<p dir="ltr">This thesis presents results in Optimal Transport theory and applications to unsuper-
vised statistical learning and robust asset pricing. In unsupervised learning applications,
we assume that we observe the distribution of some data of interest which might be too
big in size, have a high-dimensional structure or be polluted with noise. We investigate the
construction of an optimal distribution that precedes the given data distribution in convex
order, which means that the given distribution is a dispersion of it. The intention is to
use this construction to estimate a concise, lower-dimensional or unpolluted version of the
given data. We provide existence and convergence results and show that popular methods
including k-means and principal curves can be unified under this model. We further investi-
gate a relaxation of the order relation that leads to similar results in terms of existence and
convergence and broadens the range of applications to include e.g. the Principal Compo-
nent Analysis and the Factor model. We relate the two versions and show that the relaxed
problem can be described as a bilinear optimization with a tractable computational method.
As examples, we apply our method to generate fixed-weight k-means, principal curves with
bounded curvature that are actual generalizations of PCA, and a latent factor structure
in a classical Gaussian setting. In robust finance applications, we investigate the Vectorial
Martingale Optimal Transport problem, the geometry of its solutions, and its application to
model-free asset pricing. We consider a multi-asset, two-period contract pricing model and
show that the solution to this problem with a sub or supermodular payoff function reduces
to a single factor in the first period in the case of two underlying assets (d = 2), but not in
general for a greater number of assets. This result for d = 2 enables the construction of a
joint distribution of prices at the first period from market data, which adds information to
the model-free pricing method and reduces the computational dimensionality. We provide
an improved version of an existing pricing method and show numerical evidence of increased
accuracy.</p>
| Identifer | oai:union.ndltd.org:purdue.edu/oai:figshare.com:article/26364592 |
| Date | 30 July 2024 |
| Creators | Marcelo Cruz de Souza (19207069) |
| Source Sets | Purdue University |
| Detected Language | English |
| Type | Text, Thesis |
| Rights | CC BY 4.0 |
| Relation | https://figshare.com/articles/thesis/Optimal_Transport_Theory_and_Applications_to_Unsupervised_Learning_and_Asset_Pricing/26364592 |
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