Multi-dimensional anticipated backward stochastic differential equations with quadratic growth
Ying Hu, Feng Li, Jiaqiang Wen
TL;DR
This work extends the theory of anticipated backward stochastic differential equations to the multi-dimensional setting with diagonally quadratic generators, addressing both bounded and unbounded terminal data. By combining fixed-point methods in a BMO framework, a priori estimates, and iterative gluing of local solutions, it establishes local solvability for bounded terminals and two global solvability regimes, plus a global result for unbounded terminals under carefully crafted growth and convexity conditions. The results broaden the solvability landscape for ABSDEs and offer tools potentially applicable to stochastic control with delay and to financial models featuring delayed information and quadratic risks. Overall, the paper provides new existence, uniqueness, and quantitative bounds for multi-dimensional ABSDEs with quadratic growth, including cases with anticipating terms and embedded expectation nonlinearities.
Abstract
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 $Z_{t+ζ_t}$.