We study portfolio choice when both interacting fast and slow components of expected returns are latent and the investor observes only prices. Starting from the full price filtration, we derive rather than impose the price-history statistic used in learning and optimal investment. At finite times, the filtered fast factor decomposes exactly into a primitive-rate generalized MACD-type component–a signed difference of two exponential moving average (EMA)-type price filters–a closed-loop deterministic response, and a linear Volterra filtering-feedback correction driven by that component. At steady filtering-error covariance, its minimal memory is instead governed by the uncancelled closed-loop Kalman poles. After separating the deterministic response, distinct real poles yield a signed difference of two contemporaneous-price-plus-EMA processes; we characterize exact one-mode cancellations, the restrictions producing conventional MACD, and the repeated- and complex-pole alternatives. The complete feedback-corrected estimate enters optimal trading demand through conditional expected drift: together with observed price and wealth, it completely determines logarithmic risky exposure and determines the common myopic component under power and exponential utility, whose hedging terms may additionally depend on the full filtered state. We verify the resulting controls on every finite horizon. Deterministic population experiments in illustrative economies quantify when simpler exponential signals preserve portfolio value, how much price history matters over finite horizons, and when intertemporal hedging makes the filter-based myopic core incomplete for wealth-normalized exposure under power utility or demand under exponential utility.

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.