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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.17702 |
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| _version_ | 1866917157436129280 |
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| author | Lai, Jou-Hua Shkolnikov, Mykhaylo Soner, H. Mete |
| author_facet | Lai, Jou-Hua Shkolnikov, Mykhaylo Soner, H. Mete |
| contents | We give a new formulation of the relative arbitrage problem from stochastic portfolio theory that asks for a time horizon beyond which arbitrage relative to the market exists in all ``sufficiently volatile'' markets. In our formulation, ``sufficiently volatile'' is interpreted as a lower bound on an ordered eigenvalue of the instantaneous covariation matrix, a quantity that has been studied extensively in the empirical finance literature. Upon framing the problem in the language of stochastic optimal control, we characterize the time horizon in question through the unique upper semicontinuous viscosity solution of a fully nonlinear elliptic partial differential equation (PDE). In a special case, this PDE amounts to the arrival time formulation of the Ambrosio-Soner co-dimension mean curvature flow. Beyond the setting of stochastic portfolio theory, the stochastic optimal control problem is analyzed for arbitrary compact, possibly non-convex, domains, thanks to a boundedness assumption on the instantaneous covariation matrix. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_17702 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Relative arbitrage problem under eigenvalue lower bounds Lai, Jou-Hua Shkolnikov, Mykhaylo Soner, H. Mete Mathematical Finance Analysis of PDEs Differential Geometry Optimization and Control Probability 91G10, 93E20, 49L25, 53E10 We give a new formulation of the relative arbitrage problem from stochastic portfolio theory that asks for a time horizon beyond which arbitrage relative to the market exists in all ``sufficiently volatile'' markets. In our formulation, ``sufficiently volatile'' is interpreted as a lower bound on an ordered eigenvalue of the instantaneous covariation matrix, a quantity that has been studied extensively in the empirical finance literature. Upon framing the problem in the language of stochastic optimal control, we characterize the time horizon in question through the unique upper semicontinuous viscosity solution of a fully nonlinear elliptic partial differential equation (PDE). In a special case, this PDE amounts to the arrival time formulation of the Ambrosio-Soner co-dimension mean curvature flow. Beyond the setting of stochastic portfolio theory, the stochastic optimal control problem is analyzed for arbitrary compact, possibly non-convex, domains, thanks to a boundedness assumption on the instantaneous covariation matrix. |
| title | Relative arbitrage problem under eigenvalue lower bounds |
| topic | Mathematical Finance Analysis of PDEs Differential Geometry Optimization and Control Probability 91G10, 93E20, 49L25, 53E10 |
| url | https://arxiv.org/abs/2512.17702 |