Quasilinear stochastic hyperbolic differential equations with nondecreasing coefficient
The authors study the hyperbolic stochastic partial differential equation (SPDE) \[ {\partial^2 \over \partial s\partial t}u= f\bigl(s,t,u(s,t) \bigr)+ g\bigl(s,t,u(s,t) \bigr) {\partial^2 \over \partial s\partial t} W, \quad u(s,0)= u(0,t)=a, \quad s,t\in [0,1], \] where \(W= \{W_z,\;z\in [0,1]^2\}\) denotes the Brownian sheet, \(a\) is a real, and \(f,g: [0,1]^2 \times R\to R\) are measurable, uniformly bounded functions. For \(g\not \equiv 1\), under the assumption that \(f(z,r)\) is nondecreasing in \(r\in R\), existence and uniqueness of a strong solution are shown. The approach bases on a monotonicity method similar to that developed by the reviewer and \textit{E. Pardoux} [in: Diffusion processes and related problems in analysis. Vol. I: Diffusions in analysis and geometry. Prog. Probab. 22, 219-233 (1990; Zbl 0722.60061)] and \textit{I. Gyöngy} and \textit{E. Pardoux} [Probab. Theory Relat. Fields 97, No. 1/2, 211-229 (1993; Zbl 0793.60064)] for elliptic and parabolic SPDEs, respectively, but with the difficulty here in the hyperbolic case that the comparison lemma does not hold anymore. In a second part of the paper, under the assumption that there is a unique solution to the above SPDE, an approximation scheme by discretization of the SPDE is introduced, similar to that recently studied by Gyöngy/Krylov for ordinary SDEs and that by \textit{I. Gyöngy} and \textit{D. Nualart} for parabolic SPDEs [Stochastic Processes Appl. 58, No. 1, 57-72 (1995; Zbl 0832.60068)].
monotonicity methods for stochastic partial differential equations
hyperbolic stochastic partial differential equation
scientific article; zbMATH DE number 1080067