Dissertation•
Chance-Constrained Optimization and Optimal Control Problems
01 Jan 2015-
TL;DR: This thesis introduces a new probabilistic risk measure—the minimax portfolio selection model and finds that the efficient frontier is superior to various efficient frontiers in the literature in the mean-variance space.
Abstract: In this thesis, four separate problems are studied. Whilst the focus of each problem is different, all four problems are modelled as either an optimization or an optimal control problem subject to probabilistic constraints. The first problem that we consider is portfolio optimization. Portfolio selection models are of great practical significance to investors around the world, and the way risk is defined and measured leads to different optimal portfolios. For this problem, we introduce a new probabilistic risk measure—the minimax portfolio selection model. The objective is to maximize the return of the portfolio while minimizing the maximum individual risk of the assets in the portfolio. This bi-criteria optimization problem is shown to be equivalent to a linear programming problem, and an analytical solution is obtained. This method is innovative and flexible, and allows for investors with different degree of risk aversion. We further show that our efficient frontier is superior to various efficient frontiers in the literature in the mean-variance space. The second problem explores a preventive-based maintenance scheduling problem to ensure reliable machine operation. Effective maintenance policies are essential for production planning purposes. The problem we investigate is to choose the continuous maintenance rate and the overhaul maintenance times to minimize the total cost of operating and maintaining the machine, where probabilistic state constraints are imposed to ensure that the machines state and output meet minimum acceptable levels with high probability. This impulsive stochastic optimal control problem can be solved using a combination of the control parameterization technique, the time-scaling transformation, and the constraint transcription method. Finally, we illustrate our approach by solving a numerical example. Our third problem explores how to optimize a pension funds cash position where the assets held include both government and corporate bonds. A typical pension fund experiences regular stochastic outgo in the form of pension benefits, death benefits as well as withdrawal benefits. Bonds are the natural assets to match the cashflows of the pension fund. The aim of the problem is to maximize the pension funds cash position at the end of the time horizon subject to an all-time probabilistic constraint that ensures all outgoing commitments can be met with sufficiently high probability. We first consider the idealistic situation of zero bond defaults and find that the optimal solution is to invest
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TL;DR: A resilience-oriented service restoration method using microgrids to restore critical load after natural disasters is proposed in this paper, and the impacts of fault locations, available generation resources, and load priority on the restoration strategy are discussed.
Abstract: A resilience-oriented service restoration method using microgrids to restore critical load after natural disasters is proposed in this paper. Considering the scarcity of power generation resources, the concept of continuous operating time (COT) is introduced to determine the availability of microgrids for critical load restoration and to assess the service time. Uncertainties induced by intermittent energy sources and load are also taken into account. The critical load restoration problem is modeled as a chance-constrained stochastic program. A Markov chain-based operation model is designed to describe the stochastic energy variations within microgrids, based on which the COT is assessed. A two-stage heuristic is developed for the critical load restoration problem. First, a strategy table containing the information of all feasible restoration paths is established. Then the critical load restoration strategy is obtained by solving a linear integer program. Numerical simulations are performed on the IEEE 123-node feeder system under several scenarios to demonstrate the effectiveness of the proposed method. The impacts of fault locations, available generation resources, and load priority on the restoration strategy are discussed.
307 citations
Cites background from "Chance-Constrained Optimization and..."
...Therefore, the α-percentile of TR c is defined [26]: Tc = sup { T|Pr ( TR c ≥ T ) ≥ α (3)...
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11 Jul 2012
TL;DR: The fundamental principles of financial derivatives from both financial and mathematical perspectives are examined, and how the mathematical tools from stochastic calculus, differential equations and numerical methods join forces to form an essential part in modern finance is demonstrated.
Abstract: Outline: This is the first part of a two-semester sequence course on mathematical finance. In the Fall semester, we will examine the fundamental principles of financial derivatives from both financial and mathematical perspectives, and demonstrate how the mathematical tools from stochastic calculus, differential equations and numerical methods join forces to form an essential part in modern finance. The emphasis of the course is a mathematical understanding of the intrinsic relationships among various financial instruments, which serves as a basis for investment decisions and trading strategies. The central theme of the Fall semester is the classic Black-Scholes-Merton model, and we plan to give a thorough treatment of the original model, with extensive discussions on the practical extensions in response to various disadvantages of the original model. One of the most intuitive and transparent approaches to illustrate that is also extensively used in practice is the binomial tree model. It contains most of the essential idea of the Black-Scholes-Merton methodology, and it can be naturally extended to build more general continuous-time models. Time permitting, we will include as much real life examples as possible to make this a rewarding experience for those who plan to pursue a career in this direction, as well as those who are just intrigued by the subject and its impact on our society.
96 citations
01 Sep 2017
TL;DR: This work characterizes the space of velocities that would allow each robot to fulfill its share in collision avoidance with a specified probability and proposes a series of reformulations which when combined with time scaling based concepts leads to a closed form characterization of solution space of PRVO.
Abstract: We present PRVO, a probabilistic variant of Reciprocal Velocity Obstacle (RVO) for decentralized multi-robot navigation under uncertainty. PRVO characterizes the space of velocities that would allow each robot to fulfill its share in collision avoidance with a specified probability. PRVO is modeled as chance constraints over the velocity level constraints defined by RVO and takes into account the uncertainty associated with both state estimation as well as the actuation of each robot. Since chance constraints are in general computationally intractable, we propose a series of reformulations which when combined with time scaling based concepts leads to a closed form characterization of solution space of PRVO for a given probability of collision avoidance. We validate our formulation through numerical simulations in which we highlight the advantages of PRVO over the related existing formulations.
49 citations
Cites background or methods from "Chance-Constrained Optimization and..."
...Our approach combines ideas from velocity obstacle based multi-robot collision avoidance and chance constraints which are used to ensure constraint satisfaction under uncertainty [5], [6]....
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...) is a non-convex quadratic in terms of random variables and thus the techniques proposed in [6] are not applicable in our case....
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...One notable exception exists in the case in which the random variables under consideration have Gaussian distribution and the chance constraints are defined over affine inequalities [6]....
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01 Sep 2017
TL;DR: In this paper, a semi-analytical method is proposed to simplify the formulation and computation of chance constraints in the economic dispatch problem, which has lower computational cost, controllable level of risk and low information exchange between agents and the independent system operator.
Abstract: This paper addresses a computational issue of solving stochastic economic dispatch with the participation of uncertain demand response providers. In particular, we study the linear constraints in the chance constrained formulation of the dispatch problem. A semi-analytical method is proposed to simplify the formulation and computation of chance constraints in the economic dispatch problem. Compared with existing methods such as deterministic, analytical method and sample-based approaches, the advantage of the semi-analytical method lies in its lower computational cost, controllable level of risk and low information exchange between agents and the independent system operator. Numerical results on the IEEE 14 bus system with two demand response providers illustrate the potential efficacy of the proposed method.
2 citations
Cites methods from "Chance-Constrained Optimization and..."
...There are also other sample-based methods such as one proposed by [21]....
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References
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TL;DR: In this paper, the authors present and justify a set of four desirable properties for measures of risk, and call the measures satisfying these properties "coherent", and demonstrate the universality of scenario-based methods for providing coherent measures.
Abstract: In this paper we study both market risks and nonmarket risks, without complete markets assumption, and discuss methods of measurement of these risks. We present and justify a set of four desirable properties for measures of risk, and call the measures satisfying these properties “coherent.” We examine the measures of risk provided and the related actions required by SPAN, by the SEC=NASD rules, and by quantile-based methods. We demonstrate the universality of scenario-based methods for providing coherent measures. We offer suggestions concerning the SEC method. We also suggest a method to repair the failure of subadditivity of quantile-based methods.
8,651 citations
"Chance-Constrained Optimization and..." refers background in this paper
...It lacks subadditivity and convexity, and is not a coherent risk measure [1]....
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TL;DR: In this article, the authors defined asset classes technology sector stocks will diminish as the construction of the portfolio, and the construction diversification among the, same level of assets, which is right for instance among the assets.
Abstract: So it is equal to the group of portfolio will be sure. See dealing with the standard deviations. See dealing with terminal wealth investment universe. Investors are rational and return at the point. Technology fund and standard deviation of investments you. Your holding periods of time and as diversification depends. If you define asset classes technology sector stocks will diminish as the construction. I know i've left the effect. If the research studies on large cap. One or securities of risk minimize more transaction. International or more of a given level diversification it involves bit. This is used the magnitude of how to reduce stress and do change over. At an investment goals if you adjust for some cases the group. The construction diversification among the, same level. Over diversification portfolio those factors include risk. It is right for instance among the assets which implies.
6,323 citations
TL;DR: In this paper, the combined problem of optimal portfolio selection and consumption rules for an individual in a continuous-time model was examined, where his income is generated by returns on assets and these returns or instantaneous "growth rates" are stochastic.
Abstract: OST models of portfolio selection have M been one-period models. I examine the combined problem of optimal portfolio selection and consumption rules for an individual in a continuous-time model whzere his income is generated by returns on assets and these returns or instantaneous "growth rates" are stochastic. P. A. Samuelson has developed a similar model in discrete-time for more general probability distributions in a companion paper [8]. I derive the optimality equations for a multiasset problem when the rate of returns are generated by a Wiener Brownian-motion process. A particular case examined in detail is the two-asset model with constant relative riskaversion or iso-elastic marginal utility. An explicit solution is also found for the case of constant absolute risk-aversion. The general technique employed can be used to examine a wide class of intertemporal economic problems under uncertainty. In addition to the Samuelson paper [8], there is the multi-period analysis of Tobin [9]. Phelps [6] has a model used to determine the optimal consumption rule for a multi-period example where income is partly generated by an asset with an uncertain return. Mirrless [5] has developed a continuous-time optimal consumption model of the neoclassical type with technical progress a random variable.
4,908 citations
"Chance-Constrained Optimization and..." refers background in this paper
...Early studies discuss the problem of maximizing the expected utility of the investor over a single- or multi-period investment horizon, with various forms for the utility function [40, 41]....
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...42 Chance-constrained Optimization for Pension Fund Portfolios in the Presence of Default Risk models in the literature also typically assume that capital in the investment portfolio can be reallocated freely at the start of each time period [40, 41], but in practice, restructuring the portfolio is never free: there are transaction costs and potential tax liabilities to consider....
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Book•
01 Jan 1982
TL;DR: In this article, the authors present a survey of the history and varieties of probability for the laws of chance and their application in the context of Markov chains convergence of random variables.
Abstract: Events and their probabilities random variables and their distributions discrete random variables continuous random variables generating functions and their applications Markov chains convergence of random variables random processes stationary processes renewals queues Martingales diffusion processes. Appendices: Foundations and notations history and varieties of probability John Arburthnot's preface to "Of the Laws of Chance" (1692).
2,819 citations
"Chance-Constrained Optimization and..." refers background in this paper
...Then it follows from the one-sided Chebyshev’s inequality [18] that if (4....
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TL;DR: In this article, a new methodology for pricing and hedging derivative securities involving credit risk is proposed, based on the foreign currency analogy of Jarrow and Turnbull (1991) to decompose the dollar payoff from a risky security into a certain payoff and a spot exchange rate.
Abstract: This article provides a new methodology for pricing and hedging derivative securities involving credit risk. Two types of credit risks are considered. The first is where the asset underlying the derivative security may default. The second is where the writer of the derivative security may default. We apply the foreign currency analogy of Jarrow and Turnbull (1991) to decompose the dollar payoff from a risky security into a certain payoff and a "spot exchange rate." Arbitrage-free valuation techniques are then employed. This methodology can be applied to corporate debt and over the counter derivatives, such as swaps and caps. THE PURPOSE OF THIS article is to provide a new theory for pricing and hedging derivative securities involving credit risk. Two sources of credit risk are identified and analyzed. The first is where the asset underlying the derivative security may default, paying off less than promised. This is the case, for example, with imbedded options on corporate debt. The second is the credit risk introduced by the writer of the derivative security, who may also default. Examples include over-the-counter writers of options on Eurodollar futures, swaps, and swaptions. For pricing derivative securities involving credit risk, there are currently two approaches. The first views these derivatives as contingent claims not on the financial securities themselves, but as "compound options" on the assets underlying the financial securities. This is the case, for example, with the pricing of imbedded options on corporate debt (see Merton (1974, 1977), Black and Cox (1976), Ho and Singer (1982), Chance (1990), and Kim, Ramaswamy, and Sundaresan (1993)) or the pricing of vulnerable options (see Johnson and Stulz (1987)). In practice, however, this valuation methodology is difficult to use. First, the assets underlying the financial security are often not tradeable and therefore their values are not observable. This makes application of the theory and estimation of the relevant parameters problematic. Second, as in the case of corporate debt, all of the other liabilities of the firm senior to the corporate debt must first (and simultaneously) be valued. This generates significant computational difficulties. As a result, this approach has not
2,071 citations