Malliavin Calculus and Its Application to Robust Optimal Investment for an Insider
In the theory of portfolio selection, there are few methods that effectively address the combined challenge of insider information and model uncertainty, despite numerous methods proposed for each individually. This paper studies the problem of the robust optimal investment for an insider under mode...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-10-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/20/4378 |
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| author | Chao Yu Yuhan Cheng |
| author_facet | Chao Yu Yuhan Cheng |
| author_sort | Chao Yu |
| collection | DOAJ |
| description | In the theory of portfolio selection, there are few methods that effectively address the combined challenge of insider information and model uncertainty, despite numerous methods proposed for each individually. This paper studies the problem of the robust optimal investment for an insider under model uncertainty. To address this, we extend the Itô formula for forward integrals by Malliavin calculus, and use it to establish an implicit anticipating stochastic differential game model for the robust optimal investment. Since traditional stochastic control theory proves inadequate for solving anticipating control problems, we introduce a new approach. First, we employ the variational method to convert the original problem into a nonanticipative stochastic differential game problem. Then we use the stochastic maximum principle to derive the Hamiltonian system governing the robust optimal investment. In cases where the insider information filtration is of the initial enlargement type, we derive the closed-form expression for the investment by using the white noise theory when the insider is ’small’. When the insider is ’large’, we articulate a quadratic backward stochastic differential equation characterization of the investment. We present the numerical result and conduct an economic analysis of the optimal strategy across various scenarios. |
| first_indexed | 2024-03-10T21:04:45Z |
| format | Article |
| id | doaj.art-795d39f015fe4addae9b3e34fb019d59 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T21:04:45Z |
| publishDate | 2023-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-795d39f015fe4addae9b3e34fb019d592023-11-19T17:15:11ZengMDPI AGMathematics2227-73902023-10-011120437810.3390/math11204378Malliavin Calculus and Its Application to Robust Optimal Investment for an InsiderChao Yu0Yuhan Cheng1Department of Mathematical Sciences, Tsinghua University, Beijing 100084, ChinaSchool of Management, Shandong University, Jinan 250100, ChinaIn the theory of portfolio selection, there are few methods that effectively address the combined challenge of insider information and model uncertainty, despite numerous methods proposed for each individually. This paper studies the problem of the robust optimal investment for an insider under model uncertainty. To address this, we extend the Itô formula for forward integrals by Malliavin calculus, and use it to establish an implicit anticipating stochastic differential game model for the robust optimal investment. Since traditional stochastic control theory proves inadequate for solving anticipating control problems, we introduce a new approach. First, we employ the variational method to convert the original problem into a nonanticipative stochastic differential game problem. Then we use the stochastic maximum principle to derive the Hamiltonian system governing the robust optimal investment. In cases where the insider information filtration is of the initial enlargement type, we derive the closed-form expression for the investment by using the white noise theory when the insider is ’small’. When the insider is ’large’, we articulate a quadratic backward stochastic differential equation characterization of the investment. We present the numerical result and conduct an economic analysis of the optimal strategy across various scenarios.https://www.mdpi.com/2227-7390/11/20/4378Malliavin calculusforward integralrobust optimal investmentinsider informationmodel uncertaintystochastic maximum principle |
| spellingShingle | Chao Yu Yuhan Cheng Malliavin Calculus and Its Application to Robust Optimal Investment for an Insider Mathematics Malliavin calculus forward integral robust optimal investment insider information model uncertainty stochastic maximum principle |
| title | Malliavin Calculus and Its Application to Robust Optimal Investment for an Insider |
| title_full | Malliavin Calculus and Its Application to Robust Optimal Investment for an Insider |
| title_fullStr | Malliavin Calculus and Its Application to Robust Optimal Investment for an Insider |
| title_full_unstemmed | Malliavin Calculus and Its Application to Robust Optimal Investment for an Insider |
| title_short | Malliavin Calculus and Its Application to Robust Optimal Investment for an Insider |
| title_sort | malliavin calculus and its application to robust optimal investment for an insider |
| topic | Malliavin calculus forward integral robust optimal investment insider information model uncertainty stochastic maximum principle |
| url | https://www.mdpi.com/2227-7390/11/20/4378 |
| work_keys_str_mv | AT chaoyu malliavincalculusanditsapplicationtorobustoptimalinvestmentforaninsider AT yuhancheng malliavincalculusanditsapplicationtorobustoptimalinvestmentforaninsider |