arXiv subjects

Rúben Sousa

Publications and source records attributed to Rúben Sousa.

9 recordsLinked to original sources

arXiv2020

On the construction of convolution-like operators associated with multidimensional diffusion processes

When is it possible to interpret a given Markov process as a Lévy-like process? Since the class of Lévy processes can be defined by the relation between transition probabilities and convolutions, the answer to this question lies in the existence of a convolution-like operator satisfying the same relation with the transition probabilities of the process. It is known that the so-called Sturm-Liouville convolutions have the desired properties and therefore the question above has a positive answer for a certain class of one-dimensional diffusions. However, more general processes have never been systematically treated in the literature. This study addresses this gap by considering the general problem of constructing a convolution-like operator for a given strong Feller process on a general locally compact metric space. Both necessary and sufficient conditions for the existence of such convolution-like structures are determined, which reveal a connection between the answer to the above question and certain analytical and geometrical properties of the eigenfunctions of the transition semigroup. The case of reflected Brownian motions on bounded domains of R d and compact Riemannian manifolds is considered in greater detail: various special cases are analysed, and a general discussion on the existence of appropriate convolution-like structures is presented.

math.PR↗

arXiv2020

Product formulas and convolutions for two-dimensional Laplace-Beltrami operators: beyond the trivial case

We introduce the notion of a family of convolution operators associated with a given elliptic partial differential operator. Such a convolution structure is shown to exist for a general class of Laplace-Beltrami operators on two-dimensional manifolds endowed with cone-like metrics. This structure gives rise to a convolution semigroup representation for the Markovian semigroup generated by the Laplace-Beltrami operator. In the particular case of the operator $\mathcal{L} = \partial_x^2 + {1 \over 2x} \partial_x + {1 \over x} \partial_θ^2$ on $\mathbb{R}^+ \times \mathbb{T}$, we deduce the existence of a convolution structure for a two-dimensional integral transform whose kernel and inversion formula can be written in closed form in terms of confluent hypergeometric functions. The results of this paper can be interpreted as a natural extension of the theory of one-dimensional generalized convolutions to the framework of multiparameter eigenvalue problems.

math.AP↗

arXiv2019

On the product formula and convolution associated with the index Whittaker transform

We deduce a product formula for the Whittaker $W$ function whose kernel does not depend on the second parameter. Making use of this formula, we define the positivity-preserving convolution operator associated with the index Whittaker transform, which is seen to be a direct generalization of the Kontorovich-Lebedev convolution. The mapping properties of this convolution operator are investigated; in particular, a Banach algebra property is established and then applied to yield an analogue of the Wiener-Lévy theorem for the index Whittaker transform. We show how our results can be used to prove the existence of a unique solution for a class of convolution-type integral equations.

math.CA↗

arXiv2019

Sturm-Liouville hypergroups without the compactness axiom

We establish a positive product formula for the solutions of the Sturm-Liouville equation $\ell(u) = λu$, where $\ell$ belongs to a general class which includes singular and degenerate Sturm-Liouville operators. Our technique relies on a positivity theorem for possibly degenerate hyperbolic Cauchy problems and on a regularization method which makes use of the properties of the diffusion semigroup generated by the Sturm-Liouville operator. We show that the product formula gives rise to a convolution algebra structure on the space of finite measures, and we discuss whether this structure satisfies the basic axioms of the theory of hypergroups. We introduce the notion of a degenerate hypergroup of full support and improve the known existence theorems for Sturm-Liouville hypergroups. Convolution-type integral equations on weighted Lebesgue spaces are also introduced, and a solvability condition is established.

math.CA↗

arXiv2019

The hyperbolic maximum principle approach to the construction of generalized convolutions

We introduce a unified framework for the construction of convolutions and product formulas associated with a general class of regular and singular Sturm-Liouville boundary value problems. Our approach is based on the application of the Sturm-Liouville spectral theory to the study of the associated hyperbolic equation. As a by-product, an existence and uniqueness theorem for degenerate hyperbolic Cauchy problems with initial data at a parabolic line is established. The mapping properties of convolution operators generated by Sturm-Liouville operators are studied. Analogues of various notions and facts from probabilistic harmonic analysis are developed on the convolution measure algebra. Various examples are presented which show that many known convolution-type operators --- including those associated with the Hankel, Jacobi and index Whittaker integral transforms --- can be constructed using this general approach.

math.CA↗

arXiv2018

Lévy processes with respect to the index Whittaker convolution

The index Whittaker convolution operator, recently introduced by the authors, gives rise to a convolution measure algebra having the property that the convolution of probability measures is a probability measure. In this paper, we introduce the class of Lévy processes with respect to the index Whittaker convolution and study their basic properties. We prove that the square root of the Shiryaev process belongs to our family of Lévy process, and this is shown to yield a martingale characterization of the Shiryaev process analogous to Lévy's characterization of Brownian motion. Our results demonstrate that a nice theory of Lévy processes with respect to generalized convolutions can be developed even if the usual compactness assumption on the support of the convolution fails, shedding light into the connection between the properties of the convolution algebra and the nature of the singularities of the associated differential operator.

math.PR↗

arXiv2017

Barrier Option Pricing under the 2-Hypergeometric Stochastic Volatility Model

We investigate the pricing of financial options under the 2-hypergeometric stochastic volatility model. This is an analytically tractable model that reproduces the volatility smile and skew effects observed in empirical market data. Using a regular perturbation method from asymptotic analysis of partial differential equations, we derive an explicit and easily computable approximate formula for the pricing of barrier options under the 2-hypergeometric stochastic volatility model. The asymptotic convergence of the method is proved under appropriate regularity conditions, and a multi-stage method for improving the quality of the approximation is discussed. Numerical examples are also provided.

math.PR↗

arXiv2017

The spectral expansion approach to index transforms and connections with the theory of diffusion processes

Many important index transforms can be constructed via the spectral theory of Sturm-Liouville differential operators. Using the spectral expansion method, we investigate the general connection between the index transforms and the associated parabolic partial differential equations. We show that the notion of Yor integral, recently introduced by the second author, can be extended to the class of Sturm-Liouville integral transforms. We furthermore show that, by means of the Feynman-Kac theorem, index transforms can be used for studying Markovian diffusion processes. This gives rise to new applications of index transforms to problems in mathematical finance.

math.CA↗

arXiv2014

Pretty good state transfer of entangled states through quantum spin chains

The XX model with uniform couplings represents the most natural choice for quantum state transfer through spin chains. Given that it has long been established that single-qubit states cannot be transferred with perfect fidelity in this model, the notion of pretty good state transfer has been recently introduced as a relaxation of the constraints on fidelity. In this paper, we study the transfer of multi-qubit entangled and unentangled states through unmodulated spin chains, and we prove that it is possible to have pretty good state transfer of any multi-particle state. This significantly generalizes the previous results on single-qubit state transfer, and opens way to using uniformly coupled spin chains as quantum channels for the transfer of arbitrary states of any dimension. Our results could be tested with current technology.

quant-ph↗