The use of stochastic differential equations has been popularized in different areas such as Physics, Mathematical biology, and Finance. The analytic solution of this type of equations is in most cases extremely difficult or impossible and numerical methods must be used for their solution. In this work, we use the discontinuous Galerkin method (DG) for the solution of conservation laws with multiplicative white noise, and the local discontinuous Galerkin (LDG) for the solution of parabolic equations with multiplicative white noise. We first do a revision of classic, weak and entropy solutions for the conservation laws and parabolic PDEs. Once understood the deterministic problems we study the concept of Brownian motion, stochastic differential equations and add a multiplicative white noise in time to the conservation laws and parabolic problems. Then we proceed to study the discontinuous Galerkin method for deterministic conservation laws and the local discontinuous Galerkin for parabolic problems. In both cases, we include numerical examples to study their performance and order of convergence. In addition, we compare the performance of the LDG method with other methods for the convective-dominated convection-diffusion equations. As part of the numerical experimentation, we observe the oscillatory behavior that the DG method has for discontinuous solutions. To overcome this problem we study the use of a two step process called limiters. First, we detect elements of the domain that have oscillations and second we reconstruct the solutions in elements with oscillatory behavior. We work with the TVB and BDF limiters and propose a modified version of these limiters, the ATVB and the M-BDF limiters. A great disadvantage the DG and LDG methods have is the use of analytic integration in their weak formulations. We work with a nodal entropy DG that uses the Gauss-Lobatto quadrature to avoid analytical integration which ensures entropy stable solutions. Then we apply the DG for the stochastic convection equation and the stochastic Burgers equation, including the use of limiters and entropy stable solutions. The stochastic heat equation and linear convection-diffusion equations are solved with the LDG method. The inclusion of limiters and entropy stable solutions for stochastic conservation laws are a contribution made in the thesis, they had not been previously studied