On stochastic control problems with higher-order moments
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866912364267307008 |
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| author | Wang, Yike Liu, Jingzhen Bensoussan, Alain Yiu, Ka-Fai Cedric Wei, Jiaqin |
| author_facet | Wang, Yike Liu, Jingzhen Bensoussan, Alain Yiu, Ka-Fai Cedric Wei, Jiaqin |
| contents | In this paper, we focus on a class of time-inconsistent stochastic control problems, where the objective function includes the mean and several higher-order central moments of the terminal value of state. To tackle the time-inconsistency, we seek both the closed-loop and the open-loop Nash equilibrium controls as time-consistent solutions. We establish a partial differential equation (PDE) system for deriving a closed-loop Nash equilibrium control, which does not include the equilibrium value function and is different from the extended Hamilton-Jacobi-Bellman (HJB) equations as in Björk et al. (Finance Stoch. 21: 331-360, 2017). We show that our PDE system is equivalent to the extended HJB equations that seems difficult to be solved for our higher-order moment problems. In deriving an open-loop Nash equilibrium control, due to the non-separable higher-order moments in the objective function, we make some moment estimates in addition to the standard perturbation argument for developing a maximum principle. Then, the problem is reduced to solving a flow of forward-backward stochastic differential equations. In particular, we investigate linear controlled dynamics and some objective functions affine in the mean. The closed-loop and the open-loop Nash equilibrium controls are identical, which are independent of the state value, random path and the preference on the odd-order central moments. By sending the highest order of central moments to infinity, we obtain the time-consistent solutions to some control problems whose objective functions include some penalty functions for deviation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_13521 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On stochastic control problems with higher-order moments Wang, Yike Liu, Jingzhen Bensoussan, Alain Yiu, Ka-Fai Cedric Wei, Jiaqin Mathematical Finance Optimization and Control Primary: 93E20, 91G80, Secondary: 91B08, 49N90 In this paper, we focus on a class of time-inconsistent stochastic control problems, where the objective function includes the mean and several higher-order central moments of the terminal value of state. To tackle the time-inconsistency, we seek both the closed-loop and the open-loop Nash equilibrium controls as time-consistent solutions. We establish a partial differential equation (PDE) system for deriving a closed-loop Nash equilibrium control, which does not include the equilibrium value function and is different from the extended Hamilton-Jacobi-Bellman (HJB) equations as in Björk et al. (Finance Stoch. 21: 331-360, 2017). We show that our PDE system is equivalent to the extended HJB equations that seems difficult to be solved for our higher-order moment problems. In deriving an open-loop Nash equilibrium control, due to the non-separable higher-order moments in the objective function, we make some moment estimates in addition to the standard perturbation argument for developing a maximum principle. Then, the problem is reduced to solving a flow of forward-backward stochastic differential equations. In particular, we investigate linear controlled dynamics and some objective functions affine in the mean. The closed-loop and the open-loop Nash equilibrium controls are identical, which are independent of the state value, random path and the preference on the odd-order central moments. By sending the highest order of central moments to infinity, we obtain the time-consistent solutions to some control problems whose objective functions include some penalty functions for deviation. |
| title | On stochastic control problems with higher-order moments |
| topic | Mathematical Finance Optimization and Control Primary: 93E20, 91G80, Secondary: 91B08, 49N90 |
| url | https://arxiv.org/abs/2412.13521 |