One-dimensional BSDEs with left-continuous, lower semi-continuous and linear-growth generators
backward stochastic differential equations
The authors examine the backward stochastic differential equation NEWLINE\[NEWLINEy_t =\xi +\int^{T}_{t} g(s, y_{s}, z_{s})\, ds - \int^{T}_{t} z_{s}\, dB_{s}\,,\;\;\; t\in T.NEWLINE\]NEWLINE It is shown that if \(g\) is of linear growth in \((y,z)\) and if \(g(t,0,0)\) satisfies certain continuity conditions, then for each square-integrable terminal condition \(\xi\) the above equation has at least one solution, and such a solution is minimal. Levi- and comparison-type theorems are proved for the minimal solutions.
scientific article; zbMATH DE number 6085254