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In mathematics and physics, the Kardar-Parisi-Zhang equation or quasilinear stationary version of a time-dependent viscous Hamilton-Jacobi equation in growing interface and universality classes is also known as the quasilinear Riccati type equation. The existence of solutions to this type of equations still remains an interesting open problem. In previous studies [36,38], we obtained global bounds and gradient estimates for quasilinear elliptic equations with measure data. The main goal of this article is to obtain the existence of a renormalized solution to the quasilinear stationary solution for the degenerate diffusive Hamilton-Jacobi equation with finite measure data in Lorentz-Morrey spaces.