Stochastic integration and stochastic PDEs driven by jumps on the dual of a nuclear space

Fonseca-Mora, C. A.

doi:10.1007/s40072-018-0117-x

Cited by 18 publications

(51 citation statements)

References 33 publications

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“…We hope that with our work more applications will appear. Second, in [15] a new theory of stochastic integration and stochastic PDE's in Φ ′ β driven by Lévy noise has been introduced. Much of the work on this article is motivated to show convergence of solutions of these stochastic PDE's.…”

Section: Introductionmentioning

confidence: 99%

Tightness and weak convergence of probabilities on the Skorokhod space on the dual of a nuclear space and applications

Fonseca-Mora

2020

Studia Math.

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Let Φ ′ β denotes the strong dual of a nuclear space Φ and let D T (Φ ′ β ) be the Skorokhod space of right-continuous with left limits (càdlàg) functions from [0, T ] into Φ ′ β . In this article we introduce the concepts of cylindrical random variables and cylindrical measures on D T (Φ ′ β ), and prove analogues of the regularization theorem and Minlos theorem for extensions of these objects to bona fide random variables and probability measures on D T (Φ ′ β ) respectively. Later, we establish analogues of Lévy's continuity theorem to provide necessary and sufficient conditions for uniform tightness of families of probability measures on D T (Φ ′ β ) and sufficient conditions for weak convergence of a sequence of probability measures on D T (Φ ′ β ). Extensions of the above results to the space D ∞ (Φ ′ β ) of càdlàg functions from [0, ∞) into Φ ′ β are also given. Afterwards, we apply our results to study weak convergence of Φ ′ β -valued càdlàg processes and in particular to Lévy processes. We finalize with an application of our theory to the study of tightness and weak convergence of probability measures on the Skorokhod space D ∞ (H) where H is a Hilbert space.

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Section: Introductionmentioning

confidence: 99%

Tightness and weak convergence of probabilities on the Skorokhod space on the dual of a nuclear space and applications

Fonseca-Mora

2020

Studia Math.

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“…Remark 4.2. Since the Hilbert space H is separable, it follows from Definition 4.1(2), and from Proposition 3.8 in Fonseca-Mora (2018), that for each g ∈ G, the map (r, ω, u) → q r,u (Φ(r, ω, u) * g) 2 is P T ⊗ B(U )/B(R + )-measurable. Therefore, because G is a separable Hilbert space, the map (r, ω, u) → ||Φ(r, ω, u)|| 2 L 2 (Hq r,u ,G) is P T ⊗ B(U )/B(R + )-measurable and the integrand in (4.1) is well-defined.…”

Section: The Stochastic Integralmentioning

confidence: 99%

“…The concept of cylindrical martingale-valued measures in a locally convex space was introduced in the work Fonseca-Mora (2018), and generalizes to locally convex spaces the martingale-valued measures introduced for the finite dimensional setting by Walsh (1986) and then extended to infinite dimensional settings such as Hilbert spaces (Applebaum, 2006) and duals of nuclear Fréchet spaces (Xie, 2001). However, in this work we will consider such objects only in the Hilbert space setting.…”

Section: Introductionmentioning

confidence: 99%

Stochastic integration in Hilbert spaces with respect to cylindrical martingale-valued measures

Alvarado-Solano¹,

Fonseca-Mora²

2021

ALEA

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In this work we introduce a theory of stochastic integration for operator-valued integrands with respect to some classes of cylindrical martingale-valued measures in Hilbert spaces. The integral is constructed via the radonification of cylindrical martingales by a Hilbert-Schmidt operator theorem and unifies several other theories of stochastic integration in Hilbert spaces. In particular, our theory covers the theory of stochastic integration with respect to a Hilbert space valued Lévy process with second moments, with respect to a cylindrical Lévy processes with (weak) second moments and with respect to a Lévy-valued random martingale measures with finite second moment. As an application of our theory of integration we prove existence and uniqueness of solutions for stochastic stochastic partial differential equations driven by multiplicative cylindrical martingale-valued measure noise with rather general coefficients. Existence and uniqueness of solutions in the presence of multiplicative Lévy noise (with no moments assumptions) is also proved.

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“…[5,6,23,53]). However, in recent years there has been an increasing interest in the usage of other classes of cylindrical semimartingales as such driving noise; we can cite, for example, the cylindrical Lévy processes [21,26,28,38,43], cylindrical martingalevalued measures [13] and cylindrical continuous local martingales [33,34,54]. To the extent of our knowledge, none of these works considers stochastic integration with respect to general cylindrical semimartingales in a general locally convex space.…”

Section: Introductionmentioning

confidence: 99%

Stochastic integration with respect to cylindrical semimartingales

Fonseca-Mora

2021

Electron. J. Probab.

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In this work we introduce a theory of stochastic integration with respect to general cylindrical semimartingales defined on a locally convex space Φ. Our construction of the stochastic integral is based on the theory of tensor products of topological vector spaces and the property of good integrators of real-valued semimartingales. This theory is further developed in the case where Φ is a complete, barrelled, nuclear space, where we obtain a complete description of the class of integrands as Φ-valued locally bounded and weakly predictable processes. Several other properties of the stochastic integral are proven, including a Riemann representation, a stochastic integration by parts formula and a stochastic Fubini theorem. Our theory is then applied to provide sufficient and necessary conditions for existence and uniqueness of solutions to linear stochastic evolution equations driven by semimartingale noise taking values in the strong dual Φ of Φ. In the last part of this article we apply our theory to define stochastic integrals with respect to a sequence of real-valued semimartingales.

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Stochastic integration and stochastic PDEs driven by jumps on the dual of a nuclear space

Cited by 18 publications

References 33 publications

Tightness and weak convergence of probabilities on the Skorokhod space on the dual of a nuclear space and applications

Tightness and weak convergence of probabilities on the Skorokhod space on the dual of a nuclear space and applications

Stochastic integration in Hilbert spaces with respect to cylindrical martingale-valued measures

Stochastic integration with respect to cylindrical semimartingales

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