\(L^{p} (p>1)\) solutions of backward stochastic differential equations with monotonic and uniformly continuous generators
Let \(p>1\). A one-dimensional backward stochastic differential equation \[ y_t=\xi+\int_t^Tg(s,y_s,z_s)\,ds-\int_t^Tz_s\cdot dB_s \] is considered, where \(\xi\) is a terminal condition in \(L^p\), the generator \(g(\omega,t,y,z)\) satisfies five hypotheses including continuity and monotonicity in \(y\) and uniform continuity in \(z\) and \(B\) is a \(d\)-dimensional Brownian motion. It is proved that the equation has a unique solution \((y,z)\) in the space \(L^p(\Omega;C[0,T])\times L^p(\Omega;L^2(0,T;\mathbb R^d))\). As a side result, a comparison principle is established for this equation.
backward stochastic differential equation
scientific article; zbMATH DE number 6145426