Thesis (Ph. D.)--Drexel University, 2006. / Includes abstract and vita. Includes bibliographical references (leaves 105-113).

This dissertation is devoted to theories of processes we call ``Walsh semimartingales" and ``Walsh diffusions", as well as to related optimization problems of control and stopping. These processes move on the plane along rays emanating from the origin; and when at the origin, the processes choose the rays of their subsequent voyage according to a fixed probability measure---in a manner described by Walsh (1978) as a direct generalization of the skew Brownian motion. We first review in Chapter 1 some key results regarding the celebrated skew Brownian motions and Walsh Brownian motions. These results include the characterization of skew Brownian motions via stochastic equations in Harrison & Shepp (1981), the construction of Walsh Brownian motions in Barlow, Pitman & Yor (1989), and the important result of Tsirel'son (1997) regarding the nature of the filtration generated by the Walsh Brownian motion. Various generalizations of Walsh Brownian motions are described in detail in Chapter 2. We formally define there Walsh semimartingales as a subclass of planar processes we call ``semimartingales on rays". We derive for such processes Freidlin-Sheu-type change-of-variable formulas, as well as two-dimensional versions of the Harrison-Shepp equations. The actual construction of Walsh semimartingales is given next. Walsh diffusions are then defined as a subclass of Walsh semimartingales, described by stochastic equations which involve local drift and dispersion characteristics. The associated local submartingale problems, strong Markov properties, existence, uniqueness, asymptotic behavior, and tests for explosions in finite time, are studied in turn. Finally, with Walsh semimartingales as state-processes, we study in Chapter 3 succesively a pure optimal stopping problem, a stochastic control problem with discretionary stopping, and a stochastic game between a controller and a stopper. We derive for these problems optimal strategies in surprisingly explicit from. Crucial for the analysis underpinning these results, are the change-of-variable formulas derived in Chapter 2. Most of the results in Chapters 2 and 3 are based on two papers, [21] and [31], both cowritten by the author of this dissertation. Some results and proofs are rearranged and rewritten here.

Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (Ph. D.)--Worcester Polytechnic Institute. / Keywords: Stochastic optimal control; dynamic programming; Hamilton-Jacobi-Bellman equation; Random vibration. Keywords: Stochastic optimal control; dynamic programming; Hamilton-Jacobi-Bellman equation; Random vibration; energy balance method. Includes bibliographical references (p. 86-89).

Thesis (Ph. D.)--Hong Kong University of Science and Technology, 2003. / Includes bibliographical references (leaves 117-130). Also available in electronic version. Access restricted to campus users.

Not available / text

As we know, the deterministic LQ problems are well-posed if the state weighting matrix and the control weighting matrix are nonnegative and positive definite in the cost function, respectively. Some practical problems, however, often include indefinite weighting matrices in their cost functions such as mean-variance portfolio selection problem. This inspires us to further study the indefinite LQ problems in detail. / In this thesis, we study indefinite stochastic linear-quadratic (LQ) control with jumps and present some financial applications of this new development. / The results of the above LQ control problems are employed to deal with a mean-variance portfolio selection model in an incomplete financial market. An optimal analytical investment strategy is directly derived and the expression of its risk is explicitly presented. In addition, a mean-variance portfolio selection model in a financial market where shorting is not allowed is investigated in detail via the stochastic LQ problem with nonnegative controls. In particular, the explicit expression of the efficient frontier enables an investor to better understand the relation between the expected terminal wealth and the risk in a stock market with no-shorting. / The weighting matrices in the cost function are allowed to be indefinite (in particular, negative) when the diffusion term linearly depends on the control variable in the state equation. In this case, indefinite stochastic LQ control problems with jumps may still be sensible and well-posed. In an infinite time horizon, solvability of coupled generalized algebraic Riccati equations (CGAREs) is sufficient for the well-posedness of the stochastic LQ control problem with jumps. Moreover, an approach algorithm is devised to solve the CGAREs via semi-definite programming over linear matrix inequalities. On the other hand, it is shown that the well-posedness of the stochastic LQ control problem in a finite time horizon with jumps is equivalent to solvability of coupled generalized Riccati equations. / Li Xun. / "November 2000." / Advisers: Cai Xiaoqiang; Zhou Xunyu. / Source: Dissertation Abstracts International, Volume: 61-10, Section: B, page: 5541. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (p. 115-122). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest dissertations and theses, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

Sequential decision making problems are ubiquitous in a number of research areas such as operations research, finance, engineering and computer science. The main challenge with these problems comes from the fact that, firstly, there is uncertainty about the future. And secondly, decisions have to be made over a period of time, sequentially. These problems, in many cases, are modeled as Markov Decision Process (MDP). Most real-life MDPs are ‘high dimensional’ in nature making them challenging from a numerical point of view. We consider a number of such high dimensional MDPs. In some cases such problems can be approximately solved using Approximate Dynamic Programming. In other cases problem specific analysis can be solved to device tractable policies that are near-optimal. In Chapter 2, we presents a novel and practical non-parametric approximate dynamic programming (ADP) algorithm that enjoys graceful, dimension-independent approximation and sample complexity guarantees. In particular, we establish both theoretically and computationally that our proposal can serve as a viable replacement to state of the art parametric ADP algorithms, freeing the designer from carefully specifying an approximation architecture. We accomplish this by ‘kernelizing’ a recent mathematical program for ADP (the ‘smoothed’ approximate LP) proposed by [Desai et al., 2011]. In Chapter 3, we consider a class of stochastic control problems where the action space at each time can be described by a class of matching or, more generally, network flow polytopes. Special cases of this class of dynamic matching problems include many problems that are well-studied in the literature, such as: (i) online keyword matching in Internet advertising (the adwords problem); (ii) the bipartite matching of donated kidneys from cadavers to recipients; and (iii) the allocation of donated kidneys through exchanges over cycles of live donor-patient pairs. We provide an approximate dynamic program (ADP) algorithm for dynamic matching with stochastic arrivals and departures. Our framework is more general than the methods prevalent in the literature in that it is applicable to a broad range of problems characterized by a variety of action polytopes and generic arrival and departure processes. In Chapter 4, we consider the problem of A-B testing when the impact of the treatment is marred by a large number of covariates. Randomization can be highly inefficient in such settings, and thus we consider the problem of optimally allocating test subjects to either treatment with a view to maximizing the efficiency of our estimate of the treatment effect. Our main contribution is a tractable algorithm for this problem in the online setting, where subjects arrive, and must be assigned, sequentially. We characterize the value of optimized allocations relative to randomized allocations and show that this value grows large as the number of covariates grows. In particular, we show that there is a lot to be gained from ‘optimizing’ the process of A-B testing relative to the simple randomized trials that are the mainstay of A-B testing in the ‘big data’ regime of modern e-commerce applications, where the number of covariates is often comparable to the number of experimental trials.

<p>Essentially this project report is a discussion of mathematical modelling in pension funds, presenting sections from Cairns, A.J.D., Blake, D., Dowd, K., Stochastic lifestyling: Optimal dynamic asset allocation for defined contribution pension plans, Journal of Economic Dynamics and Control, Volume 30, Issue 2006, Pages 843-877, with added details and background material in order to demonstrate the mathematical methods. In the investigation of the management of the investment portfolio, we only use one risky asset together with a bond and cash as other assets in a&nbsp / continuous time framework. The particular model is very much designed according to the members&rsquo / preference and then the funds are invested by the fund manager in the financial market. At the end, we are going to show various simulations of these models. Our methods include stochastic control for utility maximisation among others. The optimisation problem entails the optimal&nbsp / investment portfolio to maximise a certain power utility function. We use MATLAB and MAPLE programming languages to generate results in the form of graphs and tables</p>

Thesis (Ph. D.)--University of Missouri-Columbia, 2000. / Typescript. Vita. Includes bibliographical references (leaves 200-206). Also available on the Internet.