A general converse comparison theorem for backward stochastic differential equations (Q2756232)

Let \((Y^i,Z^i)\), \(i=1,2\), be the solutions of two backward stochastic differential equations NEWLINE\[NEWLINEY_t=\xi+ \int^T_t g_i(s,Y_s,Z_s)ds-\int^T_t Z_sdW_s,\quad t\in [0,T],\;i=1,2.NEWLINE\]NEWLINE It is a classical result that, if \(P\)-a.s., \(\forall (t,y,z)\), \(g_1(t,y,z)\geq g_2(t,y,z)\), then, \(P\)-a.s., \(\forall t\in [0,T]\), \(Y_t^1\geq Y_t^2\) [\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) or \textit{N. El Karoui}, \textit{S. Peng} and \textit{M. C. Quenez}, Math. Finance 7, No. 1, 1-71 (1997; Zbl 0884.90035)]. The authors prove here that, under convenient assumptions, a converse comparison theorem holds: The following two conditions are equivalent NEWLINENEWLINENEWLINEi) \(P\)-a.s., \(\forall(t,y,z)\), \(g_1(t,y,z)\geq g_2(t,y,z);\)NEWLINENEWLINENEWLINEii) for each \(\xi\in L^1({\mathcal F}_T)\), \({\mathcal E}_{g_1}(\xi) \geq{\mathcal E}_{g_2}(\xi)\), where \({\mathcal E}_g(\xi):=Y_0\) denotes the \(g\)-expectation of \(\xi\).