Existence of strong solutions for Itô's stochastic equations via approximations

Gyöngy, István; Крылов, Н. В.

doi:10.1007/bf01203833

Cited by 374 publications

(319 citation statements)

References 12 publications

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“…Another point of view is to relax the continuity assumptions of the coefficients σ and b, and consider the convergence of the Euler scheme. I. Gyöngy and N. Krylov [10], Gyöngy [11] and D. J. Higham et al [12] have presented results in this direction.…”

Section: 2mentioning

confidence: 78%

On irregular functionals of SDEs and the Euler scheme

Avikainen

2009

Finance Stoch

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Abstract. We prove a sharp upper bound for the approximation error |g(X) − g(X)| p in terms of moments of X −X, where X andX are random variables and the function g is a function of bounded variation. We apply the results to the approximation of a solution of a stochastic differential equation at time T by the Euler scheme, and show that the approximation of the payoff of the binary option has asymptotically sharp strong convergence rate 1/2. This has consequences for multilevel Monte Carlo methods.

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Section: 2mentioning

confidence: 78%

On irregular functionals of SDEs and the Euler scheme

Avikainen

2009

Finance Stoch

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“…Besides, due to classical Yamada-Watanabetype argument (see e.g. [14], [26]), existence of a pathwise solution follows from existence of a martingale solution together with pathwise uniqueness. The difference lies also in the way how the initial condition is posed: for pathwise solutions we are given a random variable u 0 whereas for martingale solutions we can only prescribe an initial law Λ.…”

Section: Remark 25 In Def 21 (J) the Continuity Equation Is Statementioning

confidence: 99%

“…In the twodimensional case we gain a stronger convergence result (see Theorem 2.11). This is based on the uniqueness for the system (1.2) and a new version of the Gyöngy-Krylov characterization of convergence in probability [14] which applies to the setting of quasi-Polish spaces (see Proposition A.4).…”

Section: Introductionmentioning

confidence: 99%

Incompressible Limit for Compressible Fluids with Stochastic Forcing

Breit¹,

Feireisl

Hofmanová

2016

Arch Rational Mech Anal

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Abstract. We study the asymptotic behavior of the isentropic Navier-Stokes system driven by a multiplicative stochastic forcing in the compressible regime, where the Mach number approaches zero. Our approach is based on the recently developed concept of weak martingale solution to the primitive system, uniform bounds derived from a stochastic analogue of the modulated energy inequality, and careful analysis of acoustic waves. A stochastic incompressible Navier-Stokes system is identified as the limit problem.

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“…The following criteria for convergence in probability whose proof can be found in [23] need to be highlighted.…”

Section: Homogenization Of a Stochastic Ladyzhenskaya Model For Incommentioning

confidence: 99%

Homogenization in algebras with mean value

Woukeng¹

2015

Banach J. Math. Anal.

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Abstract. In several works, the theory of strongly continuous groups is used to build a framework for solving stochastic homogenization problems. Following this idea, we construct a detailed and comprehensive theory of homogenization. This enables to solve homogenization problems in algebras with mean value, regardless of whether they are ergodic or not, thereby responding affirmatively to the question raised by Zhikov and Krivenko [V.V. Zhikov, E.V. Krivenko, Homogenization of singularly perturbed elliptic operators. Matem. Zametki, 33 (1983) 571-582 (english transl.: Math. Notes, 33 (1983) 294-300)] to know whether it is possible to homogenize problems in nonergodic algebras. We also state and prove a compactness result for Young measures in these algebras. As an important achievement we study the homogenization problem associated with a stochastic Ladyzhenskaya model for incompressible viscous flow, and we present and solve a few examples of homogenization problems related to nonergodic algebras.

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Existence of strong solutions for Itô's stochastic equations via approximations

Abstract: Given strong uniqueness for an It6's stochastic equation with discontinuous coefficients, we prove that its solution can be constructed on "any" probability space by using, for example, Euler's polygonal approximations. Stochastic equations in IR d and in domains in IR d are considered.

Cited by 374 publications

References 12 publications

On irregular functionals of SDEs and the Euler scheme

On irregular functionals of SDEs and the Euler scheme

Incompressible Limit for Compressible Fluids with Stochastic Forcing

Homogenization in algebras with mean value

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