The problem of optimal portfolio execution has become one of the most important problems in the area of financial mathematics. Over the past two decades, numerous researchers have developed a variety of different models to address this problem. In this dissertation, we extend the LOB (Limit Order Book) model proposed by Obizhaeva and Wang (2013) by incorporating a more realistic assumption on the order book depth; the amount of liquidity provided by a LOB market is finite at all times. We use an algorithmic approach to solve the problem of optimal execution under time-varying constraints on the depth of a LOB. For the simplest case where the order book depth stays at a fixed level for the entire trading horizon, we reduce the optimal execution problem into a one-dimensional root-finding problem which can be readily solved by standard numerical algorithms. When the depth of the LOB is monotone in time, we first apply the KKT (Karush-Kuhn-Tucker) conditions to narrow down the set of candidate strategies and then use a dichotomy-based search algorithm to pin down the optimal one. For the general case that the order book depth doesn't exhibit any particular pattern, we start from the optimal strategy subject to no liquidity constraints and iterate over execution strategy by sequentially adding more constraints to the problem in a specific fashion until primal feasibility is achieved. Numerical experiments indicate that our algorithms give comparable results to those of current existing convex optimization toolbox CVXOPT with significantly lower time complexity. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Spring Semester 2018. / March 22, 2018. / convex programming, limit order market, market impact model, optimal portfolio execution / Includes bibliographical references. / Arash Fahim, Professor Directing Dissertation; Jen Atkins, University Representative; Alec Kercheval, Committee Member; Giray Okten, Committee Member.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2017 / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 175-181). / In this thesis, we investigate the long time behavior of focusing mass critical nonlinear Schrödinger equation (NLS). We will focus on the singularity formation and long time asymptotics. To be specific, there are two parts in the thesis. In the first part, we give a construction of log-log blow up solutions which blow up at m prescribed points simultaneously. In the second part, we show weak convergence to ground state for certain radial blow up solutions to NLS at well chosen time sequence. We also include a lecture note on concentration compactness. Concentration compactness is one of the main tool we use in the second part of the thesis. / by Chenjie Fan. / Ph. D. / Ph.D. Massachusetts Institute of Technology, Department of Mathematics

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2017 / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 129-133). / A key feature of integrability for systems of evolution PDEs ut = F(u), where F lies in a differential algebra of functionals V and u = (U1, ... , ul) depends on one space variable x and time t, is to be part of an infinite hierarchy of generalized symmetries. Recall that V carries a Lie algebra bracket {F, G} = XF(G) - XG(F), where XF denotes the evolutionnary vector field attached to F. In all known examples, these hierarchies are constructed by means of Lenard-Magri sequences: one can find a pair of matrix differential operators (A(a), B(a)) and a sequence (G.n)>n>0,[epsilon] Vl such that ** F = B(GN) for some N >/= 0, ** {B(Gn), B(Gm)} = 0 for all n, m >/= 0, ** B(G,+1 ) = A(G) for all n,m >/= 0. We show that in the scalar case l = 1 a necessary condition for a pair of differential operators (A, B) to generate a Lenard-Magri sequence is that for all constants [lambda], the family C[lambda] = A + [lambda]B must satisfy for all F, G [epsilon]V {C[lambda](F), C[lambda](G)} [epsilon] ImC[lambda]. We call such pairs integrable. We give a sufficient condition on an integrable pair of matrix differential operators (A, B) to generate an infinite Lenard- Magri sequence when the rational matrix differential operator L = AB-1 is weakly non-local and the algebra of differential functions V is either Z or Z/2Z-graded. This is applied to many systems of evolution PDEs to prove their integrability. / by Sylvain Carpentier. / Ph. D. / Ph.D. Massachusetts Institute of Technology, Department of Mathematics

Source: Dissertation Abstracts International, Volume: 30-05, Section: B, page: 2304. / Thesis (Ph.D.)--The Florida State University, 1969.

Source: Dissertation Abstracts International, Volume: 30-05, Section: B, page: 2303. / Thesis (Ph.D.)--The Florida State University, 1969.

Source: Dissertation Abstracts International, Volume: 30-05, Section: B, page: 2284. / Thesis (Educat.D.)--The Florida State University, 1969.

Source: Dissertation Abstracts International, Volume: 30-05, Section: B, page: 2313. / Thesis (Ph.D.)--The Florida State University, 1969.

Source: Dissertation Abstracts International, Volume: 30-10, Section: B, page: 4713. / Thesis (Ph.D.)--The Florida State University, 1969.

Source: Dissertation Abstracts International, Volume: 30-12, Section: B, page: 5602. / Thesis (Ph.D.)--The Florida State University, 1969.

In financial markets under uncertainty, the classical Black-Scholes model cannot explain the empirical facts such as fat tails observed in the probability density. To overcome this drawback, during the last decade, Lévy process and stochastic volatility models were introduced to financial modeling. Today crude oil futures markets are highly volatile. It is the purpose of this dissertation to develop a mathematical framework in which American options on crude oil futures contracts are priced more effectively than by current methods. In this work, we use the Variance Gamma process to model the futures price process. To generate the underlying process, we use a random tress method so that we evaluate the option prices at each tree node. Through fifty replications of a random tree, the averaged value is taken as a true option price. Pricing performance using this method is accessed using American options on crude oil commodity contracts from December 2003 to November 2004. In comparison with the Variance Gamma model, we price using the Black-Scholes model as well. Over the entire sample period, a positive skewness and high kurtosis, especially in the short-term options, are observed. In terms of pricing errors, the Variance Gamma process performs better than the Black-Scholes model for the American options on crude oil commodities. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Degree Awarded: Summer Semester, 2008. / Date of Defense: July 10, 2008. / Variance Gamma Process, American Options on Crude Oil Futures Commodity Calibration, Random Tree Method, Lévy Process / Includes bibliographical references. / Craig A. Nolder, Professor Directing Dissertation; Fred Huffer, Outside Committee Member; Bettye Anne Case, Committee Member; Alec N. Kercheval, Committee Member; Jack Quine, Committee Member.