Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1970. / Bibliography: leaf 48. / M.S.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1972. / This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. / Vita. / Includes bibliographical references. / by Richard A. Shore. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 155-174). / The conflict between rational models of economic behavior and their systematic deviations, often referred to as behavioral economics, is one of the most hotly debated issues in social sciences. This thesis reconciles the two opposing perspectives by applying evolutionary principles to economic behavior and deriving implications that cut across species, physiology, and genetic origins. In the context of a binary-choice model, we first show that risk aversion emerges via natural selection if reproductive risk is "systematic", i.e., correlated across individuals in a given generation. The degree of risk aversion is determined by the stochastic nature of reproductive rates, and different statistical properties lead to different utility functions. More generally, irrational behaviors are not just mere divergence from rationality, but seeds necessary for successfully coping with environmental transformations. Furthermore, there is an optimal degree of irrationality in the population depending on the degree of environmental stochasticity. When applied to evolutionary biology, we show that what appears to be group selection may, in fact, simply be the consequence of natural selection occurring in stochastic environments with "systematic" risks. Those individuals with highly correlated risks will appear to form "groups", even if their actions are totally autonomous, mindless, and, prior to selection, uniformly randomly distributed in the population. Evolutionary principles can also be used to model the dynamics of financial markets. In a multiperiod model of the contagion of investment ideas, we show that heterogeneous investment styles can coexist in the long run, implying a wider variation of diverse strategies compared to traditional theories. These results may provide new insights to the survival of a wide range of hedge funds. In a model that investors maximize their relative wealth, the initial wealth plays a critical role in determining how the optimal behavior deviates from the Kelly Criterion, regardless of whether the investor is myopic or maximizing the infinite-horizon wealth. / by Ruixun Zhang. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1985. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: leaves 96-98. / by Abramo Hefez. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1992. / Includes bibliographical references (p. 59). / by Erlan E. Wheeler II. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 125-126). / This thesis consists of two results. The first result is a strong maximum principle for certain parabolic systems of equations, which, for illustrative purposes, I consider as reaction-diffusion systems. Using the theory of viscosity solutions, I give a proof which extends the previous theorem to no longer require any regularity assumptions on the boundary of the convex set in which the system takes its values. The second result is an approximation scheme for reflected stochastic differential equations (SDE) of the Stratonovich type. This is a joint result with Professor Daniel W. Stroock. We show that the distribution of the solution to such a reflected SDE is the weak limit of the distribution of the solutions of the reflected SDEs one gets by replacing the driving Brownian motion by its N-dyadic linear interpolation. In particular, we can infer geometric properties of the solutions to a Stratonovich reflected SDE from those of the solutions to the approximating reflected SDE. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1992. / Includes bibliographical references (p. 89-91). / by Matthew Ando. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1989. / Title as it appeared in M.I.T. Graduate List, Feb. 1989: Carleman inequalities and strong unique continuation. / Includes bibliographical references (leaf 59). / by Yonne Mi Kim. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004. / Includes bibliographical references (p. 129). / In this thesis we give a probabilistic proof of the Morse inequalities in the nondegenerate and degenerate case. For the nondegenerate case the kernel associated with the Witten Laplacian has an expression via the Malliavin calculus. The first step is the analysis of this heat kernel at a point away the critical set. Using Markov property, an iteration procedure and estimates on exit times from balls, everything is reduced to the estimation of a solution to a parabolic initial-boundary problem on a ball in the Euclidean space. We achieve that by constructing a supersolution. For the case the point is close to the critical set, we use an integration by parts in the Malliavin calculus and split the analysis for paths staying inside a given distance from the critical point or exiting the corresponding ball. For the paths exiting, again an iterative Markov property argument reduces the problem to a parabolic initial-boundary value problem that can be handled by the construction of the supersolution mentioned above. For the quantity involving the paths staying inside a given ball around the critical point, we can reverse the argument, this time with the Euclidean space playing the role of the original manifold and reduce the problem to one in the Euclidean settings. This turns out to be an elementary harmonic oscillator problem that finishes the argument. / (cont.) The case of the degenerate Bott-Morse function requires a bit more work due to the fact that the geometry near the critical submanifolds is in general not trivial. After some standard constructions, we have two choices of the connection around critical submanifolds. One is the Levi-Civita and the other is Bismut's connection. The main step in this analysis is to prove that the heat kernels of certain operators with respect to Levi-Civita connection and the Bismut connection stay bounded when the parameters involved become large. This is achieved by a fiberwise version of the argument given in the nondegenerate case. Using the boundedness, one can prove the basic comparison. Finally, the rest is just a fiberwise harmonic oscillator problem. / by Ionel Popescu. / Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1999. / Includes bibliographical references (p. 79-82). / by Rebecca Freja Goldin. / Ph.D.