Finding the nonnegative minimal solutions of Cauchy PDEs in a volatility-stabilized market
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909629952294912 |
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| author | Yang, Nicole Tianjiao Ichiba, Tomoyuki |
| author_facet | Yang, Nicole Tianjiao Ichiba, Tomoyuki |
| contents | The strong relative arbitrage problem in Stochastic Portfolio Theory seeks an investment strategy that almost surely outperforms a benchmark portfolio at the end of a given time horizon. The highest relative return in relative arbitrage opportunities is characterized by the smallest nonnegative continuous solution of a Cauchy problem for a partial differential equation (PDE). However, solving this type of PDE poses analytical and numerical challenges, due to the high dimensionality and its non-unique solutions. In this paper, we discuss numerical methods to address the relative arbitrage problem and the associated PDE in a volatility-stabilized market, using time-changed Bessel bridges. We present a practical algorithm and demonstrate numerical results through an example in volatility-stabilized markets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_13558 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finding the nonnegative minimal solutions of Cauchy PDEs in a volatility-stabilized market Yang, Nicole Tianjiao Ichiba, Tomoyuki Computational Finance Probability Mathematical Finance 60H10, 91G10 The strong relative arbitrage problem in Stochastic Portfolio Theory seeks an investment strategy that almost surely outperforms a benchmark portfolio at the end of a given time horizon. The highest relative return in relative arbitrage opportunities is characterized by the smallest nonnegative continuous solution of a Cauchy problem for a partial differential equation (PDE). However, solving this type of PDE poses analytical and numerical challenges, due to the high dimensionality and its non-unique solutions. In this paper, we discuss numerical methods to address the relative arbitrage problem and the associated PDE in a volatility-stabilized market, using time-changed Bessel bridges. We present a practical algorithm and demonstrate numerical results through an example in volatility-stabilized markets. |
| title | Finding the nonnegative minimal solutions of Cauchy PDEs in a volatility-stabilized market |
| topic | Computational Finance Probability Mathematical Finance 60H10, 91G10 |
| url | https://arxiv.org/abs/2411.13558 |