We present a numerical method for the direct simulation of osmotic pressure and flow. The direct simulation means a simulation in which thermal fluctuations are generated by random forces, and solute molecules appear as individual particles, as opposed to a continuous description in which solutes appear as concentration fields. The equations of motion for the fluid are the thermally fluctuating incompressible time-dependent Stokes equations. The fluid is allowed to slip through the vesicle wall thus making the membrane permeable to water. Two types of random forces are used. A Lagrangian random force is applied directly to the immersed material to achieve fluctuation-dissipation balance with the slip resistance, and an Eulerian random force is applied to the fluid to achieve fluctuation-dissipation balance with the fluid viscosity. To this end, we present simulation models of osmotic swelling by the stochastic immersed boundary method, and a pumping mechanism driven by a non-equilibrium osmotic effect.