Anticipated Backward Stochastic Differential Equations with Quadratic Growth: Multidimensional Results
Résumé
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 [21, Journal of Differential Equations, 270 (2021), 1298-1311] 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+δt , Z t+ζt ) is of general growth with respect to Y t and Y t+δt . For the global solution with bounded terminal values, the generator f (t, Y t , Z t , Y t+δt , Z t+ζt ) 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+δt ) 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+δt , Z t+ζt )] is of diagonal quadratic growth in Z t and linear growth in Z t+ζt .
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