arXiv subjects

C. A. Fonseca-Mora

Publications and source records attributed to C. A. Fonseca-Mora.

At least 19 recordsLinked to original sources

arXiv2026

Itô's formula for Lévy-Itô processes taking values in the dual of nuclear space

Using the theory of stochastic integration in duals of nuclear spaces with respect to cylindrical martingale-valued measures, a vector-valued Itô formula is proved for generalized Itô processes defined with respect to these integrals. The abstract result is then applied to prove an Itô formula for Lévy-Itô processes defined with respect to Lévy processes taking values in the dual of a reflexive nuclear space.

math.PR↗

arXiv2026

The martingale representation theorem for cylindrical martingale valued measures

We prove a martingale representation theorem for cylindrical martingale-valued measures defined on a separable Banach space. The main tool for establishing the theorem, is a new theory of non-radonifying stochastic integration in reflexive Banach spaces. A second one is the study and characterization of cylindrical white noise measure processes. As consequences of our representation theorem, we prove analogous versions for Hilbert space-valued measures and for cylindrical square integrable martingales. Finally, we apply the results to characterize the solutions to the weak martingale problem for SDEs driven by cylindrical white noise measures.

math.PR↗

arXiv2026

Itô's Formula for Itô processes defined with respect to a cylindrical-martingale valued measure

Using the authors' recently developed stochastic integration [Stoch PDE: Anal Comp, 2024], we prove an Itô formula for Hilbert space-valued Itô processes defined with respect to a cylindrical martingale-valued measure. We develop some tools from stochastic analysis, as are the predictable and optional quadratic variation of a stochastic integral, the continuous and purely discontinuous parts of an integral process, and a Riemann representation formula. As an application of our Itô formula, we prove a Burkholder inequality for the stochastic integral defined with respect to a cylindrical martingale-valued measure. Finally, we derive Itô formulas for Hilbert space-valued martingale-valued measures and for cylindrical square integrable martingales.

math.PR↗

arXiv2026

Markov property and path regularity for the solutions to SPDEs driven by cylindrical-martingale valued measures

In this paper we prove the Markov property for the solution to stochastic partial differential equations driven by a cylindrical orthogonal martingale-valued measure. We assume our coefficients are time-dependent and satisfy some growth and Lipschitz conditions. We also prove that for time-independent coefficients and under mild assumptions on the cylindrical orthogonal martingale-valued measure, the solutions to our stochastic partial differential equations are Feller. Finally, in the case that the $C_{0}$-semigroup is quasi-contraction, we show that the solution to our stochastic partial differential equation possesses a càdlàg version.

math.PR↗

arXiv2025

Vector-Valued Stochastic Integration With Respect to Semimartingales in the Dual of Nuclear Space

In this work, we investigate a theory of stochastic integration for operator-valued processes with respect to semimartingales taking values in the dual of a nuclear space. Our construction of this particular stochastic integral relies on previous results from [Electron. J. Probab., Volume 26, paper no. 147, 2021], together with specific tools which share some common features with good integrators in finite dimensions. We investigate various properties of this stochastic integral together with applications. In particular we obtain approximations by Riemann sums results, and provide an alternative proof of Üstünel's version of Itô's formula involving of distributions.

math.PR↗

arXiv2024

Riesz spaces of signed charges on semi-rings

A constructive definition of the supremum of a family of set functions is exploited in the context of Riesz spaces of signed measures and finitely additive functions (signed charges) on semi-rings. We explore applications, particularly to establish a Jordan decomposition for signed charges on semi-rings, whether the structure of Riesz space is present or not.

math.FA↗

arXiv2024

Cylindrical Martingale-Valued Measures, Stochastic Integration and SPDEs

We develop a theory of Hilbert-space valued stochastic integration with respect to cylindrical martingale-valued measures. As part of our construction, we expand the concept of quadratic variation, introduced by Veraar and Yaroslavtsev (2016), to the case of cylindrical martingale-valued measures that are allowed to have discontinuous paths (this is carried out within the context of separable Banach spaces). Our theory of stochastic integration is applied to address the existence and uniqueness of solutions to stochastic partial differential equations in Hilbert spaces.

math.PR↗

arXiv2024

Tightness and weak convergence in the topology of local uniform convergence for stochastic processes in the dual of a nuclear space

Let $Φ'$ denote the strong dual of a nuclear space $Φ$ and let $C_{\infty}(Φ')$ be the collection of all continuous mappings $x:[0,\infty) \rightarrow Φ'$ equipped with the topology of local uniform convergence. In this paper we prove sufficient conditions for tightness of probability measures on $C_{\infty}(Φ')$ and for weak convergence in $C_{\infty}(Φ')$ for a sequence of $Φ'$-valued processes. We illustrate our results with two applications. First, we show the central limit theorem for local martingales taking values in the dual of an ultrabornological nuclear space. Second, we prove sufficient conditions for the weak convergence in $C_{\infty}(Φ')$ for a sequence of solutions to stochastic partial differential equations driven by semimartingale noise.

math.PR↗

arXiv2023

Convergence Uniform on Compacts in Probability with Applications to Stochastic Analysis in Duals of Nuclear Spaces

Let $Φ'$ denote the strong dual of a nuclear space $Φ$. In this paper we introduce sufficient conditions for the convergence uniform on compacts in probability for a sequence of $Φ'$-valued processes with continuous or càdlàg paths. We illustrate the usefulness of our results by considering two applications to stochastic analysis. First, we introduce a topology on the space of $Φ'$-valued semimartingales which are good integrators and show that this topology is complete and that the stochastic integral mapping is continuous on the integrators. Second, we introduce sufficient conditions for the convergence uniform on compacts in probability of the solutions to a sequence of linear stochastic evolution equations driven by semimartingale noise.

math.PR↗

arXiv2022

Time regularity of stochastic convolutions and stochastic evolution equations in duals of nuclear spaces

Let $Φ$ a locally convex space and $Ψ$ be a quasi-complete, bornological, nuclear space (like spaces of smooth functions and distributions) with dual spaces $Φ'$ and $Ψ'$. In this work we introduce sufficient conditions for time regularity properties of the $Ψ'$-valued stochastic convolution $\int_{0}^{t} \int_{U} S(t-r)'R(r,u) M(dr,du)$, $t \in [0,T]$, where $(S(t): t \geq 0)$ is a $C_{0}$-semigroup on $Ψ$, $R(r,ω,u)$ is a suitable operator form $Φ'$ into $Ψ'$ and $M$ is a cylindrical-martingale valued measure on $Φ'$. Our result is latter applied to study time regularity of solutions to $Ψ'$-valued stochastic evolutions equations.

math.PR↗

arXiv2022

Almost Sure Uniform Convergence of Stochastic Processes in the Dual of a Nuclear Space

Let $Φ$ be a nuclear space and let $Φ'$ denote its strong dual. In this paper we introduce sufficient conditions for the almost surely uniform convergence on bounded intervals of time for a sequence of $Φ'$-valued processes having continuous (respectively càdlàg) paths. The main result is formulated first in the general setting of cylindrical processes but later specialized to other situations of interest. In particular, we establish conditions for the convergence to occur in a Hilbert space continuously embedded in $Φ'$. Furthermore, in the context of the dual of an ultrabornological nuclear space (like spaces of smooth functions and distributions) we also include applications to the convergence of a series of independent càdlàg process and to the convergence of solutions to linear evolution equations driven by Lévy noise.

math.PR↗

arXiv2021

Stochastic integration with respect to cylindrical semimartingales

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.

math.PR↗

arXiv2021

Stochastic Evolution Equations with Lévy Noise in the Dual of a Nuclear Space

In this article we give sufficient and necessary conditions for the existence of a weak and mild solution to stochastic evolution equations with (general) Lévy noise taking values in the dual of a nuclear space. As part of our approach we develop a theory of stochastic integration with respect to a Lévy process taking values in the dual of a nuclear space. We also derive further properties of the solution such as the existence of a solution with square moments, the Markov property and path regularity of the solution. In the final part of the paper we give sufficient conditions for the weak convergence of the solutions to a sequence of stochastic evolution equations with Lévy noises.

math.PR↗

arXiv2020

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

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 (which is not required to satisfy any moment condition), 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.

math.PR↗

arXiv2020

Semimartingales on Duals of Nuclear Spaces

This work is devoted to the study of semimartingales on the dual of a general nuclear space. We start by establishing conditions for a cylindrical semimartingale in the strong dual $Φ'$ of a nuclear space $Φ$ to have a $Φ'$-valued semimartingale version whose paths are right-continuous with left limits. Results of similar nature but for more specific classes of cylindrical semimartingales and examples are also provided. Later, we will show that under some general conditions every semimartingale taking values in the dual of a nuclear space has a canonical representation. The concept of predictable characteristics is introduced and is used to establish necessary and sufficient conditions for a $Φ'$-valued semimartingale to be a $Φ'$-valued Lévy process.

math.PR↗

arXiv2019

Lévy Processes and Infinitely Divisible Measures in the Dual of a Nuclear Space

Let $Φ$ be a nuclear space and let $Φ'_β$ denote its strong dual. In this work we establish the one-to-one correspondence between infinitely divisible measures on $Φ'_β$ and Lévy processes taking values in $Φ'_β$. Moreover, we prove the Lévy-Itô decomposition, the Lévy-Khintchine formula and the existence of càdlàg versions for $Φ'_β$-valued Lévy processes. A characterization for Lévy measures on $Φ'_β$ is also established. Finally, we prove the Lévy-Khintchine formula for infinitely divisible measures on $Φ'_β$.

math.PR↗

arXiv2019

Tightness and Weak Convergence of Probabilities on the Skorokhod Space on the Dual of a Nuclear Space and Applications

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 $D_{\infty}(Φ'_β)$ of càdlàg functions from $[0,\infty)$ 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 $D_{\infty}(H)$ where $H$ is a Hilbert space.

math.PR↗

arXiv2019

Regularization of Cylindrical Processes In Locally Convex Spaces

Let $Φ$ be a locally convex space and let $Φ'$ denote its strong dual. In this paper we introduce sufficient conditions for the existence of a continuous or a càdlàg $Φ'$-valued version to a cylindrical process defined on $Φ$. Our result generalizes many other known results on the literature and their different connections will be discussed. As an application, we use our results to show the existence of a $Φ'$-valued càdlàg Lévy process version to a given cylindrical Lévy process in $Φ'$.

math.PR↗