Portfolio Optimization under Fast and Slow Latent Mean-Reverting and Momentum Drift

We consider a class of partial information portfolio optimization problems in which a trader does not observe the true drift of a financial asset, driven by two latent stochastic factors operating at distinct time scales. We prove that under logarithmic, exponential, and power utility functions, the optimal strategies depend on the current price and on the difference between fast and slow exponentially weighted moving averages of prices, yielding a Moving Average Convergence Divergence (MACD) signal. The MACD structure is not imposed a priori: optimality is established over all admissible price-history-dependent strategies, and the divergence signal emerges endogenously as a parameter of the unconstrained optimal control.

Stochastic Filtering and Control in Partially Observable Diffusion Models

Optimal control theory studies the selection of admissible control policies that optimize a prescribed value functional subject to given system dynamics. In the stochastic setting, the state evolution is described by a stochastic differential equation, and admissible controls are required to be adapted to the filtration generated by the available information, ensuring that control actions depend only on past and present observations. In this proposal, we are particularly interested in systematic methods for reducing partially observed stochastic control problems to fully observable ones, using tools from stochastic filtering theory. Such problems arise when the controller has only partial information about the underlying system state and must act on the basis of indirect or noisy observations.

C*-Algebras Generated by Weakly Quasi-Lattice Ordered Groups

In 1992 Alexandru Nica published a highly influential paper examining the C^*-algebras generated by a semigroup he calls a quasi-lattice ordered group. Of particular interest to Nica were the C^*-algebras generated by the Toeplitz representation of a quasi-lattice order, and a universally defined C^*-algebra he calls C^*(G, P). Due to the concreteness of the first C^*-algebra and the universal properties of the second, Nica was interested in finding sufficient conditions to determine when these two algebras were isomorphic. He called quasi-lattice orders that satisfied this isomorphism condition amenable. Finding techniques that establish amenability is now a topic at the cutting edge of research in the study of C^*-algebras of semigroups. In this dissertation, we follow the outline set out by Nica’s paper to study the C^*-algebras generated by weakly quasi-lattice ordered groups. Most of the proofs throughout this dissertation required a substantial amount of original work to fill in details omitted by Nica, as he often only offered a proof outline.