Optimal and Robust Control for a Class of Nonlinear Stochastic Systems

This thesis focuses on theoretical research of optimal and robust control theory for a class of nonlinear stochastic systems. The nonlinearities that appear in the diffusion terms are of a square-root type. Under such systems the following problems are investigated: optimal stochastic control in both finite and infinite horizon; robust stabilization and robust H∞ control; H₂/H∞ control in both finite and infinite horizon; and risk-sensitive control. The importance of this work is that explicit optimal linear controls are obtained, which is a very rare case in the nonlinear system. This is regarded as an advantage because with explicit solutions, our work becomes easier to be applied into the real problems. Apart from the mathematical results obtained, we have also introduced some applications to finance

The History of the Quantitative Methods in Finance Conference Series. 1992-2007

This report charts the history of the Quantitative Methods in Finance (QMF) conference from its beginning in 1993 to the 15th conference in 2007. It lists alphabetically the 1037 speakers who presented at all 15 conferences and the titles of their papers.

Control and estimation with limited information: a game-theoretic approach

Modern control systems can be viewed as interconnections of spatially distributed multiple subsystems, where the individual subsystems share their information with each other through an underlying network that inherently introduces limitations on information flow. Inherent limitations on the flow of information among individual subsystems may stem from structural constraints of the network and/or communication constraints of the network. Hence, in order to design optimal control and estimation mechanisms for modern control systems, we must answer the following two practical but important questions: (1) What are the fundamental communication limits to achieve a desired control performance and stability? (2) What are the approaches one has to adopt to design a decentralized controller for a complex system to deal with structural constraints? In this thesis, we consider four different problems within a game-theoretic framework to address the above questions. The first part of the thesis considers problems of control and estimation with limited communication, which correspond to question (1) above. We first consider the minimax estimation problem with intermittent observations. In this setting, the disturbance in the dynamical system as well as the sensor noise are controlled by adversaries, and the estimator receives the sensor measurements only sporadically, with availability governed by an independent and identically distributed (i.i.d.) Bernoulli process. This problem is cast in the thesis within the framework of stochastic zero-sum dynamic games. First, a corresponding stochastic minimax state estimator (SMSE) is obtained, along with an associated generalized stochastic Riccati equation (GSRE). Then, the asymptotic behavior of the estimation error in terms of the GSRE is analyzed. We obtain threshold-type conditions on the rate of intermittent observations and the disturbance attenuation parameter, above which 1) the expected value of the GSRE is bounded from below and above by deterministic quantities, and 2) the norm of the sequence generated by the GSRE converges weakly to a unique stationary distribution. We then study the minimax control problem over unreliable communication channels. The transmission of packets from the plant output sensors to the controller, and from the controller to the plant, are over sporadically failing channels governed by two independent i.i.d. Bernoulli processes. Two different scenarios for unreliable communication channels are considered. The first one is when the communication channel provides perfect acknowledgments of successful transmissions of control packets through a clean reverse channel, which is the TCP (Transmission Control Protocol), and the second one is when there is no acknowledgment, which is the UDP (User Datagram Protocol). Under both scenarios, the thesis obtains output feedback minimax controllers; it also identifies a set of explicit existence conditions in terms of the disturbance attenuation parameter and the communication channel loss rates, above which the corresponding minimax controller achieves the desired performance and stability. In the second part of the thesis, we consider two different large-scale optimization problems via mean field game theory, which address structural constraints in the complex system stated in question (2) above. We first consider two classes of mean field games. The first problem (P1) is one where each agent minimizes an exponentiated performance index, capturing risk-sensitive behavior, whereas in the second problem (P2) each agent minimizes a worst-case risk-neutral performance index, where a fictitious agent or an adversary enters each agent's state system. For both problems, a mean field system for the corresponding problem is constructed to arrive at a best estimate of the actual mean field behavior in various senses in the large population regime. In the finite population regime, we show that there exist epsilon-Nash equilibria for both P1 and P2, where the corresponding individual Nash strategies are decentralized as functions of the local state information. In both cases, the positive parameter epsilon can be taken to be arbitrarily small as the population size grows. Finally, we show that the Nash equilibria for P1 and P2 both feature robustness due to the risk-sensitive and worst-case behaviors of the agents. In the last main chapter of the thesis, we study mean field Stackelberg differential games. There is one leader and a large number, say N, of followers. The leader holds a dominating position in the game, where he first chooses and then announces his optimal strategy, to which the N followers respond by playing a Nash game. The followers are coupled with each other through the mean field term, and are strongly influenced by the leader's strategy. From the leader's perspective, he is coupled with the N followers through the mean field term. In this setting, we characterize an approximated stochastic mean field process of the followers governed by the leader's strategy, which leads to a decentralized epsilon-Nash-Stackelberg equilibrium. As a consequence of decentralization, we subsequently show that the positive parameter epsilon can be picked arbitrarily small when the number of followers is arbitrarily large. In the thesis, we also include several numerical computations and simulations, which illustrate the theoretical results

Games in Energy Markets

We study energy markets in game theoretic framework. The energy markets consist of two types of energy producers: exhaustible producer and renewable producer. An exhaustible producer produces energy with exhaustible resources, such as oil. The resource reserves of each exhaustible producer diminish due to production, and also get replenished with costly effort to explore for new resources. This exploration activity is modeled through a controlled point process that leads to stochastic increments to reserves level. A renewable producer uses renewable resources, such as solar power, to produce energy. The renewable resources are infinite, but costly in production. Each producer chooses optimal controls of production quantity and exploration effort (exhaustible producers only), in order to maximize individual profit that equals his quantity of production multiplied by market price, minus costs of production and exploration. The producers interact with each other through the energy price that is a function of aggregate production, as one's profit does not only depend on his own production quantity, but also depends on the total quantity of all other producers. We aim to study the equilibrium total production and price. In Chapter 2 we study the game between an exhaustible producer and a renewable producer under stochastic demand that switches between different regimes. We study how the regime changes and the relative cost of production, which is a proxy for market competitiveness, affect game equilibria, and compare with the case of deterministic demand. A novel feature driven by stochasticity of demand is that production may shut down during low demand to conserve reserves. In Chapter 3 we study game with a continuum of homogeneous exhaustible producers. Mean field game approach is employed to solve for an approximate Markov Nash equilibrium of the game. We develop numerical schemes to solve the resulting system of partial differential equations: a backward Hamilton-Jacobi-Bellman (HJB) equation for the game value function of a representative producer and a forward transport equation for the distribution of the reserves levels among all producers.In Chapter 4 we study a time-stationary mean field game model, in which the reserves level remains invariant due to the counteracting effects of production and exploration. We also study the impact of uncertainty in the regime that the exploration process becomes asymptotically deterministic, so that discovery of new resources happens at high frequency with small amount of each discovery

Topics in Financial Math (Uncertain Volatility, Ross Recovery and Mean Field Games on Random Graph)

In this thesis, we discuss three new topics in Financial Mathematics using partial differential equation (PDE): uncertain volatility with stochastic bounds, Ross recovery with multivariate driving states and mean field games on the Erdos Renyi random graph, in three chapters respectively. In Chapter 1, we study a class of uncertain volatility models with stochastic bounds, over which volatility stays between two bounds, but instead of using two deterministic bounds, the uncertain volatility fluctuates between two stochastic bounds generated by its inherent stochastic volatility process. We then apply a regular perturbation analysis upon the worst-case scenario price, and derive the first order approximation in the regime of slowly varying stochastic bounds. The original problem which involves solving a fully nonlinear PDE in dimension two for the worst-case scenario price, is reduced to solving a nonlinear PDE in dimension one and a linear PDE with source, which gives a tremendous computational advantage.In Chapter 2, we address the problem of recovering the real world probability distribution from observed option prices by avoiding the intensively debated transition independence, through placing structure on the dynamics of the numeraire portfolio in a preference-free manner. We firstly utilize the Ito and Feynman-Kac theorem to derive a a uniformly elliptic operator, whose inverse is a compact linear operator, based on boundary conditions, and then apply the Krein-Rutman theorem to guarantee the uniqueness of the positive eigenfunction, which happens to generate the physical transition probability. In Chapter 3, we analyze a model of inter-bank lending and borrowing, by means of mean field games on the Erdos Renyi random graph. An open-loop Nash equilibrium is obtained using a system of fully coupled forward backward stochastic differential equations (FBSDEs), whose unique solution leads to the master equation. We explore the approximation to the finite player game equilibrium through a decoupled system of diffusion equations generated by the master equation under frozen graph, and through a weakly interacting particle system on random graph generated by the master equation under random graph, respectively

Closed-Loop Nash Equilibrium in the Class of Piecewise Constant Strategies in a Linear State Feedback Form for Stochastic LQ Games

In this paper, we examine a sampled-data Nash equilibrium strategy for a stochastic linear quadratic (LQ) differential game, in which admissible strategies are assumed to be constant on the interval between consecutive measurements. Our solution first involves transforming the problem into a linear stochastic system with finite jumps. This allows us to obtain necessary and sufficient conditions assuring the existence of a sampled-data Nash equilibrium strategy, extending earlier results to a general context with more than two players. Furthermore, we provide a numerical algorithm for calculating the feedback matrices of the Nash equilibrium strategies. Finally, we illustrate the effectiveness of the proposed algorithm by two numerical examples. As both situations highlight a stabilization effect, this confirms the efficiency of our approach.1 Decembrie 1918 Universit