Quadratic reflected BSDEs with unbounded obstacles
backward stochastic differential equation
optimal stopping for quadratic g-evaluation
Given a driving Brownian motion \(B\), a terminal condition \(\xi\) and a lower barrier process \(L\), the authors study the reflected backward stochastic differential equation (BSDE) NEWLINE\[NEWLINEdY_t=-f(t,Y_t,Z_t)dt-dK_t+Z_tdB_t,\quad t\in[0,T],\quad Y_T=\xi,NEWLINE\]NEWLINE with an adapted solution \((Y,Z,K)\) satisfying \(Y_t\geq L_t\), \((Y_t-L_t)dK_t=0\), where the increasing process \(K\) is the minimal force needed in order to keep \(Y\) above the barrier \(L\). The authors consider the case, where \(f\) can have quadratic growth in \(z\), and the barrier \(L\) and the terminal condition \(\xi\) are allowed to be unbounded. The first work to deal with BSDEs which driver can have quadratic growth in \(z\) is that by Kobylanski (2000); later she extended this work to reflected BSDEs with bounded \(\xi\) and bounded \(f\). With a localisation procedure Briand and Hu (2006) showed that, for BSDEs with quadratic growth in \(z\), the boundedness of \(\xi\) is not necessary for the existence of solutions. In 2008, they used the so-called \(\theta\)-difference method and showed that, if, in addition, \(f\) is concave in \(z\), then one has the uniqueness of solutions for the BSDE. Delbaen, Hu and Richou (2011) proved the uniqueness for solutions having a given exponential moment by passing through the Fenchel-Legendre dual of \(f\) and they studied associated PDEs, for which the uniqueness was proven by Da Lio and Ley (2011). The authors of the reviewed paper extend the above mentioned works to the case of reflected BSDEs which driving process has quadratic growth in \(z\) and which terminal condition \(\xi\) and barrier \(L\) may be unbounded. Under the additional assumption of concavity of \(f\) in \(z\), they obtain the uniqueness of solutions with an alternative approach. Moreover, they study stability results and the optimal stopping for g-evaluations. As application, the authors study associated semi-linear PDEs with obstacle with quadratic growth in the gradient of the solution and prove a comparison theorem for such PDEs when \(f\) is concave in \(z\).
scientific article; zbMATH DE number 6040279
scientific article