A converse comparison theorem for anticipated BSDEs and related non-linear expectations
The authors consider the anticipated backward stochastic differential equation \[ \begin{aligned} -dY_t& =f(t,Y_t,Z_t,Y_{t+\delta(t)})\,dt-Z_t\,dW_t,\quad t\in[0,T] \\ Y_t& =\xi_t t\in[T,T+K], \end{aligned} \] where \(\delta\) is a continuous \(\mathbb R_+\)-valued function satisfying some technical assumptions, \(Y\) is a real process, \(Z\) is \(\mathbb R^d\)-valued, \(W\) is a Brownian motion in \(\mathbb R^d\), \(\xi\) is a terminal condition process and \(f\) is, roughly speaking, a random variable with a Lipschitz-like dependence on the last three variables (the domain for the last variable being \(L^2(\mathcal F_r)\)). The underlying filtration is generated by \(W\). The paper contains two main results. The first one asserts the existence and uniqueness of solutions to the above equation, whereby \(T\) is admitted to be a finite stopping time rather than a positive constant, provided that \(f\) satisfies an eventually vanishing driver condition. For the second result, the notions of \((f,\delta)\)-expectation and conditional \((f,\delta)\)-expectation are introduced, and it is proved that such expectations exist under reasonable conditions. The \((f,\delta)\)-expectation assigns to an adapted process \(X\) the process \(\mathcal E^\delta_f[X](s)=\operatorname{E}[Y_s]\), where \(Y\) is the solution of the equation \[ \begin{aligned} Y_r &=X_t+\int_r^tf(s,Y_s,Z_s,Y_{s+\delta(s)})\,ds- \int_r^tZ_s\,dW_s,\quad r\in[0,t] \\ Y_r &=X_r\,\, r\in[t,t+K]. \end{aligned} \] The converse comparison theorem then asserts that, if two equations with drivers \(f_1\) and \(f_2\) and with the same anticipating function \(\delta\) are considered, then \(f_1(t,y,z,\theta_{t+\delta(t)})\geq f_2(t,y,z,\theta_{t+\delta(t)})\) holds for every \((t,y,z,\theta)\) if and only if \(\mathcal E^\delta_{f_1}[X](0)\geq\mathcal E^\delta_{f_2}[X](0)\) holds for every \(X\).
scientific article; zbMATH DE number 6128613