Browsing by Subject "Stochastic control"
Now showing 1 - 7 of 7
- Results Per Page
- Sort Options
Item Asymptotics for optimal investment with high-water mark fee(2015-08) Kontaxis, Andrew; Sîrbu, Mihai; Gamba, Irene M; Mendoza-Arriaga, Rafael; Zariphopoulou, Thaleia; Zitkovic, GordanThis dissertation studies the problem of optimal investment in a fund charging high-water mark fees. We consider a market consisting of a riskless money-market account and a fund charging high-water mark fees at rate λ, with share price given exogenously as a geometric Brownian motion. A small investor invests in this market on an infinite time horizon and seeks to maximize expected utility from consumption rate. Utility is taken to be constant relative risk aversion (CRRA). In this setting, we study the asymptotic behavior of the value function for small values of the fee rate λ. In particular, we determine the first and second derivatives of the value function with respect to λ. We then exhibit for each λ explicit sub-optimal feedback investment and consumption strategies with payoffs that match the value function up to second order in λ.Item Computational methods for stochastic control problems with applications in finance(2014-05) Mitchell, Daniel Allen; Muthuraman, KumarStochastic control is a broad tool with applications in several areas of academic interest. The financial literature is full of examples of decisions made under uncertainty and stochastic control is a natural framework to deal with these problems. Problems such as optimal trading, option pricing and economic policy all fall under the purview of stochastic control. These problems often face nonlinearities that make analytical solutions infeasible and thus numerical methods must be employed to find approximate solutions. In this dissertation three types of stochastic control formulations are used to model applications in finance and numerical methods are developed to solve the resulting nonlinear problems. To begin with, optimal stopping is applied to option pricing. Next, impulse control is used to study the problem of interest rate control faced by a nation's central bank, and finally a new type of hybrid control is developed and applied to an investment decision faced by money managers.Item Effects of patient heterogeneity in a first-come-first-serve kidney transplant model(2020-05-07) Chang, Chia-Hao; Shakkottai, Sanjay; Hasenbein, John J.In this thesis, we discuss how patient death rates may affect patient choice in a kidney transplant system. Specifically, the transplant system is modeled as an M/M/1 queue with reneging, where patient and kidney arrivals are modeled as independent Poisson processes and a patient's death is referred to as reneging the queue. The patients face a problem in the form of a Markov decision process, to which we found the analytical solution when it is undiscounted and when the kidney qualities have discrete distributions. With the obtained result, we were able to examine the sensitivity analysis on the overall system. Next, we show that when the kidney distributions converge (in the sense of Kolmogorov metric), so do their associated value functions, from which we are able to extend our results to continuous distributions as well. Finally, our results are substantiated by numerical simulations.Item Essays on achieving investment targets and financial stability(2013-05) Monin, Phillip James; Zariphopoulou, Thaleia, 1962-This dissertation explores the application of the techniques of mathematical finance to the achievement of investment targets and financial stability. It contains three self-contained but broadly related essays. Sharpe et al. proposed the idea of having an expected utility maximizer choose a probability distribution for future wealth as an input to her investment problem rather than a utility function. They developed the Distribution Builder as one way to elicit such a distribution. In a single-period model, they then showed how this desired distribution for terminal wealth can be used to infer the investor's risk preferences. In the first essay, we adapt their idea, namely that a desired distribution for future wealth is an alternative input attribute for investment decisions, to continuous time. In a variety of scenarios, we show how the investor's desired distribution, combined with her initial wealth and market-related input, can be used to determine the feasibility of her distribution, her implied risk preferences, and her optimal policies throughout her investment horizon. We then provide several examples. In the second essay, we consider an investor who must a priori liquidate a large position in a primary risky asset whose price is influenced by the investor's liquidation strategy. Liquidation must be complete by a terminal time T, and the investor can hedge the market risk involved with liquidation over time by investing in a liquid proxy asset that is correlated with the primary asset. We show that the optimal strategies for an investor with constant absolute risk aversion are deterministic and we find them explicitly using calculus of variations. We then analyze the strategies and determine the investor's indifference price. In the third essay, we use contingent claims analysis to study several aggregate distance-to-default measures of the S&P Financial Select Sector Index during the years leading up to and including the recent financial crisis of 2007-2009. We uncover mathematical errors in the literature concerning one of these measures, portfolio distance-to-default, and propose an alternative measure that we show has similar conceptual and in-sample econometric properties. We then compare the performance of the aggregate distance-to-default measures to other common risk indicators.Item Martingale-generated control structures and a framework for the dynamic programming principle(2017-07-17) Fayvisovich, Roman; Žitković, Gordan; Sirbu, Mihai; Zariphopoulou, Thaleia; Larsen, KasperThis thesis constructs an abstract framework in which the dynamic programming principle (DPP) can be proven for a broad range of stochastic control problems. Using a distributional formulation of stochastic control, we prove the DPP for problems that optimize over sets of martingale measures. As an application, we use the classical martingale problem to prove the DPP for weak solutions of controlled diffusions, and use it show that the value function is a viscosity solution of the associated Hamilton-Jacobi-Bellman equation.Item Optimal investment with high-watermark fee in a multi-dimensional jump diffusion model(2017-08-04) Li, Zheng, active 21st century; Sîrbu, Mihai; Mueller, Peter; Tompaidis, Efstathios; Zariphopoulou, Thaleia; Zitkovic, GordanThis dissertation studies the problem of optimal investment and consumption in a market in which there are multiple risky assets. Among those risky assets, there is a fund charging high-watermark fees and many other stocks, with share prices given exogenously as a multi-dimensional geometric Lévy process. Additionally, there is a riskless money market account in this market. A small investor invests and consumes simultaneously on an infinite time horizon, and seeks to maximize expected utility from consumption. Utility is taken to be constant relative risk aversion (CRRA). In this setting, we first employ the Dynamic Programming Principle to write down the Hamilton-Jacobi-Bellman (HJB) integro-differential equation associated with this stochastic control problem. Then, we proceed to show that a classical solution of the HJB equation corresponds to the value function of the stochastic control problem, and hence the optimal strategies are given in feedback form in terms of the value function. Moreover, we provide numerical results to investigate the impact of various parameters on the investor’s strategies.Item Quantitative and modeling aspects of optimal decision making under uncertainty(2023-08) Zhang, Luhao; Zariphopoulou, Thaleia, 1962-; Gao, Rui; Tompaidis, Efstathios; Gamba, Irene M; Sirbu, MihaiThis dissertation focuses on the problem of decision making under uncertainty, more precisely, the quantitative and modeling aspects of "how to acquire and, in turn, exploit information optimally for decision-making in stochastic environments". To address the challenges posed by different types of uncertainty, a range of methods have been developed in the fields of stochastic control under partial information, dynamic information acquisition, data-driven optimization, model uncertainty, and robust optimization. Specifically, this dissertation is composed by two parts: The first part focuses on an offline data-driven decision-making problem with side information. With abundant data routinely collected in many industries to support decision-making, historical data with numerous side information–temporal, spatial, social, or economical–are available prior to the decision making and reveals partial information on the randomness of the problem. The challenge for these high-dimensional problems is that the empirical distribution constructed from the observed data is not representative of the underlying true distribution between contextual information and decisions, and strategies solely based on the empirical data can lead to poor performance when implemented. Therefore, a fundamental problem in data-driven decision-making under uncertainty, as well as in statistical learning, is finding solutions that perform well not only on the observed data but also on new and previously unseen data. To hedge against the distributional uncertainty of the offline dataset, this dissertation provides an end-to-end learning framework, based on distributionally robust stochastic optimization (DRSO), that prescribes a non-parametric policy with certified robustness, provable optimality, and efficient implementation. Specifically, we study policy optimization for a series of feature-based decision-making problems, which seeks an end-to-end policy that renders an explicit mapping from features to decisions. In this dissertation, we first consider a Wasserstein robust optimization framework, where we highlight our contribution in finding an optimal robust policy without restricting onto a parametric family while still maintaining computational efficiency and interpretability. More specifically, by exploiting the structure of the optimal policy, we identify a new class of policies that are proven to be robust optimal and can be computed by linear programming. We apply our work in newsvendor problem. Furthermore, we propose a new uncertainty set based on causal transport distance which contains distributions that share a similar conditional information structure with the nominal distribution. We derive a tractable dual reformulation for evaluating the worst-case expected cost and show that the worst-case distribution has a similar conditional information structure as the nominal distribution. We identify tractable cases to find the optimal decision rules over an affine class or the entire nonparametric class, and apply our work in conditional regression, incumbent pricing and portfolio selection. The second part is concerned with dynamic information acquisition with sequential decision-making and differential information sources. When involving dynamic learning to facilitate decision making, since the decision makers often have imperfect and costly information, they encounter a trade-off between the information learning and the expected payoff, given the limited information. For example, when comparing new technologies, the firm often spends a considerable amount of funds and time on research and development (R&D) to identify the best technology to adopt. Other examples include investors designing algorithms to learn about the return of different assets, scientists conducting research to investigate the validity of different hypotheses, etc. From the viewpoint of dynamic information acquisition, the practically important features are the choice of "what to learn", as well as "when to learn and stop learning". Most of the decision-making problems considered in this line of work are static (i.e. a single, irreversible decision) problems which, however, over-simplify the structure of many real-world applications that require dynamic or sequential decisions. Moreover, the information acquisition source in these studies typically remains constant (e.g. a single noise signal) throughout the decision process, failing to capture the adaptive nature of decision makers in response to stochastically changing environments. Herein, we introduce a general framework in which we allow for both sequential (possibly reversible) decisions and dynamically changing information sources (distinct signals), and it also includes the cost of acquiring information across time. We analyze a benchmark example, motivated by the return/exchange policies in e-commerce platforms. Specifically, we introduce a sequential decision-making problem that allows decision makers to reverse their initial decisions and their costly information acquisition setting to change accordingly. We investigate the optimal strategies for information acquisition and decision reversal, and carry out a complete sensitivity and asymptotic analysis on how decision makers can effectively adapt their learning behavior to ultimately achieve the best decision-making outcomes. In what follows, we describe each approach separately. For each part, we introduce the corresponding model, construct solutions, and provide a detailed analytical methodology.