1. 1
The main aim of this thesis is to introduce an algorithm to asses optimal weighted distri-
butions as the solution of an optimization problem. In particular we want to minimize the
following distance
d(F, FX) = EP
R
F(u) − FX(u)
2
du (1)
where F = N
i=1 wi1{Y i
T ≤u} is a weighted sample distribution, {Y i
T }N
i=1 is a vector of
observations, FX is a target distribution, wi ∈ ∆N
are the object of our minimization
and ∆N
= w ∈ RN
: wi ≥ 0 for any i = 1, ..., N and N
i=1 wi = 1 . Using the optimal
weights that we obtain from the optimization problem we will built the iterative algorithm
that we will use to perform the optimal weighted distributions. We will apply our algorithm
to the case of two populations Ω1 and Ω2 with assigned distribution functions F1 and F2.
Since the optimal weights depends on distribution functions F1, F2 and FX, we have to
impose an initial guess for the target distribution function FX. The distinctive features
of the method we have developed consist in the fact that the underlying mathematical
model is not known a priori and in the fact that the optimal weighted distributions does
not depend on the initial guess. With this new approach the optimal weighted distribution
is updated as new information becomes available, since we can update the distribution
functions F1 and F2. As population Ω1 and Ω2 we will consider some data from specific
financial time series, in particular we will consider Ω2 as ”stress population”.
A stress test is a simulation technique used on asset and liability portfolios to determine
their reactions to different financial situations, in general it is a useful method
• for determining how a portfolio will fare during a period of financial crisis.
• to evaluate how certain stressors will affect a company or industry.
• to evaluate the strength of institutions, for example a bank stress test is an analy-
sis conducted under unfavorable economic scenarios which is designed to determine
whether a bank has enough capital to withstand the impact of adverse developments.
It can either be carried out internally by banks as part of their own risk management,
or by supervisory authorities as part of their regulatory oversight of the banking sec-
tor.
In the particular case of bank stress tests, they are meant to detect weak spots in the
banking system at an early stage, so that preventive action can be taken by the banks
and regulators. These tests are usually computer-generated simulation models that test
hypothetical scenarios and they focus on a few key risks, such as credit risk, market risk,
and liquidity risk, to test banks financial health in crisis situations. The results of these
tests depend on the assumptions made in various economic scenarios, which are described
as ”unlikely but plausible.”
We give now a description of the principal arguments of each chapter.
In Chapter 1 we will review some theoretical results related to the Strong Law of
Large Number and the Glivenko-Cantelli Theorem., in particular we will consider the

2. 2
works of Tuker (1959) and Koul (1970). The work of Tuker (1959) can be considered as a
generalization of the theorem of Glivenko-Cantelli, where the assumption of independence
is replaced by strictly stationarity and ergodicity of the sequence of random variables. We
will also present the work of Ursula et al (2009) that we use as a starting point about our
study on weighted distributions.
In Chapter 2 we will presents our theoretical results about the minimization of problem
(1). Section 2.1 deals with the case of two populations Ω1 and Ω2 with assigned distribution
functions F1 and F2. In this particular case we will assume the weighted sample distribution
given by
F = F1 + F2 =
n1
i=1
w11{Wi
T ≤u} +
n2
j=1
w21{Zj
T ≤u}
where F1 and F2 are the sample distributions of the two populations Ω1 and Ω2, {Wi
T }n1
i=1
and {Zj
T }n2
j=1 are the sequences of observations in the two populations, with n1 and n2 the
total number of observations in each population, and w1, w2 ∈ [0, 1] are the two weights,
with n1w1 + n2w2 = 1, that we want to find. We will assume that the observations
inside each population Ω1 (or Ω2) are i.i.d but we will not impose any conditions on the
observations between the two populations. In section 2.2 we will extend the previous results
to the general case of N populations, in particular, contrary to the case of two populations,
we will obtain an implicit expression for the optimal weights.
Chapter 3 starts we the study of particular financial time series (FTSE100 index, Google
stock quotes and Mediobanca Milano stock quotes) and with the application of some well
known techniques about time series. We will also present the structure of our algorithm
with some theoretical considerations, and we will apply this algorithm to the time series
considered. For the numerical studies we will consider, for simplicity, the case of two
populations and we will assume the two populations uncorrelated. Given a time interval
[0,T], where T ≥ 1 is the expiration time, we will assume to know the evolution of a time
series from time zero to an arbitrary time ˆt, 0 ≤ ˆt < T. For simplicity we will assume
to have a known underlying model on the time interval [0, ˆt] and to have uncertainty in
period [ˆt, T]. We will also suppose to have an initial guess for the target distribution FX
that we will call F0
X.
Considering the optimal weights w1 and w2 as function of the distribution Fi
X, the iterative
method that we will obtain is the following
Fi+1
X (u) = w1 Fi
X F1(u) + w1 Fi
X F2(u), u ∈ R for i = 0, 1, 2, . . . ,
where F1 and F2 are the distribution functions of the populations Ω1 and Ω2. In the
final part of the chapter we will present the numerical results and we will show that the
algorithm does not depend on the initial guess for the target distribution function and
there exists a limit distribution function, given by precise optimal weights.