Author

Yulong Zhou

Year of Award

2017

Degree Type

Thesis

Degree Name

Doctor of Philosophy (PhD)

Department

Department of Mathematics.

Principal Supervisor

Zeng, Tieyong

Keywords

Boltzmann, Ludwig, 1844-1906; Probability measures; Stochastic control theory; Transport theory

Language

English

Abstract

In this thesis we study two problems concerning probability. The first is stochastic control problem, which essentially amounts to find an optimal probability in order to optimize some reward function of probability. The second is to approximate the solution of the Boltzmann equation. Thanks to conservation of mass, the solution can be regarded as a family of probability indexed by time. In the first part, we prove a dynamic programming principle for stochastic optimal control problem with expectation constraint by measurable selection approach. Since state constraint, drawdown constraint, target constraint, quantile hedging and floor constraint can all be reformulated into expectation constraint, we apply our results to prove the corresponding dynamic programming principles for these five classes of stochastic control problems in a continuous but non-Markovian setting. In order to solve the Boltzmann equation numerically, in the second part, we propose a new model equation to approximate the Boltzmann equation without angular cutoff. Here the approximate equation incorporates Boltzmann collision operator with angular cutoff and the Landau collision operator. As a first step, we prove the well-posedness theory for our approximate equation. Then in the next step, we show the error estimate between the solutions to the approximate equation and the original equation. Compared to the standard angular cutoff approximation method, our method results in higher order of accuracy.

Comments

Principal supervisor: Dr. Zeng Tieyong. Thesis submitted to the Department of Mathematics.; Thesis (Ph.D.)--Hong Kong Baptist University, 2017.

Bibliography

Includes bibliographical references (pages 95-104).

Available for download on Wednesday, October 10, 2018



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