A converse comparison theorem for BSDEs and related properties of \(g\)-expectation
Let \(W= (W_t)_{t\in [0,T]}\) denote a \(d\)-dimensional Brownian motion. Given two drivers \(g^1\) and \(g^2\) satisfying standard assumptions, and two square integrable random variables \(\xi^1\) and \(\xi^2\), the authors compare the solution processes \(Y^1= Y^1(\xi^1)\), \(Y^2= Y^2(\xi^2)\) of the backward stochastic differential equations (for short, BSDEs) \[ dY^i_t= -g^i(t, Y^i_t, Z^i_t)dt+ Z^i_t dW_t,\quad t\in [0,T],\quad Y^i_T= \xi^i,\quad i= 1,2. \] Nonlinear BSDEs of this type have been introduced by \textit{E. Pardoux} and \textit{S. Peng} [Syst. Control Lett. 14, No. 1, 55-61 (1990; Zbl 0692.93064)]. A key role in the theory of BSDEs and its various applications is played by the comparison theorem due to S. Peng (1992), and \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)]. It allows to compare the solutions \(Y^1(\xi^1)\), \(Y^2(\xi^2)\) whenever one can compare the drivers \(g^1\), \(g^2\) and the terminal values \(\xi^1\), \(\xi^2\). In the present paper the authors study an inverse comparison problem. They show that if \(Y^1_t(\xi)\leq Y^2_t(\xi)\), for all \(t\in [0,T]\) and all \(\xi\in L^2(\sigma\{W\})\), then it must hold that \(g^1(t,y,z)\leq g^2(t,y,z)\), for all \((t,x,y)\). After that they apply this converse comparison theorem to PDEs and to discuss Jensen's inequality for the \(g\)-expectation, a notion introduced by \textit{S. Peng} [in: Backward stochastic differential equations. Pitman Res. Notes Math. Ser. 364, 141-159 (1997; Zbl 0892.60066)]. Their converse comparison theorem generalizes a recent work by \textit{Z. Chen} [C. R. Acad. Sci., Paris, Sér. I, Math. 326, No. 4, 483-488 (1998; Zbl 0914.60025)] who studied assumptions which imply \(g^1(t,x,y)= g^2(t,x,y)\).
backward stochastic differential equations
scientific article; zbMATH DE number 1522368