In the first part of this thesis, we study the non-parametric

methods for estimation and optimization. In particular, a new

non-parametric method, objective operational statistics, is proposed

to inventory control problems, where the only information available

is the sample data and we do not assume any relationship between

demands and order quantities. A kernel algorithm based on objective

operational statistics is constructed to approximate the objective

function directly from sample data. Moreover, we give conditions

under which the operational statistics approximation function

converges to the true objective. Numerical results of the algorithm

with applications to newsvendor problem show that the objective

operational statistics approach works well for small amount of data

and outperforms the previous parametric and non-parametric methods.

In the second part of this thesis, we present a robust hedging

problem under model uncertainty and the bounds of the optimal

objective value are derived by duality analysis.

Modern engineering and social systems are often too complex to be managed by a centralized agent. Instead, such systems are commonly structured with multiple decentralized agents each responsible for managing a subset of the system, but the resulting system performance depends on the aggregate of the decisions made by decentralized agents. Local agents' decision makings often exhibit selfish behavior as they seek to optimize their own objectives under their localized models, which if left uncoordinated can lead to substantial loss of efficiency compared with the system that can be optimized by a single (hypothetical) centralized agent. In this dissertation, we seek to study the fundamental issues of how to efficiently manage large-scale and multi-agent stochastic dynamic systems, especially on how to device efficient coordination mechanisms that would optimize system performance under various constraints that are unique to decentralized systems.

In the first part of this dissertation we study decentralized control of a general class of stochastic dynamic resource allocation problems that have many applications. We consider a stochastic system in which multiple decentralized agents allocate shared system resources in response to customer requests that arrive stochastically over time. Each agent is responsible for a subset of the allocation decisions which it makes according to a dynamic allocation policy obtained by maximizing his own expected profit subject to a potentially mis-specified model of the way in which shared resources are consumed by other agents. We introduce the notion of a transfer contract which specifies how agents compensate one another whenever resources are consumed and establish the existence of contracts under which the decentralized system has no efficiency loss relative to centralized optimality. We also show that this property is insensitive to mis-specification by each agent of the dynamics of resource consumption by others in the system. An explicit characterization of the optimal transfer contract and an iterative decentralized algorithm for computing it is also provided. In the language of duality, contracts are analogous to shadow prices and the iterative algorithm has the favor of a dual update method, but strong duality and convergence of the iterative algorithm to the set of optimal contracts are guaranteed without assumptions of convexity.

In the second part of this dissertation we study a class of related decentralized control problems but specialize to portfolio and risk management. Many financial institutions typically trade in multiple correlated markets. While centralized portfolio optimization over all trading decisions is ideal, it is generally not possible due to the complexity of each market, and firms typically adopt a decentralized setup in which trading in each market the responsibility of a particular desk. Decentralized portfolio optimization, however, is complicated by the fact that different agents are commonly only well informed about their own investment universe (proprietary research and forecasts, etc) and prefer to keep this private, and have their own incentives which they optimize on the basis of their limited models. It is well known, however, that the aggregate performance of such a system can be extremely inefficient due to the loss of diversification. In this dissertation, we formulate a multi-agent dynamic portfolio choice problem and study how to improve its efficiency. We show that an internal system of swap contracts, which define internal cash transfers between agents, can be used to facilitate risk sharing and induce agents to choose portfolios that as a collection are optimal for the firm. Conceptually using swap contracts is similar to performance benchmarking that is often employed in the finance literature for decentralized portfolio management, but our new approach offers a significant advantage in that the swap contracts can be constructed in decentralized manner without requiring an all-knowing central agent. We provide an explicit characterization of the optimal swap contracts and an iterative algorithm for computing them that can be implemented without compromising proprietary agent level data.

Throughout this dissertation, we also discuss various important issues surrounding decentralized control of stochastic dynamic systems, including but not limited to approximation methods, performance attribution, sensitivity analysis, and fairness issues, etc.

We study two important generalizations of dynamic portfolio choice problems: a portfolio choice problem with market impact costs and a portfolio choice problem under the Hidden Markov Model.

In the first problem, we allow the presence of market impact and illiquidity. Illiquidity and market impact refer to the situation where it may be costly or difficult to trade a desired quantity of assets over a desire period of time. In this work, we formulate a simple model of dynamic portfolio choice that incorporates liquidity effects. The resulting problem is a stochastic linear quadratic control problem where liquidity costs are modeled as a quadratic penalty on the trading rate. Though easily computable via Riccati equations, we also derive a multiple time scale asymptotic expansion of the value function and optimal trading rate in the regime of vanishing market impact costs. This expansion reveals an interesting but intuitive relationship between the optimal trading rate for the illiquid problem and the classical Merton model for dynamic portfolio selection in perfectly liquid markets. It also gives rise to the notion of a liquidity time scale. Furthermore, the solution to our illiquid portfolio problem shows promising performance and robustness properties.

In the second problem, we study dynamic portfolio choice problems under regime switching market. We assume the market follows the Hidden Markov Model with unknown transition probabilities and unknown observation statistics. The main difficulty of this dynamic programming problem is its high-dimensional state variables. The joint probability density function of the hidden regimes and the unknown quantities is part of the state variables, and this makes the problem suffer from the curse of dimensionality. Though the problem cannot be solved by any standard fashions, we propose approximate methods that tractably solve the problem. The key is to approximate the value function by that of a simpler problem where the regime is not hidden and the parameters are observable (the C-problem). This approximation allows the optimal portfolio to be computed in a semi-explicit way. The approximate solution shares the same structure with the solution of C-problem, but at the same time it provides clear insight into the unobservable extension. In addition, the performance of the proposed methods is reasonably close to the upper-bound obtained from the information relaxation problem.

We develop two dynamic Bayesian portfolio allocation models that address questions of learning and model uncertainty by taking model-specific shortcomings into account.

In our first model, we formulate a multi-period portfolio choice problem in which the investor is uncertain about parameters of the model, can learn these parameters over time from observing asset returns, but is also concerned about robustness. To address these concerns, we introduce an objective function which can be regarded as a Bayesian version of relative regret. The optimal portfolio is characterized and shown to involve a ``tilted'' posterior, where the tilting is defined in terms of a family of stochastic benchmarks. We have found this model to perform at least as well as a benchmark given the true market parameters, while outperforming it when the market assets have the same trend.

Our next model extends the Black-Litterman portfolio choice model by taking several potential errors into account. We extend Black-Litterman to multiple periods, which allows for us to take into account the pairs of expert forecasts and the realized return. By doing so, we can then perform inference on these experts and discover whether they may have any bias for or against any specific assets. We can also perform similar inference on the market equilibrium distribution, which is typically represented by the capital asset pricing model (CAPM). The result is a model that is analytically intractable but may be solved numerically via Gibbs sampling. Controlled tests show our model performs favorably when Black-Litterman's model assumptions about the market equilibrium and expert views are violated. Backtests shed light on the model's ability to account for CAPM's shortcomings.