

Magnetic Schrödinger operators; Dirichlet forms: Additive functionals | GENERALIZED SCHRODINGER SEMIGROUPS | FIELDS | MATHEMATICS, APPLIED | NONCOMPACT MANIFOLDS | CONTINUITY PROPERTIES | STATISTICS & PROBABILITY | Dirichlet forms: Additive functionals | Magnetic Schrodinger operators | STOCHASTIC OSCILLATORY INTEGRALS | METRIC MEASURE-SPACES | DIRICHLET FORMS | HEAT KERNEL | OPERATORS | ESSENTIAL SELF-ADJOINTNESS | Kernels | Approximation | Metric space | Mathematical analysis | Markov processes | Electromagnetic fields | Density | Symmetry

Magnetic Schrödinger operators; Dirichlet forms: Additive functionals | GENERALIZED SCHRODINGER SEMIGROUPS | FIELDS | MATHEMATICS, APPLIED | NONCOMPACT MANIFOLDS | CONTINUITY PROPERTIES | STATISTICS & PROBABILITY | Dirichlet forms: Additive functionals | Magnetic Schrodinger operators | STOCHASTIC OSCILLATORY INTEGRALS | METRIC MEASURE-SPACES | DIRICHLET FORMS | HEAT KERNEL | OPERATORS | ESSENTIAL SELF-ADJOINTNESS | Kernels | Approximation | Metric space | Mathematical analysis | Markov processes | Electromagnetic fields | Density | Symmetry

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

Siberian Advances in Mathematics, ISSN 1055-1344, 1/2016, Volume 26, Issue 1, pp. 17 - 29

For the first time we present a complete proof (from the standpoint of stochastic analysis) of the generalized Itô–Venttsel’ formula whose ideas were adduced...

Itô–Venttsel’ formula | Poisson measure | δ-sequence | mean-square convergence | generalized Itô equation | Mathematics, general | Mathematics | Methods | Differential equations

Itô–Venttsel’ formula | Poisson measure | δ-sequence | mean-square convergence | generalized Itô equation | Mathematics, general | Mathematics | Methods | Differential equations

Journal Article

Stochastic Processes and their Applications, ISSN 0304-4149, 1999, Volume 79, Issue 1, pp. 45 - 67

Stability of stochastic differential equations with Markovian switching has recently received a lot of attention. For example, stability of linear or...

Brownian motion | M-matrix | Generalized Itô’s formula | Markov chain generator | Lyapunov exponent | Generalized Itô's formula | M -matrix | STATISTICS & PROBABILITY | generalized Ito's formula | Lyapunov exponent Generalized Ito's formula Brownian motion Markov chain generator M-matrix

Brownian motion | M-matrix | Generalized Itô’s formula | Markov chain generator | Lyapunov exponent | Generalized Itô's formula | M -matrix | STATISTICS & PROBABILITY | generalized Ito's formula | Lyapunov exponent Generalized Ito's formula Brownian motion Markov chain generator M-matrix

Journal Article

Systems & Control Letters, ISSN 0167-6911, 02/2013, Volume 62, Issue 2, pp. 178 - 187

One of the important issues in the study of hybrid SDDEs is the automatic control, with consequent emphasis being placed on the asymptotic analysis of...

Brownian motion | Markov chain | Asymptotic boundedness | Generalized Itô’s formula | Exponential stability | Generalized Itô's formula | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | INTERVAL SYSTEMS | Generalized Ito's formula | AUTOMATION & CONTROL SYSTEMS | DEPENDENT STABILITY

Brownian motion | Markov chain | Asymptotic boundedness | Generalized Itô’s formula | Exponential stability | Generalized Itô's formula | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | INTERVAL SYSTEMS | Generalized Ito's formula | AUTOMATION & CONTROL SYSTEMS | DEPENDENT STABILITY

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 2011, Volume 376, Issue 1, pp. 11 - 28

In this paper, we prove that a stochastic logistic population under regime switching controlled by a Markov chain is either stochastically permanent or...

Brownian motion | Generalized Itô's formula | Markov chain | Stochastic permanence | Stochastic differential equation | MATHEMATICS, APPLIED | STABILITY | PERSISTENCE | BEHAVIOR | ENVIRONMENT | DIFFERENTIAL-EQUATIONS | NOISE | Generalized Ito's formula | RANDOM PERTURBATION | MATHEMATICS | LOTKA-VOLTERRA MODEL | DYNAMICS | SYSTEMS | Markov processes | Analysis | Extinction (Biology)

Brownian motion | Generalized Itô's formula | Markov chain | Stochastic permanence | Stochastic differential equation | MATHEMATICS, APPLIED | STABILITY | PERSISTENCE | BEHAVIOR | ENVIRONMENT | DIFFERENTIAL-EQUATIONS | NOISE | Generalized Ito's formula | RANDOM PERTURBATION | MATHEMATICS | LOTKA-VOLTERRA MODEL | DYNAMICS | SYSTEMS | Markov processes | Analysis | Extinction (Biology)

Journal Article

Journal of Theoretical Probability, ISSN 0894-9840, 6/2016, Volume 29, Issue 2, pp. 590 - 616

We prove an Itô–Tanaka formula and existence of pathwise stochastic integrals for a wide class of Gaussian processes. Motivated by financial applications, we...

Generalized Lebesgue–Stieltjes integral | Mathematical finance | Gaussian processes | 91G20 | Probability Theory and Stochastic Processes | Mathematics | Statistics, general | 60H05 | Itô–Tanaka formula | Pathwise stochastic integral | 60G15 | Föllmer integral

Generalized Lebesgue–Stieltjes integral | Mathematical finance | Gaussian processes | 91G20 | Probability Theory and Stochastic Processes | Mathematics | Statistics, general | 60H05 | Itô–Tanaka formula | Pathwise stochastic integral | 60G15 | Föllmer integral

Journal Article

Stochastic Processes and their Applications, ISSN 0304-4149, 09/2017, Volume 127, Issue 9, pp. 2816 - 2840

Within the framework of the previous paper (Bonaccorsi and Mazzucchi, 2015), we develop a generalized stochastic calculus for processes associated to higher...

Generalized Itô calculus | Probabilistic representation of solutions of PDEs | Stochastic processes on the complex plane | PDE | EQUATIONS | STATISTICS & PROBABILITY | Generalized Ito calculus | SIGNED MEASURE | FORMULA | Stochastic processes

Generalized Itô calculus | Probabilistic representation of solutions of PDEs | Stochastic processes on the complex plane | PDE | EQUATIONS | STATISTICS & PROBABILITY | Generalized Ito calculus | SIGNED MEASURE | FORMULA | Stochastic processes

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 09/2015, Volume 266, pp. 539 - 559

In this paper, we investigate the stochastic permanence and extinction of a stochastic ratio-dependent prey–predator model controlled by a Markov chain. In the...

Brownian motion | Markov chain | Generalized Itô’s formula | Ratio-dependent prey-predator model | Stochastic differential equation | Generalized Itô's formula | MATHEMATICS, APPLIED | II SCHEMES | DIFFERENTIAL-EQUATIONS | MODEL | Generalized Ito's formula | MODIFIED LESLIE-GOWER | QUALITATIVE-ANALYSIS | DYNAMICS | CONVERGENCE | Markov processes | Analysis

Brownian motion | Markov chain | Generalized Itô’s formula | Ratio-dependent prey-predator model | Stochastic differential equation | Generalized Itô's formula | MATHEMATICS, APPLIED | II SCHEMES | DIFFERENTIAL-EQUATIONS | MODEL | Generalized Ito's formula | MODIFIED LESLIE-GOWER | QUALITATIVE-ANALYSIS | DYNAMICS | CONVERGENCE | Markov processes | Analysis

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 2009, Volume 232, Issue 2, pp. 427 - 448

In this paper, we investigate a Lotka–Volterra system under regime switching d x ( t ) = diag ( x 1 ( t ) , … , x n ( t ) ) [ ( b ( r ( t ) ) + A ( r ( t ) ) x...

Brownian motion | Markov chain | Generalized Itô’s formula | Stochastic permanence | Stochastic differential equation | Generalized Itô's formula | EXISTENCE | MATHEMATICS, APPLIED | GLOBAL STABILITY | POSITIVE SOLUTIONS | PERSISTENCE | ENVIRONMENT | DIFFERENTIAL-EQUATIONS | MODEL | Generalized Ito's formula | RANDOM PERTURBATION | UNIQUENESS | LOGISTIC EQUATION | Markov processes | Analysis

Brownian motion | Markov chain | Generalized Itô’s formula | Stochastic permanence | Stochastic differential equation | Generalized Itô's formula | EXISTENCE | MATHEMATICS, APPLIED | GLOBAL STABILITY | POSITIVE SOLUTIONS | PERSISTENCE | ENVIRONMENT | DIFFERENTIAL-EQUATIONS | MODEL | Generalized Ito's formula | RANDOM PERTURBATION | UNIQUENESS | LOGISTIC EQUATION | Markov processes | Analysis

Journal Article

Stochastic Processes and their Applications, ISSN 0304-4149, 2008, Volume 118, Issue 8, pp. 1385 - 1406

The main aim of this paper is to discuss the almost surely asymptotic stability of the neutral stochastic differential delay equations (NSDDEs) with Markovian...

Brownian motion | Markov chain | Asymptotic stability | Generalized Itô formula | Exponential stability | generalized Ito formula | STABILIZATION | SYSTEMS | STATE | STATISTICS & PROBABILITY | NOISE | asymptotic stability | exponential stability | STABILIZABILITY | Asymptotic stability Exponential stability Generalized Ito formula Brownian motion Markov chain

Brownian motion | Markov chain | Asymptotic stability | Generalized Itô formula | Exponential stability | generalized Ito formula | STABILIZATION | SYSTEMS | STATE | STATISTICS & PROBABILITY | NOISE | asymptotic stability | exponential stability | STABILIZABILITY | Asymptotic stability Exponential stability Generalized Ito formula Brownian motion Markov chain

Journal Article

Siberian Advances in Mathematics, ISSN 1055-1344, 7/2015, Volume 25, Issue 3, pp. 191 - 205

We deduce an analog of the Itô–Venttsel’ formula for an Itô system of generalized stochastic differential equations (GSDE) with noncentered measure on the...

generalized stochastic differential equation | stochastic first integral | Mathematics, general | Mathematics | kernel of an integral invariant | Itô–Venttsel formula | noncentered Poisson measure

generalized stochastic differential equation | stochastic first integral | Mathematics, general | Mathematics | kernel of an integral invariant | Itô–Venttsel formula | noncentered Poisson measure

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 05/2019, Volume 348, pp. 338 - 354

This paper investigates the stability problem of complex-valued impulsive stochastic functional differential equations on networks with Markovian switching...

Impulsive effects | Complex-valued coupled systems | Razumikhin technique | Markovian switching | Complex generalized Itô’s formula | GLOBAL EXPONENTIAL STABILITY | MATHEMATICS, APPLIED | STABILIZATION | TIME-VARYING DELAYS | RAZUMIKHIN METHOD | MODEL | SYNCHRONIZATION | Complex generalized Ito's formula | NEURAL-NETWORKS | SYSTEMS | Markov processes | Numerical analysis | Analysis | Differential equations

Impulsive effects | Complex-valued coupled systems | Razumikhin technique | Markovian switching | Complex generalized Itô’s formula | GLOBAL EXPONENTIAL STABILITY | MATHEMATICS, APPLIED | STABILIZATION | TIME-VARYING DELAYS | RAZUMIKHIN METHOD | MODEL | SYNCHRONIZATION | Complex generalized Ito's formula | NEURAL-NETWORKS | SYSTEMS | Markov processes | Numerical analysis | Analysis | Differential equations

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 | 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 | BACKWARD | Functional Ito calculus | VALUED PROCESSES | REPRESENTATION | STATISTICS & PROBABILITY | FORMULA | PHYSICS, MATHEMATICAL | GENERALIZED COVARIATION

Journal Article

IEEE Transactions on Automatic Control, ISSN 0018-9286, 09/2013, Volume 58, Issue 9, pp. 2319 - 2332

One of the important issues in the study of hybrid stochastic differential delay equations (SDDEs) is the automatic control, with consequent emphasis being...

Robust stability | Stability analysis | Equations | exponential stability | generalized Itô's formula | Asymptotic boundedness | Brownian motion | Asymptotic stability | stochastic differential delay equations | Robustness | Delays | Mathematical model | Markovian switching | generalized Ito's formula | RADII | JUMP LINEAR-SYSTEMS | AUTOMATION & CONTROL SYSTEMS | ENGINEERING, ELECTRICAL & ELECTRONIC | Robust statistics | Linear systems | Usage | Stability | Analysis | Differential equations | Innovations

Robust stability | Stability analysis | Equations | exponential stability | generalized Itô's formula | Asymptotic boundedness | Brownian motion | Asymptotic stability | stochastic differential delay equations | Robustness | Delays | Mathematical model | Markovian switching | generalized Ito's formula | RADII | JUMP LINEAR-SYSTEMS | AUTOMATION & CONTROL SYSTEMS | ENGINEERING, ELECTRICAL & ELECTRONIC | Robust statistics | Linear systems | Usage | Stability | Analysis | Differential equations | Innovations

Journal Article

Stochastic Processes and their Applications, ISSN 0304-4149, 2003, Volume 103, Issue 2, pp. 277 - 291

Stability of stochastic differential equations with Markovian switching has recently been discussed by many authors, for example, Basak et al. (J. Math. Anal....

Generalized Itô's formula | Brownian motion | Markov chain | Asymptotic stability in distribution | STATISTICS & PROBABILITY | generalized Ito's formula | asymptotic stability in distribution | Generalized Ito's formula Brownian motion Markov chain Asymptotic stability in distribution

Generalized Itô's formula | Brownian motion | Markov chain | Asymptotic stability in distribution | STATISTICS & PROBABILITY | generalized Ito's formula | asymptotic stability in distribution | Generalized Ito's formula Brownian motion Markov chain Asymptotic stability in distribution

Journal Article

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