Book of abstracts

Event funded by Fundação para a Ciência e a Tecnologia (FCT), through national
funds within the Projects CEMAPRE/REM - UIDB/05069/2020 and
EXPL/EGEIND/0351/2021.

Organizing Committee:
Nuno M. Brites (Chair)
CEMAPRE/ISEG – University of Lisbon

Manuel Guerra
CEMAPRE/ISEG – University of LisbonJoão Janela
CEMAPRE/ISEG – University of Lisbon

Miguel Reis
CEMAPRE/ISEG – University of Lisbon

Rui J. Oliveira
CEMAPRE – University of Évora

Scientific Committee:João M. Guerra (Chair)
CEMAPRE/ISEG – University of Lisbon

Wenceslao González Manteiga
University of Santiago de Compostela

Paula Milheiro-Oliveira
CMUP – University of Porto

Manuel Guerra
CEMAPRE/ISEG – University of Lisbon

Nuno M. Brites
CEMAPRE/ISEG – University of Lisbon

Contents
PROGRAMvii

INVITED SPEAKERS AND TUTORIAL

1

CONTRIBUTED TALKS

13

SPEAKERS

41

AUTHORS

42

v

vi

PROGRAMvii

Thursday, May 25th, 2023

viii

Friday, May 26th, 2023

ix

x

INVITED SPEAKERS AND
TUTORIAL

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STOCHASTIC DIFFERENTIAL EQUATIONMODELS OF INDIVIDUAL GROWTH WITH
APPLICATION TO OPTIMIZATION IN CATTLERAISING
Carlos A. Braumann
CIMA, Departamento de Matemática, Escola de Ciências e Tecnologia, Universidadede Évora
braumann@uevora.pt
Gonçalo Jacinto
CIMA, Departamento de Matemática, Escola de Ciências e Tecnologia, Universidadede Évora
gjcj@uevora.pt
Patrícia A. Filipe
CIMA, Departamento de Métodos Quantitativos para Gestão e Economia, Iscte Business School, Iscte-Instituto Universitário de Lisboa
patricia.filipe@iscte-iul.pt

Random fluctuations in the internal and external environmental conditions affect
the growth rate of individual animals, but can not be captured by the classically used
regression models based on an average growth curve. Still, the classical average growth
curves that have been fitted to different types of animals and plants can be written
as solutions of ordinary differential equation (ODE) dynamical models that describe
the average growth dynamics. The problem is that classical regression models assume
that the random environmental fluctuations affect directly the animal size (i.e., affect
the ODE solution, causing deviations from the average growth curve at the time theyoccur but with no further repercussions) rather than affecting the dynamics of the growth
process (i.e. the average growth rates used in the ODE). So, we need stochastic differential
equation (SDE) models, as proposed by [6] for tree growth and later used for the growthof several animals and plants, including cattle growth (see [1, 2, 3, 5, 7, 8] for models andparameter estimation).
For a general class of average growth curve models for the size X(t) of an individual
animal at time t, which includes the ones typically used, one can find a model-specificstrictly increasing C1 function h such that the transformed size Y (t) = h(X(t)) is thesolution of the simple ODE dY (t) = β(α − Y (t))dt, where α = h(A) is the asymptotic
transformed size (A > 0 being the real asymptotic or maturity size of the animal) and
β > 0 is the rate of approach to it. If the effect of environmental fluctuations on thegrowth rate is approximated by a white noise, its cumulative effect is σW (t), where W (t)is a standard Wiener process and σ > 0 is a noise intensity parameter. We obtain theSDE dY (t) = β(α − Y (t))dt + σdW (t).
These more realistic models can help farmers optimize the profit obtained by raisingand selling an animal, i.e. to determine the selling age that maximizes the expected profit.
In [4], for a simple profit structure, the profit probability distribution, mean, standard
deviation, and other quantities of interest were obtained and the optimization problem
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was solved. Similar results but considering a more realistic, challenging, and generalprofit structure, studied in [8], are presented here. Now, the selling price per kg paid tofarmers depends on the animal’s age and weight categories and the raising costs, besides
fixed costs and maintenance costs per unit time, consider feeding costs per unit time
proportional to the randomly varying animal size.
Using the stochastic Gompertz (h(x) = ln x) and Bertalanffy-Richards (h(x) = x1/3 )models, we have applied the results to real (and abundant) weight data of Mertolengo
breed cattle males from the producers association database Genpro. We conclude that
farmers are selling the animals a little earlier than the optimal age. Sensitivity analysis
for small changes on parameter estimates shows that these have a small effect on the
optimal expected profit and a negligible effect on the optimal selling age.
Our results led to a user friendly optimization software for farmers by RuralBit. For
mixed models (α and/or β varying randomly among animals to account for genetic andother differences), see [9].

References[1] P. A. Filipe, C. A. Braumann (2008). Modelling individual animal growth in random
environments. In: Proc. 23rd Int. Workshop on Statistical Modelling, ed. P. H. C.
Eilers, Utrecht, 232–237.
[2] P. A. Filipe, C. A. Braumann, N. M. Brites, C. J. Roquete (2010). Modelling animal
growth in random environments: An application using nonparametric estimation.
Biometrical Journal 52, nr. 5, 653–666.
[3] P. A. Filipe, C. A. Braumann, N. M. Brites, C. J. Roquete (2013). Prediction for
individual growth in a random environment. In: Recent developments in modelingand applications in statistics, eds.: P. E. Oliveira, M. Graça Temido, C. Henriques,
M. Vichi. Springer, 193-201.
[4] P. A. Filipe, C. A. Braumann, C. Carlos (2015). Profit optimization for cattle growing in a randomly fluctuating environment, Optimization, 64, nr. 6, 1393–-1407.
[5] P. A. Filipe, C. A. Braumann, C. J. Roquete (2012), Multiphasic individual growthmodels in random environments. Methodology and Computing in Applied Probability
14, nr. 1, 49–56.
[6] O. García (1983). A stochastic differential equation model for the height of forest
stands. Biometrics 3, nr. 9, 1059–1072.
[7] G. Jacinto, P. A. Filipe, C. A. Braumann (2022). Weighted maximum likelihood
estimation for individual growth models. Optimization 71, nr. 11, 3295-3311.
[8] G. Jacinto, P. A. Filipe, C. A. Braumann (2022). Profit optimization of cattle growthwith variable prices. Methodology and Computing in Applied Probability 24, 1917–
1952.
[9] N. T. Jamba, G. Jacinto, P. A. Filipe, C. A. Braumann (2022). Likelihood function
through the delta approximation in mixed SDE models. Mathematics 10, nr. 3, 385.
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CAPS, FLOORS AND COLLARS CONTRACTUAL
AGREEMENTS
AND OPTIMAL INVESTMENT
1DECISIONS
José Carlos Dias
BRU-IUL, Iscte - Instituto Universitário de Lisboa, Lisbon, Portugal
jose.carlos.dias@iscte.pt
We offer novel analytical solutions for evaluating perpetual caps, floors and collars on
continuous flows under the constant elasticity of variance (CEV) model. We demonstratethat the inclusion of a perpetual bubble value is required to avoid arbitrage opportunities
in the case of the CEV process with upward-sloping volatility skews. We then extend theprevious literature on caps, floors and collars arrangements by providing new analyticalformulae for valuing finite maturity caps, floors and collars that are contingent on continuous flows. Practical applications of these contractual agreements arising within thecontext of executive management decisions are also discussed.
In addition, we also present an analytical representation for evaluating optimal investment decisions associated to a feed-in tariff (FIT) contract with a minimum priceguarantee (i.e., a price-floor regime) under the CEV model. The proposed analytic solutions can be used to optimally design FIT contractual schemes with both perpetualand finite maturity guarantees. We show that the argument that a perpetual guarantee
only induces investment for prices below the price floor when offering a risk-free investment opportunity is still valid under the CEV process. We also demonstrate that the
optimal price-floor level triggering immediate investment in the presence of a perpetual
guarantee is independent of the elasticity parameter of the CEV model. By contrast, weshow that such independence is not valid any more in the case of FIT contracts with afinite maturity guarantee. Our results provide evidence that care must be taken whena policymaker aims to design a given instrument to induce investment decisions withFIT contracts because the differences between trigger points under alternative modelingassumptions are quite significant and the excessive rents are usually paid at the expenseof tax payers.
This talk is based on the work recently developed by [1] and [2].

References[1] J. C. Dias, J. P. V. Nunes, F. C. Silva (2023). Finite maturity caps and floors on
continuous flows under the constant elasticity of variance process. Working Paper,
Iscte Business School.
[2] J. C. Dias, J. P. V. Nunes, J. P. Ruas, F. C. Silva (2023). Optimal investment
decisions with minimum price guarantees under the constant elasticity of variance
process. Working Paper, Iscte Business School.

1This work was supported by the Fundação para a Ciência e a Tecnologia [grant number
UIDB/00315/2020]

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A PROBABILISTIC MODEL OF HYSTERESIS
Michael Grinfeld
University of Strathclyde, Glasgow, UK
m.grinfeld@strath.ac.uk
Rod Cross
University of Strathclyde, Glasgow, UK
abloch@umich.edu
Andrew Wade
University of Durham, Durham, UK
ljcolomb@umich.edu
In this talk I will discuss the reasons hysteresis [4] is important in economics, inparticular in the theory of the firm.
However, the classical framework for modelling hysteretic effects in economics, which
is heavily indebted to micromagnetics, is based on assumptions (of infinite durationof memory effects and of rate-independence) that are clearly not realistic in economic
situations. Hence we introduce a new probabilistic framework for modelling a hysteretic
economic agent, which owes much to the work of R. Brockett [1], which is based on
Poisson counters. We use an elementary supply-demand fluid dynamics model, whichis quite rich already in its simplest form. This framework, in our opinion, resolves thediﬀiculties of the classical approach and is applicable also, for example, to problems of
exploitation of renewable resources.
Following Horsthemke and Lefevre [3], we compute the relevant Fokker-Planck equation and the steady state pdf in the case of a single agent (which can be considered as
an aggregate picture of the market) and discuss sunk cost accumulation rates.
The resulting framework is very easy to simulate. I will present some simulation
results and will finish by stating what seem to be interesting and relevant open problems.

References[1] R. Brockett, Stochastic Control, Notes 2009.
[2] R. Cross, M. Grinfeld, and H. Lamba, Hysteresis in economics, IEEE CSM 29 (2009),
30–43.
[3] W. Horsthemke and R. Lefever, Noise-induced Transitions: Theory and Applications
in Physics, Chemistry, and Biology, Springer 2010.
[4] I. D. Mayergoyz, Mathematical Models of Hysteresis and Their Applications, Academic Press 2003.

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STATISTICAL INFERENCE FOR SDE:
ESTIMATION AND SPECIFICATION TESTS
Wenceslao González-Manteiga
CITMAga, Department of Statistics, Mathematical Analysis and Optimization, Universidade de Santiago de Compostela, Spain
wenceslao.gonzalez@usc.es
Alejandra López-Pérez
Department of Statistics, Mathematical Analysis and Optimization, Universidade deSantiago de Compostela, Spain
alejandralopez.perez@usc.es
Manuel Febrero-Bande
CITMAga, Department of Statistics, Mathematical Analysis and Optimization, Universidade de Santiago de Compostela, Spain
manuel.febrero@usc.es
Due to their analytical tractability, stochastic differential equations (SDE) have become a centerpiece in the financial literature. In this talk, new developments on goodnessof-fit test for SDE are discussed,considering SDE with deterministic and stochastic volatility and Itô diffusions as functional time series.The finite sample behaviour of the proposed
tests, regarding power and size, are illustrated by means of a simulation study, along anapplication to real financial data.
Notwithstanding the importance of goodness-of-fit tools, latent factors and a continuoustime setting with observations occurring at discrete time points challenge the estimationof the models. Therefore, the estimation problem is addressed, as it hinders the goodnessof-fit procedures, discussing the intricacies of different estimation implementations prior
to the methodological contribution of the test procedures.

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MATHEMATICS DRIVEN BY EPIDEMICS 1
Paula Patrício
NOVA Math, Department of Mathematics, NOVA SST
pcpr@fct.unl.pt
The COVID-19 pandemic has highlighted the importance of mathematical models in
understanding the dynamics of the disease and assessing the impact of control measures.
We will discuss possible contributions from mathematics to the understanding of an
epidemic, from the most theoretical to the most practical aspects. We will visit someexamples starting from the most basic model, the SIR model. Different epidemiologicalquestions will lead to different mathematical formulations with corresponding techniques.
I will show some examples where we try to choose an appropriate formulation to answer
these questions or develop new ones [1], [2], [3] and [4].

References[1] Caetano, C., Morgado, M. L., Patrício, P., Leite, A., Machado, A., Torres, A., ...
and Nunes, B. (2022). Measuring the impact of COVID-19 vaccination and immunity
waning: A modelling study for Portugal. Vaccine, 40(49), 7115-7121[2] Esquível, M. L., Patrício, P., and Guerreiro, G. R. (2020). From ODE to OpenMarkov Chains, via SDE: an application to models for infections in individuals andpopulations. Computational and Mathematical Biophysics, 8(1), 180-197.
[3] Doutor, P., Rodrigues, P., Soares, M. D. C., and Chalub, F. A. (2016). Optimal vaccination strategies and rational behaviour in seasonal epidemics. Journal of mathematical biology, 73, 1437-1465.
[4] Rodrigues, P., Silva, C. J., and Torres, D. F. (2014). Cost-effectiveness analysis
of optimal control measures for tuberculosis. Bulletin of mathematical biology, 76,
2627-2645.

1

This work is funded by national funds through the FCT - Fundação para a Ciência e a Tecnologia,
I.P., under the scope of the projects UIDB/00297/2020 and UIDP/00297/2020 (Center for Mathematics
and Applications)7SDE 2023
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MODELLING FISH POPULATION DYNAMICS ANDFISHING STRATEGIES TO PROVIDE ADVICE FORFISHERIES MANAGEMENT
Alexandra A. Silva
Instituto Português do Mar e da Atmosfera (IPMA), Marine and Environmental Sciences Centre (MARE)
asilva@ipma.pt
Laura Wise
Instituto Português do Mar e da Atmosfera (IPMA)
lwise@ipma.pt

European fisheries are normally managed by total allowable catches (TACs) and quotas set annually for the stocks of exploited marine species [1]. TACs are based on advice
provided by the International Council for the Exploration of the Sea (ICES) to the European Commission. ICES advice on catch opportunities is the culmination of a fairly
standardised procedure which starts with collaborative work of fishery scientists frominterested countries, such as IPMA scientists, and progresses through two review panels
of independent ICES scientists. Catch opportunities are calculated based on the Maximum Sustainable Yield and on the Precautionary Approach [2, 3]. Ideally, there is a rule
(harvest control rule, HCR) which establishes the sustainable catch or fishing effort as afunction of the present/recent abundance of the stock. When suﬀicient data on historical stock abundance and harvest levels are available, the history of the stock up to thepresent may be estimated using a (stock assessment) model. The HCR CR needs to beevaluated against sustainability criteria adopted by ICES using stochastic simulations, a
procedure known as Management Strategy Evaluation (MSE).
In the first part of this talk we briefly introduce two types of stock assessment models
commonly used in ICES for data-limited and data-rich stocks: surplus production models(SPM) and age-based models (ABM) and explain how MSY reference points are derived[4].
An application of SPiCT [5], a stochastic surplus production model in continuous
time, to the stock of anchovy (Engraulis encrasicolus) in the western Iberian waters ispresented. SPMs were fitted to anchovy catches and biomass indices obtained in scientificsurveys [6]. Priors on the shape of the production curve, the initial biomass depletionand the intrinsic growth rate of the population were combined such that models varied
from nearly unconstrained to increasingly constrained. The outputs were screened forcompliance with ICES guidelines to accept an assessment model, which include criteria of
convergence, goodness-of-fit, parameter uncertainty and retrospective behaviour. Several
models complied with ICES guidelines although levels of uncertainty of relative MSY reference points were higher than the usually acceptable for long-lived stocks. The challengeof assessing anchovy is partly explained by its highly variable dynamics associated witha short life-span.

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In the second part of the talk we introduce management strategy evaluation andpresent the case study of rebuilding the purse-seine sardine fishery. MSE is an approachfor evaluating and implementing fishery management strategies robust to several types of
uncertainty while balancing economic, social and biological objectives [7]. A MSE should
result in clear and measurable objectives with a robust process for achieving them. Ideallythis should involve all participants in the fishery. It incorporates a number of interlinked
model structures and include: population dynamics; data collection; data analysis andstock assessment; a HCR that defines a specific management action (e.g., the TAC); andimplementation of the HCR. In an MSE the full management cycle is modelled. It allows
us to test the effect of changing any part of the management cycle including changes tothe operating model, assumptions about variability, etc. Alternative management procedures can be compared by running many stochastic simulations, each for several years,
to identify the performance of a rule according to different metrics under the likely rangeof conditions.
Following a decrease in the abundance of the Iberian sardine and the perceived impossibility of rebuilding the stock in the short-term (2 years), a MSE was implemented to test
the performance of different fishing mortality based sliding simple linear HCRs against a
set of statistics [8, 9]. The achievement of management objectives and compliance withICES precautionary criterion were quantified. The conditioning of the operating model
was based on the stock assessment and included recruitment stochasticity. Since no highrecruitment had been observed in the last years, the uncertainty if this was the result ofa general decline or a shift of productivity regime was a matter of concern. Therefore,
several scenarios with different Operating Models according to alternative scenarios ofproductivity and different Management Procedures were simulated for 1000 populations
in a period of thirty years. The management cycle, included observation error and thecurrent assessment model in each simulation loop. Performance statistics (such as catch,spawning stock biomass and fishing mortality) were estimated for three time periods:
initial, short and long-term. Given the uncertain recruitment productivity of the stock,
the selected HCRs are risk averse to the scenario of poorest productivity.
Finally, we highlight areas of transdisciplinary work in stock assessment, namely towards the application of new models beyond the typical single-species/-stock/-sustainabilitypillar.

References[1] Council regulation 1380/2013 on the Common Fisheries Policy, amending Council Regulations (EC) No 1954/2003 and (EC) No 1224/2009 and repealing Council Regulations (EC) No 2371/2002 and (EC) No 639/2004 and Council Decision
2004/585/EC (2013). Oﬀicial Journal L 354, page 22.
[2] ICES (2016). Technical Guidelines - Advice on fishing opportunities. ICES Technical
Guidelines. Report. https://doi.org/10.17895/ices.advice.8339
9

SDE 2023
First SDE - Stochastic Days Encounters[3] ICES (2022). ICES technical guidance for harvest control rules and stock assessments for stocks in categories 2 and 3. ICES Technical Guidelines. Report.
https://doi.org/10.17895/ices.advice.19801564.v1[4] M. Haddon (2011). Modelling and Quantitative Methods in Fisheries. Chapman &Hall, London.
[5] M. W. Pedersen, C. W. Berg (2017). A stochastic surplus production modelin continuous time. Fish and Fisheries, 18: 226–243. Blackwell Publishing Ltd.
https://doi.org/10.1111/faf.12174.
[6] ICES(2021).
Workshopon
Data-Limited
StocksSpecies(WKDLSSLS3).
ICES
Scientific
Reports.
https://doi.org/10.17895/ices.pub.8145

ofShort-Lived
3:86.
60
pp.

[7] D. S. Holland (2010). Management Strategy Evaluation and Management Procedures: Tools for Rebuilding and Sustaining Fisheries. OECD Food, Agriculture andFisheries Working Papers, No. 25, OECD Publishing. doi: 10.1787/5kmd77jhvkjf-en[8] ICES (2019). Workshop on the Iberian Sardine Management
covery Plan (WKSARMP). ICES Scientific Reports, 1:18.
http://doi.org/10.17895/ices.pub.5251

and
168

Repp.

[9] ICES (2021). Request from Portugal and Spain to evaluate a new Harvest Control Rule for the management of the Iberian sardine stock (divisions 8.c and 9.a).
In Report of the ICES Advisory Committee, 2021. ICES Advice 2021, sr.2021.05.
https://doi.org/10.17895/ices.advice.8163

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ROUGH VOLATILITY MODELSJoão Guerra
REM, ISEG - School of Economics and Management, Universidade de Lisboa
jguerra@iseg.ulisboa.pt

Rough volatility models have enjoyed a tremendous focus of interest by the academic
community over the last years, since it was introduced in the seminal papers [1] and [3].
In these models, the log-volatility behaves like a fractional Brownian motion with Hurst
parameter H < 1/2. Rough volatility models can explain important features observed involatility time series and in the implied volatility of option prices. Indeed, these models
are remarkably consistent with empirical data while remaining parsimonious. Moreover,rough volatility can also be related to the microstructure of financial markets as explainedin [2]. A particular important example of a rough volatility model is the rBergomi model,
which adjusts very well to the SP500 index with a small number of parameters. Thismodel also produces a power-law decaying at the money skew, an important featurethat is not satisfied by most conventional stochastic volatility models. The rBergomi
model is first obtained by modeling the log-variance as a truncated Brownian semistationary process (TBSS), which is motivated by the empirical finding that incrementsof log-volatility behave similarly to those of fBm. Afterwards, a deterministic changeof measure is applied, preserving analytic tractability. Unfortunately, by virtue of thedeterministic change of measure, the rBergomi model produces flat smiles for the VIX
index. This feature is very inconsistent with the market data, where the VIX smile
is upward slopping. In order to circumvent this problem, multiple solutions have been
proposed. One possibility is to use a stochastic volatility for the TBSS (which acts asa stochastic vol-of-vol), as in [4]. Another approach is to propose a fractional OrnsteinUhlenbeck stochastic change of measure for the rBergomi model that produces upward
slopping VIX smiles whilst maintaining analytic tractability, as in [5].
In this short course, we will introduce some of the most important rough volatility
models and applications, with a particular focus on the rBergomi model and its generalizations. We also discuss how to use Monte-Carlo methods for some rough volatility
models (following [4]) and recent results on the rBergomi model under a regime switchingchange of measure obtained in [5].

References[1] C. Bayer, P. Friz, J. Gatheral (2016). Pricing under rough volatility. Quantitative
Finance, 16, no. 6, 887–904.
[2] O. El Euch, M. Fukasawa, M. Rosenbaum (2018). The microstructural foundationsof leverage effect and rough volatility. Finance and Stochastics, 22, no. 2, 241–280.
[3] J. Gatheral, T. Jaisson, M. Rosenbaum (2018). Volatility is rough. Quantitative
Finance, 18, no. 6, 933–949.
11

SDE 2023
First SDE - Stochastic Days Encounters[4] H. Guerreiro, J. Guerra (2022). Least squares Monte Carlo methods in stochastic
Volterra rough volatility models. Journal of Computational Finance, 26, no.3, 73–
101.
[5] H. Guerreiro, J. Guerra (2023). VIX pricing in the rBergomi model under a regime switching change of measure. Quantitative Finance, DOI:
10.1080/14697688.2023.2178321.

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CONTRIBUTED TALKS

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DYNAMICS OF A STOCHASTIC SIRI EPIDEMIC
MODEL WITH MEDIA COVERAGE
Tomás Caraballo 1
Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, 41012-Sevilla (Spain)
caraball@us.es
This work is devoted to investigate the existence and uniqueness of a global positive
solution for a stochastic epidemic model with relapse and media coverage. We also
study the dynamical properties of the solution around both disease-free and endemic
equilibria points of the deterministic model. Furthermore, we show the existence of astationary distribution. Numerical simulations are presented to confirm the theoreticalresults. Some comments concerning the analysis from the point of view of the Random
Dynamical Systems theory will also be provided.

References[1] T. Caraballo & R. Colucci (2017). A comparison between random and stochastic
modeling for a SIR model. Commun. Pure and Applied Analysis 16 (1), 151-162[2] E. T. Caraballo & X. Han Applied Nonautonomous and Random Dynamical Systems.
Springerbriefs (2017)[3] T. Caraballo, M. El Fatini, R. Petterson & R. Taki (2018). A stochastic SIRI epidemic model with relapse and media coverage. Discrete and Continuous Dynamical
Systems, Series B 23, 3483-3501

1

The author acknowledges financial support from Ministerio de Ciencia e Innovación, Agencia Estatalde Investigación (Spain) under project PID2021-122991NB-C21.

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ON ESTIMATION IN SUCCESSIVE MORE 1
RESTRICT CLASSES OF MIXED MODELS
Carla Santos
NOVAMATH, Polytechnic Institute of Beja, Portugal
carla.santos@ipbeja.pt
Cristina Dias
NOVAMATH, Polytechnic Institute of Portalegre, Portugal
cpsd@ipportalegre.pt
Célia Nunes
Department of Mathematics and Center of Mathematics and Applications, Universityof Beira Interior, Portugal
celian@ubi.ptJoão T. Mexia
NOVAMATH, Department of Mathematics, New University of Lisbon, Portugal
jtm@fct.unl.pt
Linear mixed models are widely used as they provide a flexible approach to correlateddata, such as those resulting from repeated measurements, as found in medical research,
agriculture research, and in many other fields. Addressing the accuracy of the estimators, more specific classes of mixed models have been studied. Linear mixed models whose
covariance matrices are the positive definite linear combinations of known pairwise orthogonal projection matrices (POPM) that add up to the identity matrix constitute theclass of models with orthogonal block structure (OBS). OBS have estimators with good
behaviour for estimable vectors and variance components, however we can achieve best
linear unbiased estimators, for estimable vectors, considering models with commutative
orthogonal block structure (COBS). COBS are based on the commutativity between theorthogonal projection matrix on the space spanned by the mean vector, and the POPM
belonging to the principal basis of the commutative Jordan algebra of symmetric matrices,
associated to the model. Regarding estimation in COBS, this commutativity condition
can be also achieved resorting to U-matrices and using the fundamental partition of theobservations vector, constituted by the sub-vectors corresponding to the different sets of
the levels of the fixed effects factors.
Keywords: Best linear unbiased estimators, Orthogonal block structure, U-matrices.

1

This work is funded by national funds through the FCT – Fundação para a Ciência e a Tecnologia, I.P.
(Portuguese Foundation for Science and Technology) under the scope of the projects UIDB/00297/2020
and UIDB/00212/2020.

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GENOTYPE X ENVIRONMENT INTERACTION
AND YIELD STABILITY ANALYSIS OF1 WHEAT(TRITICUM AESTIVUM) GENOTYPES
Cristina Dias
NOVAMATH, Polytechnic Institute of Portalegre, Portugal
cpsd@ipportalegre.pt
Carla Santos
NOVAMATH, Polytechnic Institute of Beja, Portugal
carla.santos@ipbeja.ptJoão T. Mexia
NOVAMATH, Department of Mathematics, New University of Lisbon, Portugal
jtm@fct.unl.pt
Fifteen wheat genotypes were evaluated at four locations (Elvas, Beja, Benavila
and Revilheira) using RCBD with two replications during 2015/2016, 2016/2017, and2017/2018. The objective was to study the G x E interaction, adaptability, and phenotypic stability of wheat genotypes at four locations. Data was collected on quantitative
traits and wheat quality productions. The analyses of variance showed significant differences among genotypes in the four environments and when locations were combined. The
Genotype x environment interaction (GEI) was also significant for all trials, indicating
consistency of performance of the genotypes over the four locations. The genotypes C1,
C6, C11, C12, and C15 are the most stable, with the highest yield, revealing considerable adaptability to the different regions. Cultivars 7, 9, and 13, are also quite stablewith high medium yield. The cultivars, 1, 6, and 12 are highly stable, with production
above average, revealing wide adaptability to the region, the remaining cultivars are moreunstable with production below average. The mean performance of genotypes in wheatyield was also analyzed using univariate and multivariate stability parameters. The cultivars C1, C6, and C12 were identified as stable and widely adapted genotypes by five
univariate stability parameters and also by AMMI-II. Moreover, GGE identified C1 andC6 as specifically adapted to the locations of Elvas and Benavila while C7, C11, and C15
were specifically adapted to the locations of Revilheira and Beja.
Keywords: Production stability, Genotype-environment interaction, AMMI analysis, Linear regression analysis.

References[1] D. G. Pereira and J.T. Mexia (2003). Reproducibility of joint regression analysis.
Biometryczne, 33, 279–293.
[2] F .A. Van Eeuwijk, (1995). Linear and bilinear models for the analysis of multienvironment trials: An inventory of models. Euphytica, 84, 1-7.
1This work is funded by national funds through the FCT - Fundação para a Ciência e a Tecnologia,
I.P., under the scope of the project UIDB/00297/2020 (Center for Mathematics and Applications).

16

SDE 2023
First SDE - Stochastic Days Encounters[3] H. P. Gauch and P. Piepho (2008). Statistical Analysis of Yield Trials by AMMI andGGE: Further Considerations. Crop Science . 48, 866- 889.
[4] H. Scheffé (1959). The Analysis of Variance. John Wiley and Sons. New York.

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SDE 2023
First SDE - Stochastic Days Encounters

CENSORED QUANTILE REGRESSION WITH
AUXILIARY COVARIATES 1
Zhaozhi Fan
Memorial University of Newfoundland, NL, Canada
zhaozhi@mun.ca
Asokan Variyath
Memorial University of Newfoundland, NL, Canada
variyath@mun.ca
Chithran Vasudevan
Memorial University of Newfoundland, NL, Canada
chithran@mun.ca
In the analysis of survival data using quantile regression models (Koenker and Bassett
1978, Koenker, 2005), severe censoring could trigger problems such as the existence of an
estimator of the regression coeﬀicients for extreme quantiles. There is a need of having
samples with large size as to have more event times included in the data.
In epidemiological studies, there is often times only a small portion of the whole study
cohort accurately observed with some key exposures – the validation sample, while therest of the cohort was measured inaccurately, or only some auxiliary information wascollected. Censored regression models accommodating error-prone covariates have been
extensively investigated, including the cases when a validation sample is available, seeZhou and Pepe (1995), Fan and Wang (2009) and the references therein.
In this paper we propose a method to include the non-validation sample into themodeling through regression calibration, and other methods of smoothing, and conduct
quantile regression analysis based on the whole study cohort. The method is based on aquantile regression model proposed by Peng and Huang (2008).
For the ith subject, let Ti be the logarithm of the failure time and Xi the p-vector
covariate, i = 1, 2, . . . , n. Let Ci be the logarithm of the right censoring time and let Yi =min(Ti , Ci ) be the logarithm of the observed survival time. Define an event indicator, δi =I(Ti ≤ Ci ). Assume independent censoring, that is, conditional on Xi , Ci is independentof Ti . Let QTi (τ |Xi ) = inf{t : P (Ti ≤ t|Xi ) ≥ τ }.
The quantile regression model proposed in Peng and Huang (2008) is
QTi (τ |Xi ) = exp{X⊤i β(τ )},
for a specific quantile, τ ∈ (0, 1). Because of the monotonicity of the quantile function,
the quantile regression model based on the AFT model is a special case of this more
flexible model.
Define the counting process, Ni (t) = I(Yi ≤ t, δi = 1). Let H(u) = − log(1 − u) for0 ≤ u < 1. When the Xi ’s are available for the entire cohort, the observed data are thetriplet {Yi , Xi , δi }. Peng and Huang (2008) introduced the censored quantile regression
estimator as a generalization of the Nelson-Aalen estimator of the cumulative hazard
1

The research was partially support by Discovery Grants from the National Sciences and Engineering
Research Council Canada.

18

SDE 2023
First SDE - Stochastic Days Encounters
function of Ti . For a fixed τ ∈ (0, 1), the regression coeﬀicient, β(τ ) can be estimated bysolving the estimating equation√n Sn (β, τ ) = 0,
where
)
( (
n
]
) ∫ τ [1∑
⊤ β(u)
XX⊤
β(τ)dH(u) .
(1)−
I Yi ≥ e i
X i Ni e i
Sn (β, τ ) =
n i=1
0
(
) ∫ [
]
τ
X⊤
β(τ )
X⊤
β(u)
i
i
Here Ni e− 0 I Yi ≥ edH(u) is a martingale associated with the counting process, Ni (t). The martingale property ensures that E{Sn (β0 , τ )} = 0, where β0 (τ )is the true value of the censored quantile regression parameter.
In some epidemiological studies, some key exposures are accurately measured onlywith a small portion of the study cohort, the remaining with just auxiliary values collected. The observed data are composed of {Yj , Wj , X1j , X2j , δj }, j ∈ V, the validation
sample and {Yl , Wl , X2l , δl }, l ∈ NV, the non-validation sample, where W is the auxiliary
covariate which is observed for the whole study cohort.
We propose to estimate the estimating function Sn (β, τ ) over the non-validation partthrough regression calibration and plug-in or kernel smoothing directly on the estimating
function.
√
Let Z = (X1 , X2 ) and Ẑ = (X̂1 , X2 ); then β̂ is the generalized solution to nSˆn (β, τ ) =0, where{ (}) ∫ τ [
]
ρn ∑
X⊤
β(τ )
X⊤
β(u)
ˆ
j
j
X j Nj e
Sn (β, τ ) =−I Yj ≥ edH(u)mv j∈V0}{ (]) ∫ τ [
1 − ρn ∑⊤ β(u)
Ẑ
Ẑ⊤
β(τ
)
+
dH(u)
Ẑl Nl e l−
I Yl ≥ e l
mn
0
l∈V̄

= ρn ΩVmv (β, τ ) + (1 − ρn )Ω̂V̄mn (β, τ ),(2)

where ρn = mv /n. The above estimating equation is based on the regression calibration. It can also be done through kernel smoothing. The proposed estimator is consistentand has an asymptotically normal distribution under regularity conditions.
The eﬀiciency gain versus the one only based on the validation sample is remarkable.
The method also provides us possibilities of looking into higher (lower) extreme quantilesof the failure distribution.

19

SDE 2023
First SDE - Stochastic Days Encounters

COMPARATIVE ANALYSIS ANALYTICAL AND
SIMULATIVE METHODS OF SOLVINGEVOLUTION DIFFERENCE EQUATION
Irada Dzhalladova
University of Bialystok, Faculty of Mathematics, Bialystok, Poland
idzhalladova@gmail.com
Tomasz Czyżycki
University of Bialystok, Faculty of Mathematics, Bialystok, Poland
tomczyz@math.uwb.edu.pl
On the probability space (Ω, F, P) [1] we consider an initial problem formulated forthe system of evolution difference equations with random coeﬀicients in the form
xn+1 = A(ξn+1 , ξn )xn ,x0 = φ(ω)

n = 1, 2, . . . ,

(1)
(2)

where A is an m × m matrix with random elements, ξn is the Markov chain offinite number of the states θ1 , θ2 , . . ., θq with the probabilities pk (n) = P { ξn = θk },k = 1, 2, . . . , q, n = 1, 2, . . . that satisfy the system of difference equations [2]p(n + 1) = Πp(n),(3)( )qp(n) = (p1 (n), p2 (n), . . . , pq (n))T , Π = πks k,s=1 is a q × q transition matrix and φ : Ω →Rm .
If the random variable ξn+1 is in the state θk , k = 1, 2, . . . , q and the random variableξn is in the state θs , s = 1, 2, . . . , q, we denote
Aks = A(θk , θs ),k, s = 1, 2, . . . , q

and assume that there exist inverse matrices A−1
ks .
The m-dimensional column vector–function xn , n = 1, 2, . . . is called a strong solutionof system (1) if it satisfies (1) with initial condition (2) [3].
Description of exact solutions of (1)–(2) and study their behaviour directly is a very
challenging question, therefore we use moments of the proces xn and dependences between
them to reduce stochastic analysis to deterministic case. Our hypothesis is the following:stability in mean of solutions of (1)–(2) should be represented in stability of solutions ofappropriate moment equations.
We introduce functions fk (n, z), n = 1, 2, . . . , k = 1, 2, ..., q, z ∈ R which are theparticular probability density functions of xn , n = 1, 2, . . . determined by the formula∫{}fk (n, z) dz = P xn ∈ R, ξ = θk ,(4)Rand satisfy the following equations
fk (n + 1, z) =

q
∑

(
)
πks fs n, A−1(z)
det A−1
ks
ks .

s=1

20

(5)

SDE 2023
First SDE - Stochastic Days Encounterswhere z = xn+1 |ξn = θk , n = 1, 2, ..., k = 1, 2, ..., q.
We proved the following theorems:
Theorem 1. System of moment equations of the first order for a solution xn , n = 1, 2, . . .
of (1) is of the formq∑(1)
Ek {xn+1 } =πks Aks Es(1) {xn },(6)s=1

where k, s - numbers of states of Markov chain ξn and the functions∫(1)
Ek {xn } =zfk (n, z) dz,
k = 1, 2, ..., q,(7)R

are called particular moments of the first order for a solution xn , n = 1, 2, . . . of (1) [4].
Definition 1. 1) Moment of the first order for a solution xn , n = 1, 2, . . . of (1) is thefunctionq∑(1)
Ek {xn }.
E (1) {xn } =k=12) The trivial solution of the equation (1) we call stable in mean if for any solution
xn , n = 1, 2, ... of (1) holds lim E (1) {|xn |} = 0.
n→+∞Theorem 2. Stability conditions for the solutions of (1)–(2) obtained by averaging xn ,where xn is modelled stochastically by using random numbers generator, are equivalent
with stability in mean for such solutions expressed by conditions for corresponding system
of moment equations (7).
We have solved deterministic equations (6) numerically for certain data and compared
the solution with stochastic simulations of the process xn accordingly to the studied
equation (1). Obtained results confirm our hypothesis that stability in mean of a solutionof the system (1) is equivalent with stability of a solution of corresponding moment
equations (6).

References[1] Cheng-Der Fuh (2004) Uniform Markov Renewal Theory and Ruin Probabilities inMarkov Random Walks, The Annals of Applied Probability 2004, Vol. 14, No. 3,
1202–1241.
[2] L. Socha (1993) Równania momentów w stochastycznych układach dynamicznych,
Wydawnictwo Naukowe PWN, Warszawa, in Polish.
[3] R. Z. Hasminski (1980) Stochastic Stability of Differential Equations, Sijthoff &Noordhoff, Alphen aan den Rijn.
[4] I. A. Dzhalladova, M. Ružičková,�(2020) Dynamical Systems with Random Structure
and their Applications (Stability, Oscillations and Optimizations of Systems) (1sted.), Cambridge Scientific Publishers.

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SDE 2023
First SDE - Stochastic Days Encounters

ON A COUPLED PRICE-VOLUME EQUITY MODELWITH AUTO-INDUCED REGIME SWITCHING
M. L. Esquível
Nova SST and CMA UNL, Portugal
mle@fct.unl.pt
N. P. Krasii
DSTU Rostov-on-Don, Russian Federation
krasnad@yandex.ru
P. P. Mota
Nova SST and CMA UNL, Portugal
pjpm@fct.unl.pt
A. V. Shamraeva
Financial University, Russian Federation
vvshamraeva@fa.ru

0.1 Introduction
We present a rigorous development of a model for the couple (Price, Volume of transactions) first introduced in [2]. For this development we rely on the precise formulation ofdiffusion auto-induced regime switching models presented in [1]. The auto-induced regime
switching models referred to may be those based on a finite set of stochastic differential
equations (SDE)—all defined on the same bounded time interval—and, a sequence ofinterlacing stopping times defined by the hitting threshold times of the trajectories of thesolutions of the SDE. The regimes may be defined parametrically—that is, the SDE coeﬀicients keep the same functional form with varying parameters—or the functional formof the SDE coeﬀicients may change with each regime. By using the same noise sourcefor both the price and the liquidity regime switching models—volume (liquidity) that,in general, is not a tradable asset—we ensure that despite incorporating information on
liquidity, the price part of the coupled model can be assumed to be arbitrage free andcomplete.
We consider a price-liquidity coupled model with regimes and thresholds given, in a
first approximation, by:{
dSt = µ(t, St , θ)dt + σ(t, St , θ)dBt1 , S0 ∈ R+
(1)
dLt = ν(t, Lt , λ)dt + η(t, Lt , λ)dBt2 , L0 ∈ R+
where (Bt1 )t≥0 and (Bt2 )t≥0 are Brownian processes with correlation matrix,
[
]
1 ρ
/=
Σ
ρ 1
and the — lower and upper — thresholds for the price and liquidity being denoted,
respectively, by Sm , SM for the price and Lm , LM , for the liquidity. Let us state two
22

SDE 2023
First SDE - Stochastic Days Encounters

hypothesis that rule the regime changes according to the parameters and thresholds. The
first hypothesis concerns the SDE for the process describing the time evolution of theprice in Formula (1).
A The interplay of the regimes and the threshold, for the process (St )t≥0 , is given by
the following relations satisfied by the parameter values and consequently the pricedrift coeﬀicient,


h
h


if Lt > LMθµ(t, St , θ ) if Lt > LM
θ = θ s if Lm ≤ Lt ≤ LM , µ(t, St , θ) = µ(t, St , θ s ) if Lm ≤ Lt ≤ LM


 d

θ
if Lt < Lmµ(t, St , θ d ) if Lt < Lm ,(2)with a similar relation for the price volatility coeﬀicient.
B The interplay of the regimes and the thresholds for the process (Lt )t≥0 , that is, theliquidity process is given by:


h
h


λ
if
S
>
S
t
M
ν(t, Lt , λ ) if St > SMλ = λs if Sm ≤ St ≤ SM , ν(t, Lt , λ) = ν(t, Lt , λs ) if Sm ≤ St ≤ SM



 d
ν(t, Lt , λd ) if St < Sm ,λif St < Sm(3)
with a similar relation for the liquidity volatility coeﬀicient.

0.2

Main Result

The main result is the following.
Theorem 0.2.1 (On the existence of a price-liquidity process model). Suppose that for
the SDE in Formula (1) both Hypothesis A and Hypothesis B are verified. Suppose thatµ and σ — as well as ν and η — satisfy regularity hypothesis of Yamada-Watanabe type.
Suppose that the increasing sequence of regime switching F-stopping times 0 ≡ τ0 < τ1 <
τ2 < · · · < τn < · · · verify, almost surely, limn→+∞ τn = +∞ and, for any T ∈ R+ , thefunction:nT : Ω −→ N(4)
ω 7−→ # {k ≥ 1 : τk (ω) ≤ T }is almost surely finite. Then there exists a unique in law regime switching process(St , Lt )t≥0 with continuous trajectories.
We will present an application to real data together with an adequate estimation
procedure for the chosen models for Price and Volume.

References[1] M.L. Esquível, N.P. Krasii, P.P. Mota (2020). Auto and externally induced regime
switching diffusions Communications on Stochastic Analysis. 14 (1-2), 27–47.
23

SDE 2023
First SDE - Stochastic Days Encounters[2] N.S. Rianço, M.L. Esquível, P.P. Mota, C.A. Veiga (2009). On a price-liquidity
threshold regime switching model (preliminary report). In: Proceedings of the 6th
St. Petersburg Workshop on Simulation, eds.: S.M. Ermakov, V.B. Melas, A.N.
Pepelyshev. VVM com. Ltd, vol. I, 419–424.

24

SDE 2023
First SDE - Stochastic Days Encounters

INVESTMENT IN TWO ALTERNATIVE PROJECTSWITH MULTIPLE SWITCHES AND THE EXIT
OPTION
Carlos Oliveira 1
Department of Industrial Economics and Technology Management, Norwegian University of Science and Technology, 7491, Trondheim, Norway; ISEG - School of Economics
and Management, Universidade de Lisboa; REM - Research in Economics and Mathematics, CEMAPRE; Rua do Quelhas 6, 1200-781 Lisboa, Portugal
carlos.m.d.s.oliveira@ntnu.no
Igor Kravchenko 2
IST-ID and CEMAT, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
Igor.Kravchenko@tecnico.ulisboa.pt
Cláudia Nunes 3
IST-ID and CEMAT, Av. Rovisco Pais, 1049-001 Lisboa, Portugal; Instituto Superior
Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
cnunes@math.tecnico.ulisboa.pt

This paper uses an analytical framework to examine a firm’s investment and switchingstrategy under uncertainty. The context is the possibility to launch and operate two
distinct projects, one at a time, with exposure to a stochastic exogenous price. We allow
for multiple switches between the two projects, along with abandonment options fromeach. These possibilities fundamentally influence the operational strategy. We showthat under some conditions, a dichotomous waiting region may arise at the investment
stage. In this case we have an inaction region, for a range of prices in a certain boundedinterval, where the firm does not invest and waits to have more information about theprice evolution. This region vanishes for a high level of uncertainty. Additionally, the firm
may operate with a negative instantaneous profit, although we prove that investment inthis region is never optimal. We include a comparative statistics where we show that theinvestment threshold is not monotonic. This behavior is explained with the existence ofan abandonment option. An extension of the investment model allowing for time-to-build
is also included. We prove that the investment strategy is a threshold strategy and nota dichotomous strategy. This work is a direct extension of [1], [2], [3] and [4].

References[1] A. Bar-Ilan, W. C. Strange, (1996). Investment lags. The American Economic Review, 86(3), 610-622.
1

The author gratefully acknowledge also the support from the Portuguese Science Foundation (FCT)
through projects CEMAPRE/REM - UIDB/05069/2020 and CEMAT - UIDB/04621/2020.
2
The author gratefully acknowledge also the support from the Portuguese Science Foundation (FCT)
through projects CEMAPRE/REM - UIDB/05069/2020 and CEMAT - UIDB/04621/2020.
3
The author gratefully acknowledge also the support from the Portuguese Science Foundation (FCT)
through projects CEMAPRE/REM - UIDB/05069/2020 and CEMAT - UIDB/04621/2020.

25

SDE 2023
First SDE - Stochastic Days Encounters[2] J. P. Décamps, T. Mariotti, S. Villeneuve (2006). Irreversible investment in alternative projects. Economic Theory, 425–448.
[3] M. Guerra, P. Kort, C. Nunes, C. Oliveira (2018). Hysteresis due to irreversible exit:
Addressing the option to mothball. Journal of Economic Dynamics and Control, 92,
69–83.
[4] M. Zervos, C. Oliveira, K. Duckworth (2018). An investment model with switchingcosts and the option to abandon. Mathematical Methods of Operations Research, 88,
417-443.

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SDE 2023
First SDE - Stochastic Days Encounters

FUNCTIONAL SOLUTION OF STOCHASTIC
DIFFERENTIAL EQUATIONSImme van den Berg
CIMA, University of Évora, Portugal
ivdb@uevora.pt
Let Yt be an Ito process and Z be a suﬀiciently regular function of t and Yt . Ito’s
Lemma gives the coeﬀicients of the stochastic differential equation satisfied by the process
Z(t, Yt ) in terms of partial derivatives of Z. We consider the inverse problem. Let
dXt = µ(t, Xt )dt + σ(t, Xt )dYt

(1)

be a stochastic differential equation. Then, under some regularity conditions on µ andσ, the equation (1) has a functional solution
Xt = Z(t, Yt ),provided the partial derivatives of µ and σ satisfy()∂σσ2 ∂ 2σ
∂µ
∂σ
= 0.
−
+µ
+
σ
∂X
∂t
∂X
2 ∂X 2(2)(3)

Formula (3) follows from the equality of the mixed second-order partial derivatives of Z.
Formula (2) expresses path-independence of the solution of (1) and was also derived
in a discrete setting [1]. Indeed, path-independence is obvious for the Wiener walk andthe discrete Geometrical Brownian motion, because an upward movement U followed bya downward movement D gives an equal result as DU , i.e. U D − DU = 0. Now if
U D − DU = o(dt3/2 ),(4)

the process Xt happens to be (nearly-)equivalent to a functional solution, while (3) follows
from an asymptotic expansion applied to (4).

References[1] I.P. van den Berg, E. Amaro, (2008). Nearly recombining processes and the calculationof expectations, ARIMA, Vol. 9, p. 389–417 .

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SDE 2023
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STOCHASTIC DIFFERENTIAL EQUATIONSHARVESTING MODELS: SIMULATION AND
SOLUTION
Miguel M. Reis 1
ISEG/UL - Universidade de Lisboa, Department of Mathematics; REM - Research in
Economics and Mathematics, CEMAPRE
mreis@iseg.ulisboa.pt
Nuno M. Brites
ISEG/UL - Universidade de Lisboa, Department of Mathematics; REM - Research in
Economics and Mathematics, CEMAPRE
nbrites@iseg.ulisboa.pt
Harvesting is a primary and fundamental activity impacting on human society as afood supplier and on the environment as a resource consumer. An equilibrium, required
to keep environments healthy and the activity attractive, is attainable through the application of a sustainable strategy, driven by effort. In fisheries, effort is the action planmeasured by the number of vessels, workers, hours, among other factors, which impose
low flexibility to organisations. Effort is obtained through the solution of a non-linear
Partial Differential Equation, arising from the optimisation process. Optimal variable effort yields the highest profits and is crucial to evaluate the feasibility of other strategies.
However, highly volatile effort policies are inapplicable due to logistic and social issues. A
penalised effort strategy was applied to study effort’s new behaviour and profit. Impact
on profit depends on the strength of the penalisation parameter- the higher (lower) thisvalue, the higher (lower) the impact on effort and the lower (higher) the profit. Penalised
effort does not oscillate as sharply as the non-penalised, but it is constantly adjusted.
Thus, penalised effort reduces social issues imposed by low or null effort periods, leaving
the logistic subject as the source of uncertainty to its implementation.
Keywords: Stochastic Differential Equations, Generalised Logistic Model, CrankNicolson scheme, Penalised Policies.

1

The author acknowledges FCT - Fundação para a Ciênciae a Tecnologia, I.P., for the scholarship
under the scope of the project EXPL/EGE-IND/0351/2021/CEMAPRE.

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SDE 2023
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PARAMETER ESTIMATION IN A PARTIALLYOBSERVED HYPOELLIPTIC STOCHASTIC 1
LINEAR SYSTEM: A SIMULATION STUDY
Nilton Ávido 2
DMAT, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre s.n.,4169-007 Porto, Portugal
up2021118@up.pt
Paula Milheiro-Oliveira
DEC, Faculty of Engineering, University of Porto, Rua do Dr Roberto Frias s.n.,4200-462 Porto, Portugal; CMUP, DMAT, Faculty of Sciences, University of Porto, Rua
do Campo Alegre s.n., 4169-007 Porto, Portugal
ecan@subu.edu.tr
Hypoelliptic systems present some challenges when the estimation of model parameters is concerned. The diﬀiculty with hypoelliptic systems comes mainly from the factthat the diffusion matrix is noninvetible. In this work, we consider the problem of estimating the drift matrix in a hypoelliptic linear model, based on partial observations incontinuous time. The model is given by a 2-dimension stochastic differential equation of
the form
]
[
]
[ ]
[
Ut
Ut
0
d
= A(t, θ)
dt +
dWt
(1)
Vt
Vt
σ
representing the state of the system and the equation
dYt = HVt + ϵ dB̃

(2)

representing the observation process, with t ∈ [0, T ] and H 6= 0. The drift matrix
A depends on the vector of unknown parameters, denoted by θ = (θ1 , θ2 ) ∈ Θ. The
parameter σ > 0 is supposed to be known. Here Wt and B̃ are independent standard
Wiener processes, and ϵ is a positive number that approaches zero, ϵ → 0.
An estimator of θ is proposed by Koncz[1] for degenerate n-dimenion systems in thecomplete observation case. In this work, we adapt Koncz’s estimator to our problem
by combining it with the Kalman-Bucy Filter. As far as we know, properties of suchan estimator, in the partially observed framework, are not found in the literature anddeserve investigation. The results presented relate to the numerical simulation of a model(harmonic oscillator) that describes the dynamic behavior of an elementary engineering
structure subject to random vibrations. The unknown parameters represent the stifnessand dampings coeﬀicients of the oscilattor. From the literature, a considerable bias
should be expected in the estimation of the damping coeﬀicient. In the case ϵ = 0,
the properties of the estimator of θ are studied in [2]. Moreover, the LAN property isstated and proved. In the present work, the influence of the observation error on the1

This research was partially supported by CMUP (UID/294MAT/00144/2019) funded by FCT (Portugal).
2
The first author also acknowledges INAGBE - National Institute for Scholarship Management, an
institution belonging to the Ministry of Higher Education, Science, Technology and Innovation of theGovernment of Angola, for funding the doctoral scholarship.

29

SDE 2023
First SDE - Stochastic Days Encounters

performance of the estimator is analyzed, as the observation error variance vanishes,that is ϵ → 0. Finally, resorting to a large number of simulated paths, the error of theestimator, essencially in terms of the bias and mean square error, is studied.

References[1] Koncz, K. (1987). On the parameter estimation of diffusional type processes withconstant coeﬀicients (elementary gaussian processes).Analysis Mathematica, 13 , 75;[2] Melnykova, A. (2020). Parametric inference for hypoelliptic ergodic diffusions withfull observations. Statistical Inference for Stochastic Processes, 23 , 595-635. doi:
10.1007/s11203-020-09222-4;[3] Prior, A., Kleptsyna, M., Milheiro-Oliveira, P.(2017). On maximum likelihood estimation of the drift matrix of a degenerated o–u process. Statistical Inference for
Stochastic Processes, 20 ,57-78. doi: 10.1007/s11203-016-9137-1;[4] Samson, A., Thieullen, M. (2012). A contrast estimator for completely or partiallyobserved hypoelliptic diffusion. Stochastic Processes and their Applications, 122 ,
2521-2552. doi: 10.1016/ j.spa.2012.04.006.

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SDE 2023
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PATHWISE STOCHASTIC CONTROL PROBLEMS
Neeraj Bhauryal 1
Grupo de Física Matemática, Uni. de Lisboa, Portugal
nsbhauryal@fc.ul.pt
Ana Bela Cruzeiro 2
Grupo de Física Matemática, Uni. de Lisboa, Portugal; Dep. de Matemática, Instituto Superior Técnico, Lisboa, Portugal
ana.cruzeiro@tecnico.ulisboa.pt
Carlos Oliveira 3
Department of Industrial Economics and Technology Management, Norwegian University of Science and Technology, 7491, Trondheim, Norway; ISEG - School of Economics
and Management, Universidade de Lisboa; REM - Research in Economics and Mathematics, CEMAPRE; Rua do Quelhas 6, 1200-781 Lisboa, Portugal
carlos.m.d.s.oliveira@ntnu.no
We consider two sets of pathwise control problems and study the associated HamiltonJacobi-Bellman stochastic partial differential equation (SPDE) for the value function.
First, we study the SPDE arising from the study of pathwise optimal control problems
initiated by Lions and Souganidis in the late 90’s [3], when the state equation is driven
by two independent Brownian motions. Later, we study the SPDE which arises in thepresence of only one noise term in the state equation. We do this in the framework ofanticipating calculus and establish the well-posedness of viscosity solutions. The talk isbased on the submitted works [1] and [2].

References[1] N. Bhauryal, A.B. Cruzeiro and C. Oliveira (2022). On the wellposedness of a Hamilton-Jacobi-Bellman equation with transport noise,https://arxiv.org/abs/2209.06660[2] N. Bhauryal, A.B. Cruzeiro and C. Oliveira (2022). Pathwise stochastic control anda class of stochastic partial differential equations, https://arxiv.org/abs/2301.09214[3] P.-L. Lions and P.E. Souganidis (1998). Fully nonlinear stochastic partial differential
equations: non-smooth equations and applications, Comptes Rendus de l’Académiedes Sciences-Series I-Mathematics, 327(8), 735–741, Elsevier.

1

The author acknowledges the support of the FCT project UIDB/00208/2020.
The author acknowledges the support of the FCT project UIDB/00208/2020.
3
The author would like to thank the FCT project CEMAPRE/REM-UIDB/05069/2020.
2

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INFLUENCE OF INSURANCE IN MITIGATINGNEGATIVE EVENTS IN INVESTMENTSTRATEGIES
Alexandra Moura 1
ISEG - Lisbon School of Economics and Management, University of Lisbon and
CEMAPRE
amoura@iseg.ulisboa.pt
Carlos Oliveira 2
Department of Industrial Economics and Technology Management, Norwegian University of Science and Technology, 7491, Trondheim, Norway; ISEG - School of Economics
and Management, Universidade de Lisboa; REM - Research in Economics and Mathematics, CEMAPRE; Rua do Quelhas 6, 1200-781 Lisboa, Portugal
carlos.m.d.s.oliveira@ntnu.no
We consider a company aiming at designing investment strategies, namely defining
the best moment to invest, in the presence of uncertainty. There are two sources of
uncertainty: the company’s future revenue and the existence of unexpected adverse events
that reduce it. To protect against adverse occurrences, the company can buy an insurance
contract, which impacts the investment strategy. The firm has to decide the insurance
contract to be bought and moment to invest in the market. These decisions are notindependent of each other. Also, they depend on the insurance premium charged by
the insurance company and on how the firm measures its risk. We study the optimal
insurance contract together with the optimal moment to invest, such that the expectedvalue of the revenue is maximized. Optimal solutions are found by formulating the model
as a control problem. Numerical examples are presented.

1

The author is partially funded by FCT through project “Optimal-Re, Optimal reinsurance withdependencies”, with grant EXPL/EGE-ECO/0886/2021.
2
The author is partially funded by FCT through project “Optimal-Re, Optimal reinsurance withdependencies”, with grant EXPL/EGE-ECO/0886/2021.

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EXPLORING THE IMPACT OF COVID-19 ON
TOURISM DEMAND FORECASTS: EVIDENCE
FROM PORTUGAL
N. Sobreira 1
ISEG Lisbon School of Economics & Management, Universidade de Lisboa; REM Research in Economics and Mathematics, CEMAPRE
nsobreira@iseg.ulisboa.pt
R. Mueller
ISEG Lisbon School of Economics & Management, Universidade de Lisboa

Time series models have proven to be powerful tools for forecasting tourism demand.
However, the recent Covid-19 outbreak has severely impacted the tourism industry, andmodels which provided reasonably accurate forecasts may no longer display similar performance. This work aims to further analyse this situation by exploring the impact of
the Covid-19 pandemic on the ability of time series models to forecast tourism demand.
We perform a forecast competition using regional data from Portugal and analyse themost relevant changes in forecast performance. We include several modelling approaches
such as simpler forecasting methods applied to seasonally adjusted series, models fromthe (seasonal) ARIMA and Neural Network classes, different developments from the ETS
framework, and schemes that combine models included in the competition.
We find that the relatively stable seasonal patterns observed before the pandemic did
not persist in 2020 but were quickly recovered in 2021 and 2022. Additionally, the impact
of Covid-19 varied across regions, with regions having higher weights of domestic tourism
showing much lower declines in demand and less changes in seasonality. Although thischange in tourism dynamics was only temporary, we find evidence that this was enoughto cause significant forecast breakdowns in all considered time series methods. Moreover,the intensity of the break in forecast performance was found to differ across models
leading to important changes in model rankings, especially in the regions most affectedduring the Covid-19 period.

1

The author was partially supported by the Project CEMAPRE/REM - UIDB/05069/2020 - financed
by FCT/MCTES through national funds.

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MODELLING FRENCH MORTALITY RATES WITH
STOCHASTIC DIFFERENTIAL EQUATIONS
Daniel dos Santos Baptista
ISEG-School of Economics and Management, Universidade de Lisboa; REM - Research in Economics and Mathematics, CEMAPRE
danielbaptistapt99@gmail.com
Nuno M. Brites
ISEG/UL - Universidade de Lisboa, Department of Mathematics; REM - Research in
Economics and Mathematics, CEMAPRE
nbrites@iseg.ulisboa.pt
In recent years, the increasing life expectancy of the world’s population combinedwith a sharp decrease in birth rates over time, has proven to be a challenging problem
for governments worldwide (particularly in developed countries). Both of these factors
put at risk the sustainability of state-funded welfare programs (e.g., social security) andalso lead to a decrease in available workforce and tax revenue (including social benefit
contributions) in the near future. With the tendency for these problems to worsen in thenext decades, it is of paramount importance to estimate the extension of human life inorder to analyse the severity of this phenomenon. Stochastic differential equations havebeen used recently to model the evolution of mortality rates. In fact, such models havesome advantages when compared to the deterministic ones since we can input random
environmental fluctuations and evaluate the uncertainty in the forecasts. The main goalof this work is to apply and compare mortality stochastic differential equations models
separately for each age and sex and forecast French mortality rates until the year 2030.

Keywords: Mortality rates, Geometric Brownian motion, Stochastic Gompertz model,
Stochastic differential equations, Forecasting, Life insurance

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PANEL DATA MINIMUM DISTANCE MODE
REGRESSION
João Vieira
ISEG, Lisbon School of Economics and Management, Universidade de Lisboa, Rua do
Quelhas 6, 1200-781 Lisboa, Portugal; REM – Research in Economics and Mathematics,
CEMAPRE.
jvieira@iseg.ulisboa.pt
Paulo Parente
ISEG, Lisbon School of Economics and Management, Universidade de Lisboa, Rua do
Quelhas 6, 1200-781 Lisboa, Portugal; REM – Research in Economics and Mathematics,
CEMAPRE.
pparente@iseg.ulisboa.pt
We propose a new fixed effects estimator for the conditional mode for panel datausing a minimal distance approach, for the case in which the dependent variable has
a continuous conditional density with a well-defined global mode. We called this novel
estimator of minimum distance modal regression (M DM R) estimator. We establish
consistency and explicitly derive the limiting distribution of the M DM R estimator forpanels with large number of cross-sections and time-series under both sequential and joint
limits, and show that our estimator converges to the normal distribution at least as fastas other estimators already proposed. The proposed estimator is easy to implement andthe Monte Carlo simulations for finite sample suggest that our proposed estimator may beaffected by the skewness and the bandwith. We found that using a smaller penalty whenchoosing the value for the bandwith (larger value) leads to an unbiased estimator withsmall variances. Finally, we illustrate the use of the estimator with a simple application
to the impact of Foreign Direct Investment (FDI) on economic growth in the seven majoradvanced nations (the G7 countries).

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FRACTIONAL RESPONSES WITH SPATIAL
DEPENDENCE
Luís Silveira Santos 1
REM – Research in Economics and Mathematics, CEMAPRE
lsantos@iseg.ulisboa.pt

This paper introduces a specification to estimate models for spatially dependent fractional responses with additive errors. This specification, the Fractional Spatial Lag model(FSLM), extends the approach of [1], incorporating spatial dependence through the dependent variable. Moreover, it allows to handle observations at the boundaries, zero andone. Estimation is addressed by the Generalized Method of Moments, with well-known
instruments. A second specification is introduced, based on a series expansion of the
FSLM around the spatial lag parameter equal to zero, the approximated FSLM. This
procedure allows to obtain an approximated reduced form for the spatially dependent
fractional response, simplifying the estimation of the partial effects. The approximated
FSLM is also estimated by the Generalized Method of Moments. Robust inference isstandard. Closed formula expressions for the partial effects are deduced for both models.
An extensive Monte Carlo simulation study is presented to investigate the finite sample
properties of the estimators for the two approaches and the corresponding partial effects.
Experiments show that the estimators for the spatial lag parameter and the regression
coeﬀicients perform well in terms of bias and root mean square error for a great varietyof sampling designs. However, the estimates for the spatial lag parameter on the approximated FSLM exhibit a small bias when the true value is close to one, as expected.
Nevertheless, the estimation of the partial effects is much less affected.

References[1] Papke, L.E., Wooldridge, J.M., 1996. Econometric methods for fractional response
variables with an applicationto 401 (k) plan participation rates. Journal of Applied
Econometrics 11, 619–632;[2] Ramalho, E.A., Ramalho, J.J., 2017. Moment-based estimation of nonlinear regression models with boundaryoutcomes and endogeneity, with applications to nonnegative and fractional responses. EconometricReviews 36, 397–420;[3] 397–420.Ramalho, E.A., Ramalho, J.J., Henriques, P.D., 2010. Fractional regression
models for second stage DEAeﬀiciency analyses. Journal of Productivity Analysis
34, 239–255;[4] Ramalho, E.A., Ramalho, J.J., Murteira, J.M., 2011. Alternative estimating andtesting empirical strategiesfor fractional regression models. Journal of Economic Surveys 25, 19–68;
1

The author gratefully acknowledge the Portuguese Foundation for Science and Technology (Fundação
para a Ciência e a Tecnologia) for the financial support through national funds (CEMAPRE/REM UIDB/05069/2020).

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SDE 2023
First SDE - Stochastic Days Encounters[5] Silveira Santos, L., Proenc¸a, I., 2019. The inversion of the spatial lag operator in binary choice models: Fastcomputation and a closed formula approximation. Regional
Science and Urban Economics 76, 74–102.Spatial Econometrics: New Methods andApplications;[6] Xu, X., Lee, L.f., 2015. A spatial autoregressive model with a nonlinear transformation of the dependentvariable. Journal of Econometrics 186, 1–18.

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DIGIT ANALYSIS USING BENFORD’S LAW: A
BAYESIAN APPROACH
Pedro Fonseca
ISEG Lisbon School of Economics and Management
ptfonseca@iseg.ulisboa.pt
Rui Paulo
ISEG Lisbon School of Economics and Management
rui@iseg.ulisboa.pt
Naturally-occurring collections of numbers are known to often exhibit a logarithmically decaying pattern in the frequencies of leading digits, known as Benford’s law, thatcan be used to screen datasets for anomalies like erroneous or fraudulent data. However,assessing conformance to Benford’s law usually requires testing point null hypotheses,and classical significance tests of fixed dimension are known to over-reject point null hypotheses in large samples due to the high levels of power they attain, as the acceptanceregion shrinks with sample size, hence being prone to high false-positive rates. Thiscan result in suspicions being unduly raised in a large proportion of datasets. As analternative, we address digit analysis within the Bayesian hypothesis testing framework.
An empirical application with macroeconomic statistics from Eurozone countries demonstrates the applicability of the suggested methodology and explores the conflict between
classical and Bayesian measures of evidence in the context of digit analysis. We foundthat classical tests often reject conformance to Benford’s law in situations in which theBayesian measures of evidence suggest otherwise, and that even lower bounds on theBayesian measures often provide more evidence in favour of Benford’s law than what
p-values on classical test statistics seem to suggest.

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STOCHASTIC OPTIMAL CONTROL PROBLEM
WITH A GENERALIZED LOGISTICHARVESTING GROWTH MODEL
Guillermo Caballero Luján
Instituto Español de Oceanografia (IEO)
guille11caballero11@gmail.com
As an intern at the IEO, a biological research center, I used stochastic differential
equations to study the growth dynamics of a harvested fish population under constanteffort. I used this growth dynamics in a stochastic optimal control problem to compare
population estimates with those obtained by other methods at the IEO, and to provide
an estimated profit value that can be presented to the government to negotiate thepreservation of flora and fauna. Additionally, I attempted to unify the fish planning of
two or more similar species into one problem, studying the continuity of the solutions
with respect to the parameters of the stochastic differential equations. Most of this work
is based on papers [1], [2] and [3].

References[1] Nuno M. Brites , Carlos A. Braumann (2017).Fisheries management in random
environments: Comparison of harvesting policies for the logistic model. In Fisheries
Research.
[2] Bernhard Schmelzer (2010). On solutions of stochastic differential equations withparameters modeled by random sets. In International Journal of Approximate Reasoning.
[3] Ratneesh Suri (2008). Optimal Harvesting Strategies for Fisheries: A Differential
Equations Approach.

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SPEAKERS
Nilton Ávido, 29
Carlos A. Braumann, 2Imme van den Berg, 27
Neeraj Bhauryal, 31
Daniel dos Santos Baptista, 34
Tomás Caraballo, 14
José Carlos Dias, 4Cristina Dias, 16
Irada Dzhalladova, 20
M. L. Esquível, 22
Zhaozhi Fan, 18
Pedro Fonseca, 38
Michael Grinfeld, 5
Wenceslao González-Manteiga, 6João Guerra, 11
Guillermo Caballero Luján, 39
Alexandra Moura, 32
Carlos Oliveira, 25
Paula Patrício, 7Miguel M. Reis, 28
Alexandra A. Silva, 8Carla Santos, 15
N. Sobreira, 33
Luís Silveira Santos, 36João Vieira, 35
Laura Wise, 8

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AUTHORS
Nilton Ávido, 29

Paulo Parente, 35
Rui Paulo, 38

Carlos A. Braumann, 2Imme van den Berg, 27
Nuno M. Brites, 28, 34
Neeraj Bhauryal, 31
Daniel dos Santos Baptista, 34

Miguel M. Reis, 28
Alexandra A. Silva, 8Carla Santos, 15, 16
A. V. Shamraeva, 22
N. Sobreira, 33
Luís Silveira Santos, 36

Rod Cross, 5Tomás Caraballo, 14
Tomasz Czyżycki, 20
Ana Bela Cruzeiro, 31

Asokan Variyath, 18
Chithran Vasudevan, 18João Vieira, 35

José Carlos Dias, 4Cristina Dias, 15, 16
Irada Dzhalladova, 20

Andrew Wade, 5
Laura Wise, 8

M. L. Esquível, 22
Patrícia A. Filipe, 2Manuel Febrero-Bande, 6
Zhaozhi Fan, 18
Pedro Fonseca, 38
Michael Grinfeld, 5
Wenceslao González-Manteiga, 6João Guerra, 11
Gonçalo Jacinto, 2
N. P. Krasii, 22
Igor Kravechenko, 25
Alejandra López-Pérez, 6
Guillermo Caballero Luján, 39João T. Mexia, 15, 16
P. P. Mota, 22
Paula Milheiro-Oliveira, 29
Alexandra Moura, 32
R. Mueller, 33
Célia Nunes, 15
Cláudia Nunes, 25
Carlos Oliveira, 25, 31, 32
Paula Patrício, 742