Backward martingale transport maps and equilibrium with insider
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909212435546112 |
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| author | Kramkov, Dmitry Sîrbu, Mihai |
| author_facet | Kramkov, Dmitry Sîrbu, Mihai |
| contents | We consider an optimal transport problem with backward martingale constraint. The objective function is given by the scalar product of a pseudo-Euclidean space $S$. We show that the supremums over maps and plans coincide, provided that the law $ν$ of the input random variable $Y$ is atomless. An optimal map $X$ exists if $ν$ does not charge any $c-c$ surface (the graph of a difference of convex functions) with strictly positive normal vectors in the sense of the $S$-space. The optimal map $X$ is unique if $ν$ does not charge $c-c$ surfaces with nonnegative normal vectors in the $S$-space. As an application, we derive sharp conditions for the existence and uniqueness of equilibrium in a multi-asset version of the model with insider from Rochet and Vila [10]. In the linear-Gaussian case, we characterize Kyle's lambda, the sensitivity of price to trading volume, as the unique positive solution of a non-symmetric algebraic Riccati equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_08290 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Backward martingale transport maps and equilibrium with insider Kramkov, Dmitry Sîrbu, Mihai Probability We consider an optimal transport problem with backward martingale constraint. The objective function is given by the scalar product of a pseudo-Euclidean space $S$. We show that the supremums over maps and plans coincide, provided that the law $ν$ of the input random variable $Y$ is atomless. An optimal map $X$ exists if $ν$ does not charge any $c-c$ surface (the graph of a difference of convex functions) with strictly positive normal vectors in the sense of the $S$-space. The optimal map $X$ is unique if $ν$ does not charge $c-c$ surfaces with nonnegative normal vectors in the $S$-space. As an application, we derive sharp conditions for the existence and uniqueness of equilibrium in a multi-asset version of the model with insider from Rochet and Vila [10]. In the linear-Gaussian case, we characterize Kyle's lambda, the sensitivity of price to trading volume, as the unique positive solution of a non-symmetric algebraic Riccati equation. |
| title | Backward martingale transport maps and equilibrium with insider |
| topic | Probability |
| url | https://arxiv.org/abs/2304.08290 |