13 research outputs found
Online Local Volatility Calibration by Convex Regularization with Morozov's Principle and Convergence Rates
We address the inverse problem of local volatility surface calibration from
market given option prices. We integrate the ever-increasing flow of option
price information into the well-accepted local volatility model of Dupire. This
leads to considering both the local volatility surfaces and their corresponding
prices as indexed by the observed underlying stock price as time goes by in
appropriate function spaces. The resulting parameter to data map is defined in
appropriate Bochner-Sobolev spaces. Under this framework, we prove key
regularity properties. This enable us to build a calibration technique that
combines online methods with convex Tikhonov regularization tools. Such
procedure is used to solve the inverse problem of local volatility
identification. As a result, we prove convergence rates with respect to noise
and a corresponding discrepancy-based choice for the regularization parameter.
We conclude by illustrating the theoretical results by means of numerical
tests.Comment: 23 pages, 5 figure
Optimal Convergence Rates Results for Linear Inverse Problems in Hilbert Spaces
In this paper, we prove optimal convergence rates results for regularisation
methods for solving linear ill-posed operator equations in Hilbert spaces. The
result generalises existing convergence rates results on optimality to general
source conditions, such as logarithmic source conditions. Moreover, we also
provide optimality results under variational source conditions and show the
connection to approximative source conditions
On the Choice of the Tikhonov Regularization Parameter and the Discretization Level: A Discrepancy-Based Strategy
We address the classical issue of appropriate choice of the regularization
and discretization level for the Tikhonov regularization of an inverse problem
with imperfectly measured data. We focus on the fact that the proper choice of
the discretization level in the domain together with the regularization
parameter is a key feature in adequate regularization. We propose a
discrepancy-based choice for these quantities by applying a relaxed version of
Morozov's discrepancy principle. Indeed, we prove the existence of the
discretization level and the regularization parameter satisfying such
discrepancy. We also prove associated regularizing properties concerning the
Tikhonov minimizers.Comment: 28 pages, 4 figure
Structured Models for Cell Populations: Direct and Inverse Problems
Structured population models in biology lead to integro-differential equations that describe the evolution in time of the population density taking into account a given feature such as the age, the size, or the volume. These models possess interesting analytic properties and have been used extensively in a number of areas. In this article, we consider a size-structured model for cell division and revisit the question of determining the division (birth) rate from the measured stable size distribution of the population. We formulate such question as an inverse problem for an integro-differential equation posed on the half line. The focus is to compare from a computational view point the performance of different algorithms. Finally, we will discuss real data reconstructions with E. coli data
Structured Models for Cell Populations: Direct and Inverse Problems
Structured population models in biology lead to integro-differential equations that describe the evolution in time of the population density taking into account a given feature such as the age, the size, or the volume. These models possess interesting analytic properties and have been used extensively in a number of areas. In this article, we consider a size-structured model for cell division and revisit the question of determining the division (birth) rate from the measured stable size distribution of the population. We formulate such question as an inverse problem for an integro-differential equation posed on the half line. The focus is to compare from a computational view point the performance of different algorithms. Finally, we will discuss real data reconstructions with E. coli data
The Interplay between COVID-19 and the Economy in Canada
We propose a generalized susceptible-exposed-infected-removed (SEIR) model to track COVID-19 in Canadian provinces, taking into account the impact of the pandemics on unemployment. The model is based on a network representing provinces, where the contact between individuals from different locations is defined by a data-driven mixing matrix. Moreover, we use time-dependent parameters to account for the dynamical evolution of the disease incidence, as well as changes in the rates of hospitalization, intensive care unit (ICU) admission, and death. Unemployment is accounted for as a reduction in the social interaction, which translates into smaller transmission parameters. Conversely, the model assumes that higher proportions of infected individuals reduce overall economic activity and therefore increase unemployment. We tested the model using publicly available sources and found that it is able to reproduce the reported data with remarkable in-sample accuracy. We also tested the model’s ability to make short-term out-of-sample forecasts and found it very satisfactory, except in periods of rapid changes in behavior. Finally, we present long-term predictions for both epidemiological and economic variables under several future vaccination scenarios
The impact of COVID-19 vaccination delay: A data-driven modeling analysis for Chicago and New York City.
BACKGROUND: By the beginning of December 2020, some vaccines against COVID-19 already presented efficacy and security, which qualify them to be used in mass vaccination campaigns. Thus, setting up strategies of vaccination became crucial to control the COVID-19 pandemic. METHODS: We use daily COVID-19 reports from Chicago and New York City (NYC) from 01-Mar2020 to 28-Nov-2020 to estimate the parameters of an SEIR-like epidemiological model that accounts for different severity levels. To achieve data adherent predictions, we let the model parameters to be time-dependent. The model is used to forecast different vaccination scenarios, where the campaign starts at different dates, from 01-Oct-2020 to 01-Apr-2021. To generate realistic scenarios, disease control strategies are implemented whenever the number of predicted daily hospitalizations reaches a preset threshold. RESULTS: The model reproduces the empirical data with remarkable accuracy. Delaying the vaccination severely affects the mortality, hospitalization, and recovery projections. In Chicago, the disease spread was under control, reducing the mortality increment as the start of the vaccination was postponed. In NYC, the number of cases was increasing, thus, the estimated model predicted a much larger impact, despite the implementation of contention measures. The earlier the vaccination campaign begins, the larger is its potential impact in reducing the COVID-19 cases, as well as in the hospitalizations and deaths. Moreover, the rate at which cases, hospitalizations and deaths increase with the delay in the vaccination beginning strongly depends on the shape of the incidence of infection in each city