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Lyapunov exponents for stochastic Anderson models with non-Gaussian noise; portfolio optimization in discrete time with proportional transaction costs under stochastic volatility.

机译：具有非高斯噪声的随机Anderson模型的Lyapunov指数；随机波动下具有成比例交易成本的离散时间的投资组合优化。

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摘要

This dissertation consists of two different research topics: Lyapunov exponents and portfolio optimization. The second chapter states the study of Lyapunov exponents under stochastic Anderson model. The stochastic Anderson model in discrete or continuous space is defined for a class of non-Gaussian space-time potentials as solutions to the multiplicative stochastic heat equation for some diffusivity parameter and inverse-temperature parameter. The relation with the corresponding polymer model in a random environment is given. The large time exponential behavior of the solution is studied via its almost sure Lyapunov exponent, which is proved to exist, and is estimated as a function of the diffusivity parameter and the inverse-temperature parameter: positivity and non-trivial upper bounds are established, generalizing and improving existing results.;The third chapter of this dissertation presents how to evaluate (approximately) optimal self-financing strategy and optimal trading frequency for a portfolio with a risky asset and a risk-free asset. The objective is to maximize the expected future utility of the terminal wealth in a stochastic volatility setting, when transaction costs are incurred at each discrete trading time. A HARA utility function is used, allowing a simple approximation of the optimization problem, which is implementable forward in time. For each of various transaction cost rates, we find the optimal trading frequency, i.e. the one that attains the maximum of the expected utility at time zero. We study the relation between transaction cost rate and optimal trading frequency. The numerical method used is based on a stochastic volatility particle filtering algorithm, combined with a Monte-Carlo method. The filtering algorithm updates the estimate of the volatility distribution forward in time, as new stock observations arrive; these updates are used at each of these discrete times to compute the new portfolio allocation.
机译：本文包括两个不同的研究主题：李雅普诺夫指数和投资组合优化。第二章阐述了随机安德森模型下李雅普诺夫指数的研究。针对一类非高斯时空势，定义了离散或连续空间中的随机Anderson模型，作为某些扩散参数和逆温度参数的乘性随机热方程的解。给出了与随机环境中相应聚合物模型的关系。通过其几乎确定的Lyapunov指数研究了该溶液的大时间指数行为，事实证明该指数已经存在，并根据扩散率参数和逆温度参数进行估计：建立了正性和非平凡上界，论文的第三章介绍了如何评估（近似）具有风险资产和无风险资产的投资组合的最优自筹资金策略和最优交易频率。目的是在每个离散交易时间产生交易成本时，在随机波动情况下最大化终端财富的预期未来效用。使用了HARA实用程序功能，可以对优化问题进行简单近似，并且可以及时实现。对于各种交易成本率，我们找到最佳交易频率，即在零时间达到预期效用最大值的交易频率。我们研究了交易成本率与最佳交易频率之间的关系。所使用的数值方法基于随机波动性粒子滤波算法，并结合了蒙特卡洛方法。随着新库存观察的到来，过滤算法会及时更新波动率分布的估计值。这些更新在每个不连续的时间使用，以计算新的投资组合分配。

著录项

  • 作者

    Kim, Ha Young.;

  • 作者单位

    Purdue University.;

  • 授予单位 Purdue University.;
  • 学科 Applied Mathematics.;Statistics.;Mathematics.
  • 学位 Ph.D.
  • 年度 2010
  • 页码 72 p.
  • 总页数 72
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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