On the asymptotic behavior of the second moment of the Fourier transform of a random measure

The behavior at infinity of the Fourier transform of the random measures that appear in the theory of multiplicative chaos of Mandelbrot, Peyrière, and Kahane is an area quite unexplored. For context and further reference, we first present an overview of this theory and then the result, which is the main objective of this work, generalizing a result previously announced by Kahane. We establish an estimate for the asymptotic behavior of the second moment of the Fourier transform of the limit random measure in the theory of multiplicative chaos. After looking at the behavior at infinity of the Fourier transform of some remarkable functions and measures, we prove a formula essentially due to Frostman, involving the Riesz kernels

Unifying exotic option closed formulas

This paper aims to unify exotic option closed formulas by generalizing a large class of existing formulas and by setting a framework that allows for further generalizations. The formula presented covers options from the plain vanilla to most, if not all, mountain range exotic options and is developed in a multi-asset, multi-currency Black-Scholes model with time dependent parameters. The general formula not only covers existing cases but also enables the combination of diverse features from different types of exotic options. It also creates implicitly a language to describe payoffs that can be used in industrial applications to decouple the functions of payoff definition from pricing functions. Examples of several exotic options are presented, benchmarking the closed formulas' performance against Monte Carlo simulations. Results show a consistent over performance of the closed formula reducing calculation time by double digit factors

Pulled-to-par returns for zero coupon bonds : historical simulation value at risk

Due to bond prices pull-to-par, zero coupon bonds historical returns are not stationary, as they tend to zero as time to maturity approaches. Given that the historical simulation method for computing Value at Risk (VaR) requires a stationary sequence of historical returns, zero coupon bonds historical returns can not be used to compute VaR by historical simulation. Their use would systematically overestimate VaR, resulting in invalid VaR sequences. In this paper we propose an adjustment of zero coupon bonds historical returns. We call the adjusted returns “pulled-to- par" returns. We prove that when the zero coupon bonds continuously compounded yields to maturity are stationary the adjusted pulled-to-par returns allow VaR computation by historical simulation. We first illustrate the VaR computation in a simulation scenario, then we apply it to real data on euro zone STRIPS.info:eu-repo/semantics/publishedVersio

Open markov type population models: From discrete to continuous time

Funding Information: Funding: For the second author, this work was done under partial financial support of RFBR (Grant n. 19-01-00451). For the first and third author this work was partially supported through the project of the Centro de Matemática e Aplicações, UID/MAT/00297/2020 financed by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology). The APC was funded by the insurance company Fidelidade.We address the problem of finding a natural continuous time Markov type process—in open populations—that best captures the information provided by an open Markov chain in discrete time which is usually the sole possible observation from data. Given the open discrete time Markov chain, we single out two main approaches: In the first one, we consider a calibration procedure of a continuous time Markov process using a transition matrix of a discrete time Markov chain and we show that, when the discrete time transition matrix is embeddable in a continuous time one, the calibration problem has optimal solutions. In the second approach, we consider semi-Markov processes—and open Markov schemes—and we propose a direct extension from the discrete time theory to the continuous time one by using a known structure representation result for semi-Markov processes that decomposes the process as a sum of terms given by the products of the random variables of a discrete time Markov chain by time functions built from an adequate increasing sequence of stopping times.publishersversionpublishe

On a parallelised diffusion induced stochastic algorithm with pure random search steps for global optimisation

Funding Information: Funding: For the second author, this work was undertaken with partial financial support of RFBR (Grant n. 19-01-00451). For the first and third author, this work was partially supported through the project of the Centro de Matemática e Aplicações, UID/MAT/00297/2020, financed by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology). The APC was by supported the New University of Lisbon through the PhD program in Statistics and Risk Management of the FCT Nova Faculty.We propose a stochastic algorithm for global optimisation of a regular function, possibly unbounded, defined on a bounded set with regular boundary; a function that attains its extremum in the boundary of its domain of definition. The algorithm is determined by a diffusion process that is associated with the function by means of a strictly elliptic operator that ensures an adequate maximum principle. In order to preclude the algorithm to be trapped in a local extremum, we add a pure random search step to the algorithm. We show that an adequate procedure of parallelisation of the algorithm can increase the rate of convergence, thus superseding the main drawback of the addition of the pure random search step.publishersversionpublishe

The multi-compartment si(Rd) model with regime switching: An application to covid-19 pandemic

Grant No. 19-01-00451 UID/MAT/00297/2020We study—with existence and unicity results—a variant of the SIR model for an infectious disease incorporating both the possibility of a death outcome—in a short period of time—and a regime switch that can account for the mitigation measures used to control the spreading of the infections, such as a total lockdown. This model is parametrised by three parameters: the basic reproduction number, the mortality rate of the infected, and the duration of the disease. We discuss a particular example of application to Portuguese COVID-19 data in two short periods just after the start of the epidemic in 4 March 2020, with the first two cases dated that day. We propose a simple and effective method for the estimation of the main parameters of the disease, namely, the basic reproduction number and the mortality rate of the infected. We correct these estimated values to take into account the asymptomatic non-diagnosed members of the population. We compare the outcome of the model in the cases of the existence, or not, of a regime switch, and under three different scenarios, with a remarkable agreement between model and data deaths in the case of our basis scenario. In a final short remark, we deal with the existence of symmetries for the proposed model.publishersversionpublishe

Calibration of transition intensities for a multistate model: Application to long-term care

UID/MAT/00297/2020We consider a non-homogeneous continuous time Markov chain model for Long-Term Care with five states: the autonomous state, three dependent states of light, moderate and severe dependence levels and the death state. For a general approach, we allow for non null intensities for all the returns from higher dependence levels to all lesser dependencies in the multi-state model. Using data from the 2015 Portuguese National Network of Continuous Care database, as the main research contribution of this paper, we propose a method to calibrate transition intensities with the one step transition probabilities estimated from data. This allows us to use non-homogeneous continuous time Markov chains for modeling Long-Term Care. We solve numerically the Kolmogorov forward differential equations in order to obtain continuous time transition probabilities. We assess the quality of the calibration using the Portuguese life expectancies. Based on reasonable monthly costs for each dependence state we compute, by Monte Carlo simulation, trajectories of the Markov chain process and derive relevant information for model validation and premium calculation.publishersversionpublishe

Advances in Regression, Survival Analysis, Extreme Values, Markov Processes and Other Statistical Applications

Selected papers of the 17th Congress of the Portuguese Statistical Society, covering recent advances in Statistics, particularly in Regression, Extreme values, Markov processes and statistical applications in several areas

Applications of Fourier methods to the analysis of some stochastic processes

Dissertação de Doutoramento em Matemática: Processos Estocástico

On a Parallelised Diffusion Induced Stochastic Algorithm with Pure Random Search Steps for Global Optimisation

We propose a stochastic algorithm for global optimisation of a regular function, possibly unbounded, defined on a bounded set with regular boundary; a function that attains its extremum in the boundary of its domain of definition. The algorithm is determined by a diffusion process that is associated with the function by means of a strictly elliptic operator that ensures an adequate maximum principle. In order to preclude the algorithm to be trapped in a local extremum, we add a pure random search step to the algorithm. We show that an adequate procedure of parallelisation of the algorithm can increase the rate of convergence, thus superseding the main drawback of the addition of the pure random search step