arXiv subjects

Manuel Guerra

Publications and source records attributed to Manuel Guerra.

16 recordsLinked to original sources

arXiv2021

Minimizing ruin probability under dependencies for insurance pricing

In this work the ruin probability of the Lundberg risk process is used as a criterion for determining the optimal security loading of premia in the presence of price-sensitive demand for insurance. Both single and aggregated claim processes are considered and the independent and the dependent cases are analyzed. For the single-risk case, we show that the optimal loading does not depend on the initial reserve. In the multiple risk case we account for arbitrary dependency structures between different risks and for dependencies between the probabilities of a client acquiring policies for different risks. In this case, the optimal loadings depend on the initial reserve. In all cases the loadings minimizing the ruin probability do not coincide with the loadings maximizing the expected profit.

q-fin.RM↗

arXiv2021

Reinsurance of multiple risks with generic dependence structures

We consider the optimal reinsurance problem from the point of view of a direct insurer owning several dependent risks, assuming a maximal expected utility criterion and independent negotiation of reinsurance for each risk. Without any particular hypothesis on the dependency structure, we show that optimal treaties exist in a class of independent randomized contracts. We derive optimality conditions and show that under mild assumptions the optimal contracts are of classical (non-randomized) type. A specific for mof the optimality conditions applies in that case. We present a numerical scheme to solve the optimality conditions.

math.PR↗

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

Option pricing in exponential Lévy models with transaction costs

We present an approach for pricing European call options in presence of proportional transaction costs, when the stock price follows a general exponential Lévy process. The model is a generalization of the celebrated work of Davis, Panas and Zariphopoulou (1993), where the value of the option is defined as the utility indifference price. This approach requires the solution of two stochastic singular control problems in finite horizon, satisfying the same Hamilton-Jacobi-Bellman equation, with different terminal conditions. We introduce a general formulation for these portfolio selection problems, and then we focus on the special case in which the probability of default is ignored. We solve numerically the optimization problems using the Markov chain approximation method and show results for diffusion, Merton and Variance Gamma processes. Option prices are computed for both the writer and the buyer.

q-fin.MF↗

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

On the stochastic Lie algebra

We study the structure of the Lie algebra $\mathfrak{s}(n,\mathbb R)$ corresponding to the so-called stochastic Lie group $\mathcal{S} (n,\mathbb R)$. We obtain the Levi decomposition of the Lie algebra, classify Levi factor and classify the representation of the factor in $\mathbb{R}^n$. We discuss isomorphism of $\mathcal{S}(n,\mathbb R)$ with the group of invertible affine maps ${\it Aff}(n-1,\mathbb R)$. We prove that $\mathfrak s(n, \mathbb R)$ is generated by two generic elements.

math.RA↗

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

Optimal stopping of one-dimensional diffusions with integral criteria

This paper provides a full characterization of the value function and solution(s) of an optimal stopping problem for a one-dimensional diffusion with an integral criterion. The results hold under very weak assumptions, namely, the diffusion is assumed to be a weak solution of stochastic differential equation satisfying the Engelbert-Schmidt conditions, while the (stochastic) discount rate and the integrand are required to satisfy only general integrability conditions.

math.PR↗

arXiv2017

On a Class of Optimal Stopping Problems with Applications to Real Option Theory

We consider an optimal stopping time problem related with many models found in real options problems. The main goal of this work is to bring for the field of real options, different and more realistic pay-off functions, and negative interest rates. Thus, we present analytical solutions for a wide class of pay-off functions, considering quite general assumptions over the model. Also, an extensive and general sensitivity analysis to the solutions, and an economic example which highlight the mathematical difficulties in the standard approaches, are provided.

math.OC↗

arXiv2014

Fre'chet Generalized Trajectories and Minimizers for Variational Problems of Low Coercivity

We address consecutively two problems. First we introduce a class of so called Fre'chet generalized controls for a multi-input control-affine system with non-commuting controlled vector fields. For each control of the class one is able to define a unique generalized trajectory,and the input-to-trajectory map turns out to be continuous with respect to the Fre'chet metric. On the other side, the class of generalized controls is broad enough to settle the second problem, which is proving existence of generalized minimizers of Lagrange variational problem with functionals of low (in particular linear) growth. Besides we study possibility of Lavrentiev-type gap between the infima of the functionals in the spaces of ordinary and generalized controls.

math.OC↗

arXiv2013

New Approach to Derive the Value Function of a Firm with Exit Option

In this paper we propose a new way of proving the value of a firm that is currently producing a certain product and faces the option to exit the market. The problem of optimal exiting is an optimal stopping problem, that can be solved using the dynamic programming principle. This approach leads to a partial differential equation, called the Hamilton-Jacobi-Bellman equation. This is a free-boundary problem, and therefore, we propose an approximation for the original model. We prove the convergence of the solution of the approximated problem to the original one and finally, using the Implicit Function Theorem, we obtain this solution.

math.OC↗

arXiv2008

Measuring Singularity of Generalized Minimizers for Control-Affine Problems

An open question contributed by Yu. Orlov to a recently published volume "Unsolved Problems in Mathematical Systems and Control Theory", V.D. Blondel, A. Megretski (eds), Princeton Univ. Press, 2004, concerns regularization of optimal control-affine problems. These noncoercive problems in general admit 'cheap (generalized) controls' as minimizers; it has been questioned whether and under what conditions infima of the regularized problems converge to the infimum of the original problem. Starting with a study of this question we show by simple functional-theoretic reasoning that it admits, in general, positive answer. This answer does not depend on commutativity/noncommtativity of controlled vector fields. It depends instead on presence or absence of a Lavrentiev gap. We set an alternative question of measuring "singularity" of minimizing sequences for control-affine optimal control problems by so-called degree of singularity. It is shown that, in the particular case of singular linear-quadratic problems, this degree is tightly related to the "order of singularity" of the problem. We formulate a similar question for nonlinear control-affine problem and establish partial results. Some conjectures and open questions are formulated.

math.OC↗