Multi-dimensional anticipated backward stochastic differential equations with quadratic growth
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909617791959040 |
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| author | Hu, Ying Li, Feng Wen, Jiaqiang |
| author_facet | Hu, Ying Li, Feng Wen, Jiaqiang |
| contents | This paper is devoted to the general solvability of anticipated backward stochastic differential equations with quadratic growth by relaxing the assumptions made by Hu, Li, and Wen \cite[Journal of Differential Equations, 270 (2021), 1298--1311]{hu2021anticipated} from the one-dimensional case with bounded terminal values to the multi-dimensional situation with bounded/unbounded terminal values. Three new results regarding the existence and uniqueness of local and global solutions are established. More precisely, for the local solution with bounded terminal values, the generator $f(t, Y_t, Z_t, Y_{t+δ_t},Z_{t+ζ_t})$ is of general growth with respect to $Y_t$ and $Y_{t+δ_{t}}$. For the global solution with bounded terminal values, the generator $f(t, Y_t, Z_t, Y_{t+δ_t},Z_{t+ζ_t})$ is of skew sub-quadratic but also ``strictly and diagonally" quadratic growth in $Z_t$. For the global solution with unbounded terminal values, the generator $f(t, Y_t, Z_t, Y_{t+δ_t})$ is of diagonal quadratic growth in $Z_t$ in the first case; and in the second case, the generator $f(t, Z_t)$+$E[g(t, Y_t,Z_t, Y_{t+δ_t},Z_{t+ζ_t})]$ is of diagonal quadratic growth in $Z_t$ and linear growth in $Z_{t+ζ_t}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_20255 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multi-dimensional anticipated backward stochastic differential equations with quadratic growth Hu, Ying Li, Feng Wen, Jiaqiang Probability 60H10, 60H30 This paper is devoted to the general solvability of anticipated backward stochastic differential equations with quadratic growth by relaxing the assumptions made by Hu, Li, and Wen \cite[Journal of Differential Equations, 270 (2021), 1298--1311]{hu2021anticipated} from the one-dimensional case with bounded terminal values to the multi-dimensional situation with bounded/unbounded terminal values. Three new results regarding the existence and uniqueness of local and global solutions are established. More precisely, for the local solution with bounded terminal values, the generator $f(t, Y_t, Z_t, Y_{t+δ_t},Z_{t+ζ_t})$ is of general growth with respect to $Y_t$ and $Y_{t+δ_{t}}$. For the global solution with bounded terminal values, the generator $f(t, Y_t, Z_t, Y_{t+δ_t},Z_{t+ζ_t})$ is of skew sub-quadratic but also ``strictly and diagonally" quadratic growth in $Z_t$. For the global solution with unbounded terminal values, the generator $f(t, Y_t, Z_t, Y_{t+δ_t})$ is of diagonal quadratic growth in $Z_t$ in the first case; and in the second case, the generator $f(t, Z_t)$+$E[g(t, Y_t,Z_t, Y_{t+δ_t},Z_{t+ζ_t})]$ is of diagonal quadratic growth in $Z_t$ and linear growth in $Z_{t+ζ_t}$. |
| title | Multi-dimensional anticipated backward stochastic differential equations with quadratic growth |
| topic | Probability 60H10, 60H30 |
| url | https://arxiv.org/abs/2503.20255 |