Invariance for stochastic equations with regular coefficients

Milian, Anna

doi:10.1080/07362999708809465

Cited by 13 publications

(9 citation statements)

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“…ii)⇒i): Let (t 0 , x 0 ) ∈ R + × M and let φ : V → U ∩ M be a parametrization in x 0 satisfying (12). As in the proof of Theorem 2 (setting τ 0 = τ 1 = 0) we get a unique continuous strong solution Y to …”

Section: Proof Of Theorem 3 I)⇒ii): Fix (Tmentioning

confidence: 94%

“…Further recent results can be found in [2], [12] for finite dimensional systems, and in [11], [16] for infinite dimensional ones. See also the references therein.…”

Section: Introductionmentioning

confidence: 92%

“…Now assume X to be a local weak solution with lifetime τ . Let φ : V → U ∩ M be a parametrization in X 0 satisfying (12). By (A1) there exists a stopping time…”

Section: ξ X T (ω) Is a Linear Functional On D(a ) Which Is Boundedmentioning

confidence: 99%

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Invariant manifolds for weak solutions to stochastic equations

Filipović¹

2000

Probab Theory Relat Fields

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Abstract. Viability and invariance problems related to a stochastic equation in a Hilbert space H are studied. Finite dimensional invariant C 2 submanifolds of H are characterized. We derive Nagumo type conditions and prove a regularity result: Any weak solution, which is viable in a finite dimensional C 2 submanifold, is a strong solution.These results are related to finding finite dimensional realizations for stochastic equations. There has recently been increased interest in connection with a model for the stochastic evolution of forward rate curves.

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Section: Proof Of Theorem 3 I)⇒ii): Fix (Tmentioning

confidence: 94%

“…Further recent results can be found in [2], [12] for finite dimensional systems, and in [11], [16] for infinite dimensional ones. See also the references therein.…”

Section: Introductionmentioning

confidence: 92%

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Invariant manifolds for weak solutions to stochastic equations

Filipović¹

2000

Probab Theory Relat Fields

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“…For stochastic control systems and differential inclusions different authors used stochastic contingent sets [4,5], viscosity solutions of second-order partial differential equations [8][9][10] and derivatives of the distance function [12], see also [15][16][17][19][20][21] for several other approaches.…”

Section: Introductionmentioning

confidence: 99%

Invariance of stochastic control systems with deterministic arguments

Prato

Frankowska

2004

Journal of Differential Equations

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We prove that a closed set K of a finite-dimensional space is invariant under the stochastic control system dX ¼ bðX ; vðtÞÞ dt þ sðX ; vðtÞÞ dW ðtÞ; vðtÞAU;if and only if it is invariant under the deterministic control system with two controlsDs j ðx; vðtÞÞs j ðx; vðtÞÞ þ sðx; vðtÞÞuðtÞ; uðtÞAH 1 ; vðtÞAU:This extends the well-known result of stochastic differential equations to stochastic control systems. Furthermore, we ask only C 1;1 regularity of the diffusion s instead of the usual assumption sAC 2 :In this way our result is new even for stochastic differential equations. The arguments of the proof are based on estimates between solutions of the stochastic control system with time independent controls and families of solutions fx o ðÁÞg oAO to the deterministic control system x 0 ¼ sðx; v o Þu o ðtÞ; u o ðtÞAH 1 with appropriately chosen controls u o ðtÞ and v o AU: r

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“…The literature on stochastic invariance is rather extensive, both for finite dimensional systems, see [14], [21], and [4] and references therein, and for infinite dimensional ones, see e.g. [34], [17] and [41].…”

Section: 2mentioning

confidence: 99%

Wong-Zakai approximations of stochastic evolution equations

Tessitore

Zabczyk

2006

J. evol. equ.

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Abstract. Theorems on weak convergence of the laws of the Wong-Zakai approximations for evolution equationare proved. The operator A in the equation generates an analytic semigroup of linear operators on a Hilbert space H. The tightness of the approximating sequence is established using the stochastic factorization formula. Applications to strongly damped wave and plate equations as well as to stochastic invariance are discussed.

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Invariance for stochastic equations with regular coefficients

Abstract: We give necessary and sufficient conditions for the invariance property of closed subsets of IRm by It6 and Stratonovich equations. The conditions are expressed in terms of properly defined contingent cones and in a form suitable for applications.

Cited by 13 publications

References 1 publication

Invariant manifolds for weak solutions to stochastic equations

Invariant manifolds for weak solutions to stochastic equations

Invariance of stochastic control systems with deterministic arguments

Wong-Zakai approximations of stochastic evolution equations

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