BSDEs with polynomial growth generators
In the pioneer's papers on backward stochastic differential equations (BSDE) [\textit{E. Pardoux} and \textit{S. Peng}, in: Stochastic partial differential equations and their applications. Lect. Notes Control Inf. Sci. 176, 200-217 (1992; Zbl 0766.60079) and Syst. Control Lett. 14, No. 1, 55-61 (1990; Zbl 0692.93064)] the random function \(f(t,y,z)\) that makes the coefficient in the BSDE \[ Y_t= \xi+ \int^T_t f(s, Y_s, Z_s) ds- \int^T_t Z_s dW_s,\quad 0\leq t\leq T,\tag{1} \] has to be Lipschitz in \(y\) and \(z\), to guarantee existence and uniqueness of the solution \((Y,Z)\). Since, many people have worked on weakening this assumption [see \textit{R. W. R. Darling} and \textit{E. Pardoux}, Ann. Probab. 25, No. 3, 1135-1159 (1997; Zbl 0895.60067), \textit{S. Hamadène}, Ann. Inst. Henri Poincaré, Probab. Stat. 32, No. 5, 645-659 (1997; Zbl 0893.60031), \textit{M. Kobylanski}, C. R. Acad. Sci., Paris, Sér. I 324, No. 1, 81-86 (1997; Zbl 0880.60061), and \textit{J.-P. Lepeltier} and \textit{J. San Martín} [Stochastics Stochastics Rep. 63, No. 3-4, 227-240 (1998; Zbl 0910.60046) and Stat. Probab. Lett. 32, No. 4, 425-430 (1997; Zbl 0904.60042)]. The first part of the present paper proves existence and uniqueness of the solution \((Y,Z)\) of (1) when the Lipschitz condition on \(y\) is replaced by polynomial growth: \[ \exists C> 0,\;\forall t,y, z, |f(t,y,z)|\leq|f(t, 0,z)|+ C(1+|y|^p) \] and a monotonicity condition: \[ \exists\mu\in \mathbb{R},\;\forall t, y, y', z, (y- y')\cdot (f(t, y,z)- f(t, y',z))\leq -\mu|y- y'|^2. \] In the second part, the authors prove that existence and uniqueness of the solution hold under close assumptions, when the deterministic terminal time \(T>0\) is replaced by some random terminal time.
backward stochastic differential equation
scientific article; zbMATH DE number 1569490