Photocopy of typescript. / Digitized by Kansas Correctional Industries

Neural systems are often stochastic, non-linear, and non-autonomous. The complex manifestation of these aspects hinders the interpretation of neuroscientific data. Neuroscience thus benefits from the inclusion of theoretical models in its methodology. Detailed biophysical models of neural systems, however, are often plagued by high-dimensional and poorly constrained parameter spaces. As an alternative, data-driven effective models can often explain the core dynamical features of a dataset with few underlying assumptions. By lumping high-dimensional fluctuations into low-dimensional stochastic terms, observed time-series can be well-represented by stochastic dynamical systems. Here, I apply this approach to two datasets from weakly electric fish. The rate of electrosensory sampling of freely behaving fish displays spontaneous transitions between two preferred values: an active exploratory state and a resting state. I show that, over a long timescale, this rate can be modelled with a stochastic double-well system where a slow external agent modulates the relative depth of the wells. On a shorter timescale, however, fish exhibit abrupt and transient increases in sampling rate not consistent with a diffusion process. I develop and apply a novel inference method to construct a jump-diffusion process that fits the observed fluctuations. This same technique is successfully applied to intrinsic membrane voltage noise in pyramidal neurons of the primary electrosensory processing area, which display abrupt depolarization events along with diffusive fluctuations. I then characterize a novel sensory acquisition strategy whereby fish adopt a rhythmic movement pattern coupled with stochastic oscillations of their sampling rate. Lastly, in the context of differentiating between self-generated and external electrosensory signals, I model the sensory signature of communication signals between fish. This analysis provides supporting evidence for the presence of a sensory ambiguity associated with these signals.

Stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs) play an important role in many areas of engineering and applied sciences such as atmospheric sciences, mechanical and aerospace engineering, geosciences, and finance. Equilibrium statistics and long-time solutions of these equations are pertinent to many applications. Typically, these models contain several uncertain parameters which need to be propagated in order to facilitate uncertainty quantification and prediction. Correspondingly, in this thesis, we propose a generalization of the Polynomial Chaos (PC) framework for long-time solutions of SDEs and SPDEs driven by Brownian motion forcing. Polynomial chaos expansions (PCEs) allow us to propagate uncertainties in the coefficients of these equations to the statistics of their solutions. Their main advantages are: (i) they replace stochastic equations by systems of deterministic equations; and (ii) they provide fast convergence. Their main challenge is that the computational cost becomes prohibitive when the dimension of the parameters modeling the stochasticity is even moderately large. In particular, for equations with Brownian motion forcing, the long-time simulation by PC-based methods is notoriously difficult as the dimension of stochastic variables increases with time. With the goal in mind to deliver computationally efficient numerical algorithms for stochastic equations in the long time, our main strategy is to leverage the intrinsic sparsity in the dynamics by identifying the influential random parameters and construct spectral approximations to the solutions in terms of those relevant variables. Once this strategy is employed dynamically in time, using online constructions, approximations can retain their sparsity and accuracy; even for long times. To this end, exploiting Markov property of Brownian motion, we present a restart procedure that allows PCEs to expand the solutions at future times in terms of orthogonal polynomials of the measure describing the solution at a given time and the future Brownian motion. In case of SPDEs, the Karhunen-Loeve expansion (KLE) is applied at each restart to select the influential variables and keep the dimensionality minimal. Using frequent restarts and low degree polynomials, the algorithms are able to capture long-time solutions accurately. We will also introduce, using the same principles, a similar algorithm based on a stochastic collocation method for the solutions of SDEs. We apply the methods to the numerical simulation of linear and nonlinear SDEs, and stochastic Burgers and Navier-Stokes equations with white noise forcing. Our methods also allow us to incorporate time-independent random coefficients such as a random viscosity. We propose several numerical simulations, and show that the algorithms compare favorably with standard Monte Carlo methods in terms of accuracy and computational times. To demonstrate the efficiency of the algorithms for long-time simulations, we compute invariant measures of the solutions when they exist.

Cette thèse porte sur l'étude de quelques problèmes de contrôle stochastique dans un contexte de risque de liquidité et d'impact sur le prix des actifs. La thèse se compose de quatre chapitres.Dans le deuxième chapitre, on propose une modélisation d'un problème d'animation de marché dans un contexte de risque de liquidité en présence de contraintes d'inventaire et de changements de régime. Cette formulation peut être considérée comme étant une extension de précédentes études sur ce sujet. Le résultat principal de cette partie est la caractérisation de la fonction valeur comme solution unique, au sens de la viscosité, d'un système d'équations d'Hamilton-Jacobi-Bellman . On enrichit notre étude par la donnée de quelques résultats numériques.Dans le troisième chapitre, on propose un schéma d'approximation numérique pour résoudre un problème d'optimisation de portefeuille dans un contexte de risque de liquidité et d'impact sur le prix des actifs. On montre que la fonction valeur peut être obtenue comme limite d'une procédure itérative dont chaqueitération représente un problème d'arrêt optimal et on utilise un algorithme numérique, basé sur la quantification optimale, pour calculer la fonction valeur ainsi que la politique de contrôle. La convergence du schéma numérique est obtenue via des critères de monotonicité, stabilité et consistance.Dans le quatrième chapitre, on s'intéresse à un problème couplé de contrôle singulier et de contrôle impulsionnel dans un contexte d'illiquidité. On propose une formulation mathématique pour modéliser la distribution de dividendes et la politique d'investissement d'une entreprise sujette à des contraintes de liquidité. On montre que, sous des coûts de transaction et un impact sur le prix des actifs illiquides, la fonction valeur de l'entreprise est l'unique solution de viscosité d'une équation d'Hamilton-Jacobi-Bellman. On propose aussi une méthode numérique itérative pour calculer la stratégie optimale d'achat, de vente et de distribution de dividendes. / The purpose of this thesis is to study some stochastic control problems with liquidity risk and price impact. The thesis contains four chapters.The second chapter is devoted to the modeling aspects of a market making problem in a liquidity risk framework under inventory constraints and switching regimes. This formulation can be seen as an extension of previous studies on this subject. The main result is the characterization of the value functions as the unique viscosity solutions to the associated Hamilton-Jacobi-Bellman system. We further enrich our study with some numerical results.In the third section, we introduce a numerical scheme to solve an impulse control problem under state constraints arising from optimal portfolio selection under liquidity risk and price impact. We show that the value function could be obtained as the limit of an iterative procedure where each step is an optimal stopping problem and we use a numerical approximation algorithm based on quantization procedure to compute the value function and the optimal policy. The main result is to prove the convergence of our numerical scheme using monotonicity, stability and consistency properties.In the fourth section, we study a mixed singular and impulse control problem with liquidity risks and constraints. We propose a mathematical modeling to the dividend and investment policy of a firm operating under uncertain environment and liquidity risks. Our main contribution is to show that, under transaction costs and impact on the illiquid asset price, the firm's value function is the unique viscosity solution of a certain Hamilton-Jacobi-Bellman equation. We also formulated an iterative numerical procedure to compute the optimal dividend and investment policy.

Graduation date: 1995

Thesis (Ph. D.)--University of Hong Kong, 2006. / Title proper from title frame. Also available in printed format.

Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.

Stochastic dynamic programming is a recursive method for solving sequential or multistage decision problems. It helps economists and mathematicians construct and solve a huge variety of sequential decision making problems in stochastic cases. Research on stochastic dynamic programming is important and meaningful because stochastic dynamic programming reflects the behavior of the decision maker without risk aversion; i.e., decision making under uncertainty. In the solution process, it is extremely difficult to represent the existing or future state precisely since uncertainty is a state of having limited knowledge. Indeed, compared to the deterministic case, which is decision making under certainty, the stochastic case is more realistic and gives more accurate results because the majority of problems in reality inevitably have many unknown parameters. In addition, time scale calculus theory is applicable to any field in which a dynamic process can be described with discrete or continuous models. Many stochastic dynamic models are discrete or continuous, so the results of time scale calculus are directly applicable to them as well. The aim of this thesis is to introduce a general form of a stochastic dynamic sequence problem on complex discrete time domains and to find the optimal sequence which maximizes the sequence problem.

Stochastic Weather Generators (SWG) try to reproduce the stochastic patterns of climatological variables characterized by high dimensionality, non-normal probability density functions and non-linear dependence relationships. However, conventional SWGs usually typify weather variables with unjustified probability distributions assuming linear dependence between variables. This research proposes an alternative SWG that introduces the advantages of the Copula modeling into the reproduction of stochastic weather patterns. The Copula based SWG introduces more flexibility allowing researcher to model non-linear dependence structures independently of the marginals involved, also it is able to model tail dependence, which results in a more accurate reproduction of extreme weather events. Statistical tests on weather series simulated by the Copula based SWG show its capacity to replicate the statistical properties of the observed weather variables, along with a good performance in the reproduction of the extreme weather events. In terms of its use in crop growth models for the ratemaking process of new insurance schemes with no available historical yield data, the Copula based SWG allows one to more accurately evaluate the risk. The use of the Copula based SWG for the simulation of yields results in higher crop insurance premiums from more frequent extreme weather events, while the use of the conventional SWG for the yield estimation could lead to an underestimation of risks.

Thesis (Ph. D.)--University of Chicago, Dept. of Economics, June 2001. / Includes bibliographical references. Also available on the Internet.