OF THE DISSERTATION In this thesis, we propose the Markov tree option pricing model and subject it to large-scale empirical tests against market options and equity data to quantify its pricing and hedging performances. We begin by proposing a tree model that explicitly accounts for the dependence observed in the log-returns of underlying asset prices. The dynamics of the Markov tree model is ex...

We consider algorithms for generating Mixtures of Bagged Markov Trees, for density estimation. In problems defined over many variables and when few observations are available, those mixtures generally outperform a single Markov tree maximizing the data likelihood, but are far more expensive to compute. In this paper, we describe new algorithms for approximating such models, with the aim of spee...

Markov trees generalize naturally to bounded tree-width Markov networks, on which exact computations can still be done efficiently. However, learning the maximum likelihood Markov network with tree-width greater than 1 is NP-hard, so we discuss a few algorithms for approximating the optimal Markov network. We present a set of methods for training a density estimator. Each method is specified by...

Markov trees generalize naturally to bounded tree-width Markov networks, on which exact computations can still be done efficiently. However, learning the maximum likelihood Markov network with tree-width greater than 1 is NP-hard, so we discuss a few algorithms for approximating the optimal Markov network. We present a set of methods for training a density estimator. Each method is specified by...

We prove an apparently novel concentration of measure result for Markov tree processes. The bound we derive reduces to the known bounds for Markov processes when the tree is a chain, thus strictly generalizing the known Markov process concentration results. We employ several techniques of potential independent interest, especially for obtaining similar results for more general directed acyclic ...

The Markov Chain Tree Theorem is a classical result which expresses the stable distribution of an irreducible Markov matrix in terms of directed spanning trees of its associated graph. In this article, we present what we believe to be an original elementary proof of the theorem (Theorem 5.1). Our proof uses only linear algebra and graph theory, and in particular, it does not rely on probability...

We present a Markov mixed membership model (Markov M3) for grouped data that learns a fully connected graph structure among mixing components. A key feature of Markov M3 is that it interprets the mixed membership assignment as a Markov random walk over this graph of nodes. This is in contrast to tree-structured models in which the assignment is done according to a tree structure on the mixing c...

Dempster's rule of evidence combination is computational expensive. This paper presents a parallel approach to evidence combination on a qualitative Markov tree. Binarization algorithm transforms a qualitative Markov tree into a binary tree based on the computational workload in nodes for an exact implementation of evidence combination. A binary tree is then partitioned into clusters with each ...