Sharif Digital Repository

This thesis presents an application of risk sensitive control theory in financial decision making. A variation of Merton’s continuous-time intertemporal capital asset pricing model is devolved where the infinite horizon objective is to maximize the portfolio’s risk adjusted growth rate. The resulting model is tractable and thus provides economic insight about optimal trading strategies as well as the fact that the strategy of 100% cash is not necessarily the least risky one. For fixed income applications we utilize the concept of rolling-horizon bonds, which are stochastic process models of certain mutual fund of zero coupon bonds. We show that the optimal proportion of one’s wealth to hold... 

The purpose of this thesis is to introduce and review a recent methods in simultaneous hypothesis testing. False discovery rates, Benjamini and Hochberg’s FDR Control Algorithm, is the great success story of the new methodology. Much of what follows is an attempt to explain that success in empirical Bayes terms.The later chapters are at pains to show the limitations of current largescale statistical practice: Which cases should be combined in a single analysis? How do we account for notions of relevance between cases? What is the correct null hypothesis? How do we handle correlations? Some helpful theory is provided in answer, but much of the argumentation is by example, with graphs and... 

In this thesis we are going to introduce the notion of k-SAT problem as one of the NP-complete problems and illustrate its importance of investigation. To study and seperate the hard and easy samples of this problem, we are going to explain the satisfiability threshold conjecture and some theorems in the infinity limit cases for the phase transition threshold. At last we are going to define the notion of sharp threshold and state a fundamental theorem which gives a necessary and sufficient condition for a threshold to be coarse  

The chapter 1 begins the discussion of a theory of large-scale simultaneous hypothesis testing now under development in the statistics literature. Furthermore,this chapter introduces the False Discovery Rate (FDR) and Empirical Bayes approach. In chapter 2, the frequentist viewpoints to the simultaneous hypothesis testing is mentioned. apter 3 describes the break through paper of the Benjamini and Hochberg published in 1995. Chapter 4 provides new criteria for error and represents an outstanding method of controlling FDR by J.D. Storey. The first part of chapter 5 discusses a paper related to control of FDR for variable selection in linear model setting by E.Candes and R. Barber. In the rest... 

In the present paper we construct stock price processes with the same marginal log- normal law as that of a geometric Brownian motion and also with the same transition density (and returns’ distributions) between any two instants in a given discrete-time grid. We then illustrate how option prices based on such processes differ from Black and Scholes’, in that option prices can be either arbitrarily close to the option intrinsic value or arbitrarily close to the underlying stock price. We also explain that this is due to the particular way one models the stock-price process in between the grid time instants which are relevant for trading
 

In this thesis we have proven A variational representation for positive functionals of a Hilbert space valued Wiener process (W(.)).This representation is then used to prove a laplace principle for the family{G(W(.))}>0 where G is an appropriate family of measurable maps from the Wiener space to some Polish space  

Travel time estimation is a central issue in the urban transportation industry and is the basis of many analyses and services in businesses related to this area. In the past few years, various statistical approaches have been devised to solve this problem. The purpose of this dissertation is to review existing methods by focusing on segment-based approaches for urban travel time estimation. A big challenge is the small amount of data in hand compared to the size of the urban network. Exploring historical data and extracting correlation between urban network segments leads to modeling the urban traffic condition and travel time estimation in one specific time interval of the day  

One of the most important problems in online learning is a prediction with expert advice. In each step we make our prediction not only based on previous observation but also use expert information. In this thesis, we study the different well-known algorithms of expert advice and generalize problems when data arrival is in the two-dimensional grid. regret is a well-studied concept to evaluate online learning algorithm. online algorithm when data arrive consecutively in T time step has regret O (√(T)) . regret in two-dimensional grid with T row and P column is O(T√(P)).
2010 MSC: 68Q32 ; 68T05 ; 90C27  

Mankind according to his authority (or delusion of authority) always finds himself in a situation which need decision-¬making. Usually, he seeks to make the best possible decision. The basis for measuring the goodness of choices is different in different occasions. This measure could be level of enjoyment, economic profit, probability of reaching a goal, etc. These decisions have consequences such that the situations before and after the decisions are not the same. Most challenging decision¬-making situations are those which the decision¬maker has not the complete authority over the situation and the results of decisions are influenced by out of control factors. A significant part of... 

The multi-armed bandit problem is a popular model for studying exploration/exploitation trade-off in sequential decision problems. Many algorithms are now available for this well-studied problem. One of the earliest algorithms, given by W. R. Thompson, dates back to 1933. This algorithm,referred to as Thompson Sampling, is a natural Bayesian algorithm. The basic idea is to choose an arm to play according to its probability of being the best arm. Thompson Sampling algorithm has experimentally been shown to be close to optimal. In this dissertation several papers are being reviewed. In these papers it has been shown that Thompson Sampling algorithm achieves logarithmic expected regret for the... 

In this thesis Generative Models in Deep Learning are discussed, especially Generative Models which are based on latent variables. Deep Generative Models have key role in developing Artificial Intelligence, particularly in developments of general cognition and perception in AI. In this thesis, this role for Generative Models and their applications in cognition development, and also the mathematical foundation of generative models are discussed  

In the behavior of many convex optimization problems with random constraints in high dimensions, sudden changes or phase transitions have been observed in terms of the number of constraints. A well-known example of this is the problem of reconstructing a thin vector or a low-order matrix based on a number of random linear observations. In both cases, methods based on convex optimization have been developed, observed, and proved that when the number of observations from a certain threshold becomes more (less), the answer to the problem with a probability of close to one (zero) is correct and the original matrix is reconstructed. Recently, results have been obtained that explain why this... 

Reinforcement learning is a successful model of learning that has received a lot of attention in recent years and has had significant achievements. However, methods based on reinforcement require a lot of data. Therefore, it is important to find ideas to keep learning at a high level despite the lack of data. Many of these ideas are known as statistical regularity. In this thesis, we study methods to enhance the learning rate, including methods for sharing neural network weights between value function and policy networks. In this thesis we will try to gain a more general understanding of the regularization in reinforcement learning and increase the learning rate by implementing these methods... 

Reinforcement learning (RL) is a subfield of machine learning that expresses how to learn optimal actions in a wide range of unknown environments. Reinforcement learning problems are often phrased in terms of Markov decision processes (MDPs). However, being restricted to Markov environments to solve problems with limited state space is not an unreasonable assumption, but the main challenge is to consider these problems in as large a class of environments as possible, which includes any challenges that an agent may face in real world. Such agents are able to learn to play chess, wash dishes, invest in financial markets, and do many tasks that an intelligent human being can learn and do. In... 

Birkhoff's ergodic theorem in dynamical systems and ergodic theory, and the strong law of large numbers in probability theory are among the fundamental theorems of the two fields, which are closely related. Thus Birkhoff's ergodic theorem directly yields the strong law of large numbers. Attempts were then made to express some limit theorems in probability theory in the form of dynamic systems, such as the central limit theorem, which was expressed in the form of dynamic systems, and even generalizations of It was also obtained. In this paper, we will investigate the above and similar connections between probability limit theorems and well-known theorems in ergodic theory  

We study the expected value of stopping times in stochastic processes. Since there is no rigorous solution for computing stopping times in many processes, our approach is based on estimation using well-known methods in the Reinforcement Learning literature. The primary method in this research is the temporal difference algorithm. With some modifications, we can study the role of some state features in determining the stopping time. Moreover, without a complicated mathematical analysis, we can find functions closely enough to the goal function.Furthermore, we compare our proposed algorithm to the well-known regression methods and show our algorithm's advantages and disadvantages. The primary... 

Multi armed bandit is a simple framework for modeling sequential decision making problems. A learner should choose between some arms at every time step and gains the reward of corresponding chosen arm. The environment is unknown to the learner, so he should make a balance between staying with the option that gave highest payoffs in the past and exploring new options that might give higher payoffs in the future known as exploration vs exploitation dilemma. The goal is finding a policy that minimizes the regret, which is a performance measure of the learner policy. We can make assumptions on how the rewards are generated, like stationary stochastic model, but we abandon almost all of them and... 

Nowadays, the online markets can easily change and adjust price of the product to an optimal price to increase the profit from the sale of their products. Because of this pricing flexibility, there are many applications of online pricing in online markets and so on.We study the problem of online pricing and specifically feature-based online pricing as an online learning problem in which a seller receives highly differentiated products online and prices them with the goal of obtaining the highest possible profit.The seller does not initially know the values of the different features, but can learn the values of the features based on whether products were sold at the posted prices in the... 

Strongly Rayleigh measures are an important class of negatively dependent (repulsive) probability measures. These measures are defined via a geometric condition, called “real stability”, on their generating polynomials, and have interesting probabilistic properties. On one important property of negatively dependent measures is their rigid dependence structure. In other words, the it is impossible for these measures to have strong overall dependencies. In this thesis, we study two manifestations if this phenomenon: (1) paving property and (2) tail triviality. Informally, the paving property states that it is possible to partition the set of the components of every strongly Rayleigh random...